Abstract: An equivalent formulation of Ando's (generalization of Nagy's theorem) theorem providing a unitary dilation for a pair of commuting contractions is the statement: every contractive homomorphism of the bi-disc algebra is completely contractive. This is not true for any function algebra in three variables leaving the question of function algebras in two variables other than the bi-disc algebra open. The answer to this question for a large class will be discussed.
We shall discuss a new method of computing (integral) homotopy groups of certain manifolds in terms of the homotopy groups of spheres. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we verify a conjecture of J. C. Moore. This is joint work with Somnath Basu.
Chirp signals are frequently used in different areas of science and engineering. MCMCbased Bayesian inference is done here for purpose of one step and multiple step prediction incase of one dimensional single chirp signal with i.i.d. error structure as well as dependenterror structure with exponentially decaying covariances. We use Gibbs sampling techniqueand random walk MCMC to update the parameters. We perform five simulationstudies for illustration purpose. We also do some real data analysis to show how the methodis working in practice.
Abstract: In this talk I will briefly introduce the study of random polynomials. I will provide a brief survey of some results. Further we will focus on the study critical points (zeros of derivative) in relation to the zeros of random polynomials, initiated by Pemantle and Rivin. It can be seen from the simulations that the zeros closely pair with the critical points and we investigate this phenomenon. We also establish similar phenomenon for zeros of higher derivatives. Further we attempt to compute the pairing distance between zeros and critical points of random polynomials. We will discuss the problem in the case when all the zeros of the polynomial are real. I will also pose some questions of interest and connections to other well known problems.
Abstract: The 'Thresholding Greedy Algorithm' was introduced by Konyagin and Temlyakov in order to study some special bases in Banach spaces. This basis is known as greedy basis and is characterized by unconditional and democratic properties of the basis. Later, Albiac, Dilworth, Kalton, Wojtaszczyk and few other author's studied the weaker versions of greedy basis, namely, almost greedy basis and partially greedy basis. Almost greedy basis can be characterized by quasi-greediness and democratic property and partially greedy basis can be characterized by quasi-greediness and conservative property. In this talk we will discuss some new characterizations of almost greedy and partially greedy basis. We will also discuss generalizations of greedy basis and its relatives. First part of this talk is joint work with Stephen J. Dilworth.
I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories.
Let $\Gamma\subset \overline{\mathbb Q}^{\times}$ be a
finitely generated multiplicative group of algebraic numbers. Let
$\delta, \beta\in\overline{\mathbb Q}^\times$ be algebraic numbers
with $\beta$ irrational. In this talk, I will prove that there
exist only finitely many triples $(u, q, p)\in\Gamma\times\mathbb{Z}^2$
with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that
$$
0<|\delta qu+\beta-p|<\frac{1}{H^\vareps
$$
where $H(u)$ denotes the absolute Weil height. As an application of
this result, we also prove a transcendence result, which states as
follows: Let $\alpha>1$ be a real number. Let $\beta$ be an algebraic
irrational and $\lambda$ be a non-zero real algebraic number. For a
given real number $\varepsilon >0$, if there are infinitely many
natural numbers $n$ for which $||\lambda\alpha^n+\beta|| < 2^{-
\varepsilon n}$ holds true, then $\alpha$ is transcendental, where
$||x||$ denotes the distance from its nearest integer.
Google Meet Link: meet.google.com/rpj-qpwn-ows
We shall begin with a characterization of algebraic orthogonality in C^*-algebras in terms of order and norm. We apply this chacterization to propose a model of ``Absolutely ordered spaces'' which turns out to be a vector lattice under `commutative' conditions. This model suits to operator algebras and has potential to be applicable in quantum theory.
This talk will be in elementary level and first year Integrated MSc students should attained it.
Centerless Lie tori play an important role in explicitly constructing the extended affine Lie algebras; they play similar role as derived algebras modulo center play in the realization of affine Kac-Moody algebras. In this talk we consider the universal central extension of a centerless Lie torus and classify its irreducible integrable modules when the center acts non-trivially. They turn out to be highest weight modules for the direct sum of finitely many affine Lie algebras upto an automorphism.
An $m$-order $n$-dimensional square real tensor $\mathcal{A}$ is a multidimensional array of $n^m$ elements of the form$\mathcal{A} = (A_{i_1\dots i_m})$, $A_{i_1\dots i_m} \in \mathbb{R}$, $1 \leq i_1, \dots , i_m \leq n.$ (A square matrix of order $n$ is a $2$-order $n$-dimensional square tensor.) An $m$-order $n$-dimensional square real tensor is said to be a nonnegative (positive) tensor if all its entries are nonnegative (positive). We shall discuss the Perron-Frobenius theory for nonnegative tensors. Using these results we establish a sufficient condition for the positive semidefiniteness of homogenous multivariable polynomials.
The topology of real algebraic varieties is the study of the topology of objects that can be defined real algebraically, particularly the restrictions that the real algebraic structure imposes on the topology. The main problem that motivated interest in this area is the so called Hilbert's sixteenth problem that was suggested by Hilbert in his famous address: to classify, up to isotopy, non-singular planar real algebraic curves of a given degree. Topologically, a non-singular real algebraic curve is merely the union of circles (in fact, the fixed point set a complex conjugation on a Riemann surface), and its isotopy class is simply determined by the arrangement of these circles with respect to each other. Nevertheless, a complete classification, which would mean identifying which of these arrangements can be realized as the zero set of a real polynomial in two variables of a given degree, has as yet only been successful up to degree 7. However, it has led to several tangential questions and generalizations, which we will discuss after a brief overview of the motivating problem.
I shall discuss on control problems governed by the partial differential equations-mainly compressible Navier-Stokes equations,visco-elastic flows. I shall mention some of the basic tools applicable to study the control problems. We mainly use spectral characterization of the operator associated to the linearized PDE and Fourier series techniques to prove controllability and stabilizability results. I shall also indicate how the hyperbolic and parabolic nature of equations affects their main controllability results. Then some of our recent results obtained in this direction will be discussed.
TBA
Recent decades have witnessed significant growth and progress in spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. This series of lectures will focus on basic theory and computations of spatial statistics. Topics will include conditional and intrinsic autoregressions, connections between Markov random fields and geostatistics, variogram calculations, h-likelihood methods and matrix-free computations. Applications from agricultural variety trials, environmental sciences and geographical epidemiology will be discussed.
I will discuss the distribution of some arithmetic and geometric objects and describe the basic tools used to study questions of equidistribution.
A basis of the centralizer algebra for the action of the complex reflection group G(r, p, n) on the tensor product of its reflection representa- tion was given by Tanabe, and for p = 1, the corresponding partition algebra was studied by Orellana. In this talk, we first define the partition algebra for G(r, p, n) and call it Tanabe algebra. Along with the corresponding Schur–Weyl duality, using a confluence of ideas from Okounkov–Vershik approach, Clifford theory and higher Specht polynomials, we give a parametrization of the irreducible modules of Tanabe algebras and construct the Bratteli dia- gram. Furthermore, we give Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin basis of irreducible modules of Tanabe algebras. In the process, we also obtain some new results in the representation theory of complex reflection groups. The results presented in this talk form a part of a joint work with Dr. Shraddha Srivastava.
We will explain what it means to integrate differential forms on a manifold and state Stokes Theorem. We will see how it generalizes the fundamental theorem of calculus and Green's theorem.
The Riemann-Roch Theorem is the foundation of the theory of algebraic curves. It gives a precise answer for the dimension of the space L(D) for a divisor D of an algebraic curve. The qualitative information that a Riemann surface is an algebraic curve is seen to be equivalent to more quantitative statement of Riemann-Roch.As an application of Riemann-Roch Theorem, it can be shown that any Algebraic curve can be holomorphically embedded into projective space. Any genus zero algebraic curve is isomorphic to the Riemann Sphere(C∞). Any non-degenerate smooth projective curve X in Pn of minimal degree n is a rational normal curve.
Abstract: In this talk, we will discuss the existence of weak solutions for some elliptic boundary value problems. In particular, we will discuss the weak solutions corresponding to the Laplace operator and the Biharmonic operator.
Abstract : We discuss about the large values of the Riemann zeta-function on the line $1$ and it's consequence to the error term of the summatory function of $d^2(n)$ for $1 \le n \le x$ for large $x$ considered by Srinivasa Ramanujan where $d(n)$ denotes the sum of the positive divisors of the positive integer $n$.
The realization of the Total Character (or Gel’fand Character) τG of a finite group G, i.e. the sum of all ordinary irreducible characters of G is an old problem in character theory of finite groups. One possible approach is to try to realize τG as a polynomial in some irreducible character of G. In this vein, K. W. Johnson has asked whether it is possible to express the total character of G as a polynomial, with integer coefficients, in a single irreducible character of G. We study for several classes of finite nilpotent groups, the problem of existence of a polynomial f(x) ∈ Q[x] such that f(χ) = τG for some irreducible character χ of G. As a consequence, we completely determine the p-groups of order at most p5 (with p odd) which admit such a polynomial. Indeed, we prove that: If G is a non-abelian p-group of order p5, then G has such a polynomial if and only if Z(G) is cyclic and (G,Z(G)) is generalized Camina pair and, we conjecture that this holds good for p-groups of any order. In the talk, we also discuss about the nonlinear irreducible characters of p-groups of order at most p5.
Abstract: A well known result on translates of a function $\varphi$ in $L^{2}(\mathbb{R})$ states that the collection $\{\tau_{k}\varphi: k\in\mathbb {Z}\}$ forms an orthonormal system in $L^{2}(\mathbb{R})$ iff $p_{\varphi}(\xi)=\sum\limits_{k\in\mathbb {Z}}|\widehat{\varphi}(\xi+k)|^{2}= 1~ a.e.~ \xi\in\mathbb {T}$. Similarly in the literature there are interesting characterizations of Bessel sequence, frame, Riesz basis of a system of translates in $L^{2}(\mathbb{R})$ in terms of Fourier transform.In this talk, we aim to study frames in twisted shift-invariant spaces in $L^{2}(\mathbb{R}^{2n})$ and shift-invariant spaces on the Heisenberg group $\mathbb{H}^{n}$. First we shall define a twisted shift-invariant space $V^{t}(\varphi)$ in $L^{2}(\mathbb{R}^{2n})$ as the closed linear span of the twisted translates of $\varphi$. We shall obtain characterizations of orthonormal system, Bessel sequence, frame and Riesz basis consisting of twisted translates $\{T_{(k,l)}^{t}\varphi: k,l\in\mathbb{Z}^{n}\}$ of $\varphi\in L^{2}(\mathbb{R}^{2n})$ in terms of the kernel $K_{\varphi}$ of the Weyl transform of $\varphi$. In particular, we shall prove that if $\{T_{(k,l)}^{t}\varphi:k,l\in\mathbb{Z}^{n}\}$ is an orthonormal system in $L^{2}(\mathbb{R}^{2n})$, then $w_{\varphi}(\xi)=1$ a.e. $\xi\in\mathbb{T}^{n}$, where $w_{\varphi}(\xi)=\sum\limits_{m\in\mathbb{Z}^{n}}\int\limits_{\mathbb{R}^{n}}|K_{\varphi}(\xi+m,\eta)|^{2}d\eta,~\xi~\in\mathbb{T}^{n}$. Unlike the classical case on $\mathbb{R}^{n}$, it turns out to be a surprising fact that the converse of the above theorem need not be true. The converse is true with an additional condition, which we call "condition C". In fact, we shall see that $\{T_{(k,l)}^{t}\varphi:k,l\in\mathbb{Z}^{n}\}$ is an orthonormal system in $L^{2}(\mathbb{R}^{2n})$ if and only if $w_{\varphi}(\xi)=1$ a.e. $\xi\in\mathbb{T}^{n}$ and $\varphi$ satisfies condition C. Next we shall study shift-invariant spaces associated with countably many mutually orthogonal generators $\mathscr{A}$ on the Heisenberg group. We shall conclude the talk by providing a sampling formula on a subspace of a twisted shift-invariant space as an application of our results.
It is known that there are only finitely many perfect powers in non degenerate binary recurrencesequences. However explicitly finding them is an interesting and a difficult problem for binary recurrencesequences. A breakthrough result of Bugeaud, Mignotte and Siksek states that Fibonacci sequences$(F_n)_{n\geq 0}$ given by $F_0=0, F_1=1$ and $F_{n+2}=F_n+F_{n+1}$ for $n\geq 0$ are perfect powersonly for $F_0=0, F_1=1, F_2=1, F_6=8$ and $F_{12}=144$.In this talk, we the problem of finding perfect powers in products of terms of Recurrence Sequences. We show thatthere are only perfect powers and also give an explicit method to find them. We explicitly find the perfect powers in products of terms of some well known recurrence sequence including Fibonacci, Pell, Jacobsthal and Mersenne sequences and associated Lucas sequences.
Abstract: Let $p$ be a prime number. The two-dimensional crystalline representations of the local Galois group $\mathrm{Gal}(\bar\Q_p|\Q_p)$ are parametrized by the pairs $(k,a)$ up to twists, where $k\geq 2$ is an integer and $a\in m_{\bar\Z_p}$, the maximal ideal in the ring of integers of $\bar\Q_p$. We are interested in studying the map$(k,a)\mapsto \bar V_{k,a}$, where $\bar V_{k,a}$ denotes the semisimplified mod $p$ reduction of a typical crystalline representation $V_{k,a}$. These reductions have been computed when $k\leq 2p+1$ or when $a$ has a small $p$-adic valuation. Using the theory of Wach modules, Laurent Berger has also shown that the map displayed above is locally constant with respect to both the variables and found an explicit bound on the radius of local constancy when %$k$ is fixed and$a$ varies (2012). However, if $a$ is fixed and $k$ varies, nothing more than the existence is known about the radius of local constancy. So we ask the following simple question: for any given $p$-adic integer $a$, how close do $k$ and $k'$ need to be in the weight space to ensure that $\bar V_{k,a}\cong\bar V_{k',a}$? We give a partial answer to this question using some explicit computations in the automorphic side of the $p$-adic and mod $p$ Local Langlands Correspondences for $\GL_2(\Q_p)$.
Abstract: We shall begin with the homotopy invariance property of K-theory.After reviewing monoids and monoid algebras, we present some results which aremonoid version of the homotopy invariance property in K-theory. This answers a question ofGubeladze. Next, we will discuss the monoid version of Weibel's vanishing conjectureand some results in this direction. Finally, we will talk about the homology stability for groups.Here we present a result which improves homology stability for symplectic groups.If the time permits, some application of the homology stability will be given to the hermitian K-theory.
Cartan subgroups of Lie groups play a crucial role in the study of the structure of Lie groups, the behavior of the power maps, and also in determining whether the exponential map is surjective or has a dense image. Here we present some recent progress on `the density/surjectivity of the power maps of Lie groups ', and `the structural aspects of Cartan subgroups' and its applications on the characterization of the density of the power map in Lie groups.
Abstract: In this talk we will review on the minimal blocking sets of external, tangent and secant lines to an irreducible conic in PG(2,q).
Abstract: The theory of $M$-ideals in Banach spaces as well as in operator spaces is one of the important areas of research. It is well known from the literature that order structures and compact convex sets play a significant role in this area. Especially, the characterization of $M$-ideals in $A(K)$-spaces in terms of split face of the compact convex set $K$, established by Anderson and Alfsen, is a classical theorem of great importance.In this thesis, we have characterized $M$-ideal in order smooth $\infty$-normed spaces in terms of split faces of the quasi-state spaces. Also, we discuss the complete $M$-ideals in matricially order smooth $\infty$-normed spaces. For $p\neq \infty$, we introduce the notion of ideals, smooth $p$-order ideals to initiate the study of ideals in order smooth $p$-normedspaces.We introduce the notion of an $L^{1}$-matrix convex set in $*$-locally convex space. We show that $\{A_{0}(K_{n}, M_{n}(E))\}$(the `quantized functional space’) is a $\mathrm{C}^*$-ordered operator space. Conversely, every $\mathrm{C}^*$-ordered operator space is complete isometrically, completely isomorphic to $\{A_{0}(Q_{n}(V) M_{n}(V))\}$, where $Q_{n}(V)$ is the quasi-state space of $M_{n}(V)$ (in the matrix duality).
We start with defining binary quadratic forms and discussing some properties of them. Given an integer, we look for the forms representing that. We define a composition rule for forms and define the class group. Universal forms represent every integer. We present a theorem on the classification of universal binary quadratic forms. Finally, we discuss Ramanujan's universal quaternary diagonal quadratic forms and give formulas for the number of representations of an integer by some of those forms.
Every probability distribution has some uncertainty associated with it. The concept of Shannon’s entropy provide a quantitative measure of this uncertainty. But for some probability distributions the Shannon’s entropy measure may be negative and then it is no longer an uncertainty measure. To overcome this problem the concept of generalized entropy has been introduced in literature. In many practical situations it has been seen that when an investigator collect a sample of observations produced by nature, according to a certain model, the original distribution may not be reproduced due to various reason. For this reason it is important to consider the concept of weighted distribution. Motivated with the usefulness of the generalized entropy and the weighted lifetime distributions, we introduce the concept of weighted generalized entropy and discuss several properties of this model.
The Petersson scalar product split the space of modular forms for congruence subgroup into the space of cusp forms and the space of Eisenstein series.In this talk we show how one gets the Hecke theory for the space of Eisentein series for weight is either integer or half-integer. We also show their connection through Hecke equivalent maps.
The compact torus S1 × S1 has a structure of Riemann surfaceand therefore is a complex projective manifold. On product of odd dimensionalspheres S2p+1 × S2q+1 with p > 0 or q > 0, complex structures were obtainedby H. Hopf (1948) and Calabi-Eckmann (1953). These complex manifolds areone of the first examples of non-K¨ahler, and hence non-projective, compactcomplex manifolds.The aim of this talk is to describe a general construction of a class ofnon-K¨ahler compact complex manifolds. Let G be a complex linear algebriacgroup and let K be a maximal compact subgroup of G. Any holomorphicprincipal G-bundle EG over a complex manifold admits a smooth reduction ofthe structure group from G to K. We will show that the total space EK of thesmooth principal K-bundle, corresponding to this reduction, admits a complexstructure. In most cases, the complex manifold EK will be non-K¨ahler. Thistalk is based on a joint work with Mainak Poddar.
This will be the fourth lecture of a series of five on Finite Fields.
Abstract: This talk concerns M. P. Murthy's conjecture on the number of generators of an ideal in the polynomialring over a field. The progress on this conjecture has been quite slow. We shall give an exposition of the partialsolutions obtained so far, which includes a very recent result.
See the attachement.
TBA
Abstract: Given a holomorphic map between compact Riemann surfaces, Hurwitz's formula relates the genera of the domain and range with the degree and ramification of the map. We will see a short proof of this and some of its applications. Necessary definitions will be recalled.
TBA
The Channel Assignment problem(CAP) is a general framework focused on point-to-point communication, e.g. in radio or mobile telephone networks. One of its main threads asks for an assignment of frequencies or frequency channels to transmitters while keeping interference at an acceptable level and making use of the available channels in an efficient way. Thus the main task of the channels assignment problem is to assign channels to the station in a way that avoids interference and uses spectrum as efficient as possible. Radio k-colorings of graphs is a variation of channels assignment problem. For a simple connected graph G with diameter q, and an integer k, 1 6 k 6 q, a radio k-coloring of G is an assignment f of non-negative integers to the vertices of G such that |f (u) − f (v)| > k + 1 − d(u, v) for each pair of distinct vertices u and v of G, where d(u, v) is the distance between u and v in G. The span of a radio k-coloring f , rck (f ), is the maximum integer assigned by it to some vertex of G. The radio k-chromatic number, rck (G) of G is min{rck (f )}, where the minimum is taken over all possible radio k-colorings f of G. For k = q and k = q − 1 the radio k-chromatic number of G is termed as the radio number (rn(G)) and antipodal number (ac(G)) of G respectively. Radio k-chromatic number is known for very limited families of graphs and specific values of k (e.g. k = q, q − 1, q − 2).For this research talk only we discuss the results which are related to the radio number of graphs. For an n-vertex graph G, we shall discuss about a lower bound of rn(G) which depends on a parameter based on a Hamiltonian path in a metric closure Gc (an n-vertex complete weighted graph where weight of an edge uv is dG (u, v)). Using this result we show that how we can derive lower bounds of rn(Cn ), rn(Pm Pn ), rn(Cm Cn ), rn(Cm Pn ) and rn(Km Cn ). For any tree T an improved lower bound of radio number depending on maximum weight Hamiltonian path of T have been shown. Next we discuss about an upper bound of rn(G) by giving a coloring scheme that works for general graph and depends on the partition of the vertex set V (G) into two partite sets satisfying some conditions. We investigate the radio number of power of cycles (C r ), Toroidal Grids Tm,n. We present an algorithm which gives a radio coloring of a graph G. For an n-vertex graph the running time of this algorithm is O(n4 ).
Ricci flow and associated techniques have played pivotal roles in solving some long-standing open problems in Geometry and Topology in recent times. In this talk we will discuss some properties of an ODE related to the evolution of curvature along the Ricci flow and my recent results in this context.
We shall derive some bounds on the order of the Schur multiplier of finite p-groups. We shall also classify finite p-groups having Schur multiplier of maximum order. Finally we shall discuss the Schur multiplier and covering groups of special p-groups having derived subgroup of maximum order.
This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This is the third lecture in this series.
Abstract: The first half of the mini course will be an introducing to the two classical models of random graphs (a.k.a. Erdős-Rényi random graphs) and discuss the phenomenon of phase transition. We will also discuss thresholds for monotonic properties with examples including connectivity threshold and sub-graph containment threshold.
In the second half of the course we will consider other kind of random graphs. In particular, we will discuss various models for complex networks, including Albert-Barabási preferential attachment models. We will discuss "scale-freeness", asymptotic degree distribution and "small-world phenomenon". Properties of super and sub-linear preferential attachment models and some recent developments in de-preferential attachment models will also be discussed.
If time permits we will also introduce the random geometric graphs and discuss asymptotic of the connectivity threshold.
We estimate the dimension of the representation variety of Fuchsian groups in indefinite special orthogonal group that is of nonsplit type. In the case where the Fuchsian group is a surface group, we give an exact formula (rather that just estimate) for the dimension of the representation variety whose representation `space' is any real algebraic group.
Recent decades have witnessed significant growth and progress in spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. This series of lectures will focus on basic theory and computations of spatial statistics. Topics will include conditional and intrinsic autoregressions, connections between Markov random fields and geostatistics, variogram calculations, h-likelihood methods and matrix-free computations. Applications from agricultural variety trials, environmental sciences and geographical epidemiology will be discussed.
We consider the problem of offline changepoint estimation in a network-valued time series where the entire series is observed beforehand. We analyze a CUSUM statistic for this problem, and obtain, under minimal conditions, the rate of convergence of the resulting changepoint estimator in terms of three relevant parameters: (i) the (common) network size, (ii) the (common) network sparsity, and (iii) the total number of networks in the series. We also discuss some applications. This is based on on- going joint work with Peter Bickel, Sharmodeep Bhattacharyya, and Shirshendu Chatterjee.
This talk broadly has two parts. The first one is about the signs of Hecke eigenvalues of modular forms and the second is about some results on the magnitude of Hecke eigenvalues of modular forms, especially some Omega results.
This is the second lecture on this topic with the following:Abstact: Radon-Nikodym theorem of measure theory was proved in 1930 and A. N. Kolmogorov laid the foundation of modern probability in 1933 where the formal played a crucial role. In this Short Lecture Series in Mathematics (SLSM), we discuss the conditioning of probability theory in the light of Radon-Nikodym theorem. Upon the interest of the audience, we may introduce the notion of martingals and their convergence theorems. We have scheduled these lectures in the F-slot of Thursday and planned to go in accordance with the pace of the audience. Basic understanding of measure theory and functional analysis will be the prerequisite.
Abstract: Stream cipher plays an important role in cryptography. The basic component of stream cipher is Linear Feedback Shift Registrar. Here we will discuss about the general construction of a LFSR based stream cipher and the cryptanalysis of stream cipher.
In this talk, we shall introduce the notions of absolutely matrix ordered spaces and absolute matrix order unit spaces in the context of matrix ordered spaces. We shall prove that a unital, bijective $\ast$-linear map between absolute matrix order unit spaces is a complete isometry if, and only if, it is a completely absolute value preserving. From here, we deduce that on (unital) C$^*$-algebras such maps are precisely C$^*$-algebra isomorphisms. We shall extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We shall introduce the notion of a partial isometry in an absolute matrix order unit space to describe the comparison of order projections. We shall also discuss direct limit of absolute matrix order unit spaces to show the existence of ``Grothendieck group" through order projections and prove that ``Grothendieck group" is a functor from category of ``absolute matrix order unit spaces with morphisms as unital completely absolute preserving maps" to category of ``abelian groups". Later, we define orthogonality of complete absolute preserving maps and prove that ``Grothendieck group" functor is additive on orthogonal unital completely absolute preserving maps.
Abstract: In this talk, first I will give a brief overview of my current research activities on topological data analysis, certified geometry, optimization and level set method with applications in visualization, graphics, image analysis and simulations.Then, I will present a recent work on multivariate (or multifield) topology simplification. Topological simplification of scalar and vector fields is well-established as an effective method for analyzing and visualizing complex data sets. For multifield data, topological analysis requires simultaneous advances both mathematically and computationally. Mathematically, weshow that the projection of the Jacobi Set of multivariate data into the Reeb Space produces a Jacobi Structure that separates the Reeb Space into components. We also show that the dual graph of these components gives rise to a Reeb Skeleton that has properties similar to the scalar contour tree and Reeb Graph, for topologically simple domains. Computationally, we show how to compute Jacobi Structure, Reeb Skeleton in an approximation of the Reeb Space, and that these can be used for visualization in a fashion similar to the contour tree and Reeb Graph.
Let $f: \mathbb{C} \to \widehat{\mathbb{C}}$ be a meromorphic function (analytic everywhere except at poles) with a single essential singularity. The Fatou set of $f$, $\mathcal{F}(f)$is the subset of the plane where $\{f^n\}_{n >0}$ forms a normal family. A component $H$ of $\mathcal{F}(f)$ is called a Herman ring if there exists an analytic homeomorphism $\phi: \{z: 1<|z|< R\} \to H$ such that $\phi^{-1}(f^p (\phi(z)))$ is an irrational rotation about the origin for some natural number $p$. This $p$ is called the period of the Herman ring $H$. A component of the Fatou set is called completely invariant if $f(U) \subseteq U$ and $f^{-1}(U) \subseteq U$. If $f$ has an omitted value then it is shown that $f$ has no Herman ring of period one or two and the number of completely invariant components of $\mathcal{F}(f)$ is at most two (except for a restricted case).
The theory of pseudo-differential operators has provided a very powerful and flexible tool for treating certain problems in linear partial differential equations. The importance of the Heisenberg group in general harmonic analysis and problems involving partial differential operators on manifolds is well established. In this talk, I will introduce the pseudo-differential operators with operator-valued symbols on the Heisenberg group. I will give the necessary and sufficient conditions on the symbols for which these operators are in the Hilbert-Schmidt class. I will identify these Hilbert-Schmidt operators with the Weyl transforms with symbols in L2(R2n+1 × R2n+1). I will also provide a characterization of trace class pseudo-differential opera- tors on the Heisenberg group. A trace formula for these trace class operators would be presented.
Lecture series abstract: We will survey the von Neumann algebraic approach to locally compact quantum groups in the sense of Kustermans and Vaes, roughly following the treatment by Van Daele. We will begin with an intro to the theory of weights of von Neumann algebras. We will then proceed to describe locally compact quantum groups. If there is any time left, we will present some recent progress in the field. The prerequisite for attending the talks is some basic knowledge of C*-algebras (and perhaps also von Neumann algebras, depending on the audience).
In general, inverse problems are those where one needs to recover the unknown parameter of a system from the knowledge of the external observation. In this talk, I will mainly give an overview of Calder\'on’s inverse problems arising in several linear and nonlinear partial differential equations. I will discuss two different kinds of inverse problems for the Maxwell system and p-Laplace equation. One is the parameter identification problems and another one is the shape reconstruction issues. In particular, I will concentrate on the problem of determining the conductivity of a medium and the shape of an inclusion from the knowledge of boundary voltage or current measurements.
Gorenstein rings are very common and significant in many areas of mathematics. The classification, up to analytic isomorphism, of Gorenstein local K-algebras plays an important role in commutative algebra and algebraic geometry. The problem is difficult even if we restrict the K-algebras to the Artinian. One of the most significant information on the structure of K-algebra is given by its Hilbert function. Recently, jointly with M. E. Rossi, we characterized the Hilbert functions of Gorenstein K-algebras in some cases (K-algebras of socle degree 4). In this talk, we will discuss this new development.
Interpolation is an essential tool to approximate the unknown function from its samples. Many situations, when data arises from various complex phenomena, the data exhibits irregularity. Giving smooth approximation for these data may not be suitable. Fractal interpolation provides a non-smooth approximation for the interpolation data. Iterated function system is a basic tool to construct the fractal interpolation function and the graph of the fractal interpolation function is the attractor of the iterated function system. Data visualization is an essential subject because when data comes from scientific experiments or natural phenomena, the data may contains certain shape properties and preserving the shapes of these data are needed. Though traditional interpolation methods (polynomial, spline etc.) are good for shape preserving, these methods may not be good for irregular representation of the unknown functions. Fractal interpolation functions can be used both in shape preserving and irregular representation of the unknown functions. Also, fractal interpolation can be used to develop the numerical methods for the boundary-value problems of the ordinary differential equations. These numerical methods provide the multiple numerical solutions for the same boundary-value problem.
Transport operators have a range of intervention options available to improve or enhance their networks. Such interventions are often made in the absence of sound evidence on resulting outcomes. Cycling superhighways were promoted as a sustainable and healthy travel mode, one of the aims of which was to reduce traffic congestion. Estimating the impacts that cycle superhighways have on congestion is complicated due to the non-random assignment of such intervention over the transport network. In this paper, we analyse the causal effect of cycle superhighways utilising pre-intervention and post-intervention information on traffic and road characteristics along with socio-economic factors. We propose a modeling framework based on the propensity score and outcome regression model. The method is also extended to the doubly robust set-up. Simulation results show the superiority of the performance of the proposed method over existing competitors. The method is applied to analyse a real dataset on the London transport network. The methodology proposed can assist in effective decision making to improve network performance.
This is a planned series of informal lectures, where we develop the theory of smooth manifolds. We will start from scratch keeping prerequisites to a bare minimum.
Abstract: In this talk, we discuss the transcendence of special values of some Hurwitzzeta type series. Moreover, we find a linear independence criteria of these series undersome mild conditions. We also show that, for any a, b ∈ (0, 1) ∩ Q with a + b = 1, atleastone of the ζ(2k, a) or ζ(2k, b) must be transcendental.
Abstract: The aim of the talk is to establish equivalence between vector bundles and locally free sheaves in algebraic geometry . We will first define geometric vector bundle and locally free sheaf separately, then we will show that both of these terms are essentially the same.
We consider a weighted quasilinear eigenvalue problem in the exterior domain. We establish the existence of a positive principal eigenvalue and discuss the regularity and the asymptotic behaviour at infinity of the first eigenfunctions. A local and the global antimaximum principle is also presented.
The curvature of a contraction T in the Cowen-Douglasclass is bounded above by the curvature of the backward shiftoperator. However, in general, an operator satisfying thecurvature inequality need not be contractive. In this talk wecharacterize a slightly smaller class of contractions using astronger form of the curvature inequality. Along the way, we findconditions on the metric of the holomorphic Hermitian vectorbundle E corresponding to the operator T in the Cowen-Douglasclass which ensures negative definiteness of the curvaturefunction. We obtain a generalization for commuting tuples ofoperators in the Cowen-Douglas class.
In the first part of this talk, we present a formula for the coefficients of a weight 3/2Cohen-Eisenstein series of squarefree level N. This formula generalizes a result of Gross and it provesin particular a conjecture of Quattrini. In the second part, we explain the proof of this formula andas an application, we discuss the distribution of the order of the Tate-Shafarevich groups #ShD ofquadratic twists of elliptic curves.
Abstract: The analysis of incomplete contingency tables is an interesting problem, which is also of practical interest. In this talk, we propose various missing data models for analyzing arbitrary three-way and multidimensional incomplete tables, and study estimation and testing under them. We also explore the problem of boundary solutions in such tables, proposing their various forms, connections among them and establishing sufficient and necessary conditions for their occurrence (using only the observed cell counts), which prove useful for model selection. Finally, we suggest methods for assessment of the missing data mechanisms of variables in the above tables in terms of only the observed cell counts.
Simple unital nuclear C*-algebras, assuming a minor finiteness condition and the so-called UCT, can be classified up to *-homomorphisms by an invariant consisting of K-theory, traces, and a pairing between these objects. I will discuss the classification of such C*-algebras by focussing on the classification for C*-algebras arising from two interesting classes of dynamical systems: minimal homeomorphisms with mean dimension zero and mixing Smale spaces.In my fist lecture I will introduce the classification programme as well as the construction of C*-algebra from dynamical systems via crossed products and groupoids.In my second lecture, I will discuss properties and detail the classification of the two classes mentioned above.
Holomorphic eta quotients' are certain explicit classicalmodular forms on suitable Hecke subgroups of the full modular group.We call a holomorphic eta quotient $f$ 'reducible' if for someholomorphic eta quotient $g$ (other than 1 and $f$), the eta quotient$f/g$ is holomorphic. An eta quotient or a modular form in generalhas two parameters: Weight and level. We shall show that for anypositive integer $N$, there are only finitely many irreducible holomorphiceta quotients of level $N$. In particular, the weights of such eta quotientsare bounded above by a function of $N$. We shall provide such an explicitupper bound. This is an analog of a conjecture of Zagier which says thatfor any positive integer $k$, there are only finitely many irreducibleholomorphic eta quotients of weight $k/2$ which are not integral rescalingsof some other eta quotients. This conjecture was established in 1991 byMersmann. We shall sketch a short proof of Mersmann's theorem and weshall show that these results have their applications in factorizingholomorphic eta quotient. In particular, due to Zagier and Mersmann's work,holomorphic eta quotients of weight $1/2$ have been completely classified.We shall see some applications of this classification and we shall discussa few seemingly accessible yet longstanding open problems about etaquotients. This talk will be suitable also for non-experts: We shall define all therelevant terms and we shall clearly state the classical results which we use.
Abstract: Secure Multi-party Computation (MPC), the standard-bearer and holy-grail problem in Cryptography, permits a collection of data-owners to compute a collaborative result, without any of them gaining any knowledge about the data provided by the other, except what is derivable from the final result of the computation. The first half of the talk will discuss garbled circuits and Yao's two-party computation. in the second half, if time permits, then I will discuss one of our recent works on MPC with small population and honest majority that builds on garbled circuits.-----Brief Bio: Dr Arpita Patra is an Assistant Professor at Indian Institute of Science. Her area of interest is Cryptography, focusing on theoretical and practical aspects of secure multiparty computation protocols. She received her PhD from Indian Institute of Technology (IIT), Madras and held post-doctoral positions at University of Bristol, UK, ETH Zurich, Switzerland and Aarhus University, Denmark. Her research has been recognized with an NASI Young Scientist Platinum Jubilee Award, a SERB Women Excellence award, an INAE Young Engineer award and associateships with various scientific bodies such as Indian Academy of Sciences (IAS), National Academy of Engineering (INAE ), The World Academy of Sciences (TWAS). She is a council member of Indian Association for Research in Computing Science (IARCS) since December 2017.
This talk is divided into two parts: Let g be a Borcherds-Kac-Moody-Lie (super)algebra with the associated Quasi-Dynkin diagram G. In the first part, I will prove that the generalized chromatic polynomial of the graph G can be recovered from the Weyl denominator identity of g. From this result, I will deduce a closed formula for certain root multiplicities of g. Also, we construct a basis for these root spaces of g. In the second part, we are interested in the chromatic symmetric function of the graph G. I will prove an expression for the chromatic symmetric function of G in terms of root multiplicities of g. As an application, we will see Lie theoretic proof of many results of Stanley on chromatic symmetric functions. Stanley's tree conjecture is an important conjecture in the theory of chromatic symmetric functions which states that non-isomorphic trees are distinguished by their chromatic symmetric functions. We propose a Lie theoretic method to approach this conjecture. Finally, I will briefly discuss some future directions.
Abstract: L-functions are generating functions formed out of local data associated with either an arithmetic object or with an automorphic form. These functions are special examples of so-called Dirichlet series. Some L-functions have interesting properties like analytic or meromorphic continuation to the whole complex plane, functional equation, Euler product, non-vanishing in a certain region etc.In this talk we shall try to understand some classical L-functions such as Riemann zeta function, Dirichlet L-function, L-function arises from Modular forms and their properties.
Abstract: Employing solution of heat equation, we prove Taylor's theorem with Peano form of the remainder. In addition, we derive the Taylor series of an infinitely differentiable function under the additional assumption that the n'th derivative does not grow faster than the n'th power of some fixed positive constant.
Googlemeets link: https://meet.google.com/pdg-ymra-wkv
All are cordially invited.
Finding formulas for the number of representations of integers as sums of integer squares has a long history. In this talk we give a brief history of the problem and indicate the use of modular forms in obtaining certain formulas. We also discuss about some related questions.
An asymptotic framework for optimal control of multi-class stochastic processing networks, using formal diffusion approximations under suitable temporal and spatial scaling, by Brownian control problems (BCP) and their equivalent workload formulations (EWF), has beendeveloped by Harrison (1988). This framework has been implemented in many works for constructing asymptotically optimal control policies for a broad range of stochastic network models.To date all asymptotic optimality results for such networks correspond to settings where the solution of the EWF is a reflected Brownian motion in the positive orthant with normal reflections.In this work we consider a well studied stochastic network which is perhaps the simplest example of a model with more than one dimensional workload process. In the regime considered here, the singular control problem corresponding to the EWF does not have a simple form explicit solution,however by considering an associated free boundary problem one can give a representation for an optimal controlled process as a two dimensional reflected Brownian motion in a Lipschitz domain whose boundary is determined by the solution of the free boundary problem. Using the form of the optimal solution we propose a sequence of control policies, given in terms of suitable thresholds, for the scaled stochastic network control problems and prove that this sequence of policies is asymptotically optimal. As suggested by the solution of the EWF, the policy we propose requires a server to idle under certain conditions which are specified in terms of the thresholds determined from the free boundary. This is a joint work with A. Budhiraja and X. Liu.
A brief history of classical Hardy inequalities and related results will be presented. Further, we talk about Generalized Hardy-Sobolev inequalities and its applications.
Functional Data Analysis is one of the frontline areas of research in statistics. The field has grown considerably mainly due to the plethora of data types that cannot be handled and analyzed by using conventional multivariate statistical techniques. Such data are very common in areas of meteorology, chemometrics, biomedical sciences, linguistics, finance etc .The lecture series will primarily aim at introducing the field of functional data analysis. Since functional data analysis is broadly defined as the statistical analysis of data, which are in the form of curves or functions, we will start with probability distributions and random elements in infinite dimensional Hilbert spaces, concepts of mean and covariance kernel/operator, the associated Karhunen-Loeve expansion and some standard limit theorems in Hilbert spaces. We will then discuss some selected statistical inference problems involving functional data like inference for mean and covariance operators, functional principal component analysis, functional linear models, classification problem with functional data, robust inference techniques for functional data etc. We will recall some results from functional analysis as and when required during the lectures.
This talk will retrace the main steps of the modern theory of prime numbers and in particular how the combinatorial sieve combined with the Dirichlet series theory to give birth to the modern representation of the primes via a linear combination of terms, some of which being "linear", while the other ones are "bilinear". This will lead us to the recent developments of Green \& Tao, Mauduit \& Rivat, Tao, Helfgott, and Bourgain, Sarnak \& Ziegler.
Potts Model is a model of interacting spins on a crystalline lattice. In this talk we try to define the notion of Quantum Symmetry for Potts Model and try to explore the relations between Quantum Symmetry on Potts Model and Quantum Symmetry of the underlying graph structure. This is based On Joint work with Prof. Debashish Goswami.
Abstract:- See the attached file.
Google Meet Link: meet.google.com/zxe-csox-oyb
In this talk, we will prove Caporaso-Harris formula for counting plan curves of any genus. This formula gives the answer of the following problem:How many degree d curves are there in CP2 having δ nodes and passing through d(d+3)/2 − δ generic points?
This talk will be focused on the non-classical solutions to some hyperbolic systems of conservation laws and scalar conservation laws with discontinuous flux. We use the method of vanishing pressure limit and vanishing viscosity limit to construct solutions. For scalar conservation laws with discontinuous flux, we propose a generalized weak solution based on the vanishing viscosity limit. We then introduce a numerical scheme for this scalar conservation law that effectively captures the solution and we also do its convergence analysis. Next, we establish a Hopf-Lax type formula for the solution to the initial-boundary value problem for the 1D pressureless gas dynamics model by introducing generalized potentials.
This talk will be in elementary level and first year Integrated MSc students should attained it.
To any strongly continuous orthogonal representation of real line R on a real Hilbert space H , Hiai constructed q-deformed Araki-Woods von Neumann algebras for −1 < q <1, which are von Neumann algebras arising from non tracial representations of the q-commutation relations, the latter yielding an interpolation between Bosonic and Fermionic statistics. We settle that these von Neumann algebras are factors whenever dim(H ) ≥ 3 and completely determine their type. In the process we obtain and discuss ‘generator masas’ in these factors and establish that they are strongly mixing.
In my presentation, I will give some brief introduction to biharmonic submanifold. Also, I will introduce the Chen's conjecture of biharmonic submanifold and present some recent developments of conjecture.
Gini index (G), a scaled version of Gini’s mean difference (GMD), is a U-statistic of degree two. In this presentation, I will present a general approach to construct a new class of inequality measures called ad-hoc inequality measures (AIM) which are based on U-statistics of degree higher than two. The new AIMs satisfy anonymity, scale invariance, and population independence.We illustrate situations where delicate internal features with income disparities are more clearly explained and motivated from elementary economic persuasions of population dynamics, but those delicate features may be incorrectly missed by G. That is, one or more newly proposed AIMs are more apt to capture intricatefeatures than G in some instances. Without assuming any specific nature of the population distribution for the data, we have derived (i) the asymptotic mean square error of a general AIM; and also (ii) the asymptotic distribution of AIM: These results have provided useful inference methodologies which have beensupplemented with extensive sets of simulations and analyses of real economic data.
We formalize a simple minded notion of ``punctual gluing'' of t-structures which is nevertheless powerful enough to streamline or improve several relative motivic constructions in the literature. Examples include that of relative Artin motive, relative Picard motive, relative analogue of Bondarko's weight structure or the relative motivic t-structure on (compact) 1-motives.As a completely novel construction, we recover analogue of certain S.Morel's weight truncations in the motivic setting. As an application we can construct the analogue of an intersection complex for an arbitrary threefold in (Voevodsky's) triangulated category of mixed motives. Even more strongly, for several Shimura varieties (including all Shimura threefolds, most Shimura fourfolds, the Siegel six fold) we can construct the intersection motive in the category of relative Chow motives.
Decided later
Typically an extremal problem deals with the problem of estimating the maximum or mini- mum possible cardinality of a collection of finite objects that satisfies certain requirements. In my talk I am going to present my most recent research works related to extremal problems. For a finite abelian group G and A ⊂ [1, exp(G) − 1], the A-weighted Davenport Constant DA(G) is defined to be the least positive integer k such that any sequence S with length k over G has a non-empty A-weighted zero-sum subsequence. The original motivation for introducing Daven- port Constant was to study the problem of non-unique factorization domain over number fields. The precise value of this invariant for any group and for any set A is still unknown. In first half of my talk, I will present an Extremal Problem related to Weighted Davenport Constant, which we introduced and discuss several exciting results for any finite abelian group. It is a joint work with Prof. Niranjan Balachandran. In second part of my talk, I will discuss how to improve the Plu ̈nnecke inequality for iterated sumsets over any abelian group G. While doing so we estab- lished a bridge between almost a century old Macaulay’s theorem in commutative algebra and iterated sumsets in additive combinatorics. This process leads us to define an extremal problem as well. This is a joint work with Prof. Shalom Eliahou.
We will explain what it means to integrate a top form on a manifold. We will then state the second version of Stokes Theorem. After that, we will give the statement of Poincare Duality for smooth manifolds.
Given a Hopf algebra H in a braided category \mathcal{C} and a projection H\longrightarrow A to a Hopf subalgebra, one can construct a Hopf algebra r_{A}(H), called the partial dualization of H , with a projection to Hopf algebra dual to A. A non-degenerate Hopf pairing \omega :A\otimes B \longrightarrow 1 induces a braided equivalence between the Yetter-Drinfeld modules over a Hopf algebra and its partial dualization. In this seminar, we shall discuss this procedure in the general setting of C*-Quantum groups.
Reference:
1. Alexander Barvels, Simon Lentner, Christoph Schweigert, Partially dualized Hopf algebras have equivalent Yetter–Drinfel’d modules, Journal of Algebra 430 (2015) 303–342
2. Ralf Meyer, Sutanu Roy, Stanislaw Lech Woronowicz, Quantum group-twisted tensor products of C*-algebras II, J. Noncommut. Geom., 10 (2016), no. 3, 859-888.
Abstract: First I shall introduce differential of forms and integration over chains and finally I give proof of Stokes theorem.
Spatial heterogeneity in terms of trends or periodicity in blocks may affect the outcome of experiments, especially in agricultural experiments. The conventional randomized allocation of treatments to plots in a block may result in lesser precision. It has been commonly found that the experimental units which are neighbouring to each other within a block are correlated, or there may be existing significant trends even within small block. Use of some methods of local control, called spatial or nearest neighbour (NN) methods for analyzing the observations in the presence of significant trends in adjacent plots of field data, help in increasing precision. Other approaches to guard the effects from neighbouring plots in experimental designs leads to construction of useful optimal neighbour balanced block designs which are not only efficient under standard intra-block incomplete block design type analysis but also provide protection against the effects of correlated observations or potentially unknown trends which are highly correlated with plot positions within blocks (Keifer and Wynn, 1981, Cheng, 1983; Stroup and Mulitze, 1991; Jackroux, 1998 etc). Neighbour balanced designs are designs, wherein the allocation of treatments is such that every treatment occurs equally often with every other treatment as immediate neighbours. All Ordered Neighbour design (AONBD) are those designs where allocation of treatments is done such a way that neighbor balance is obtained at every order of neighbor, given immediate neighbor is first order.
In this lectures, we would discuss probabilistic model of primes leading to heuristics about their distribution. We would see many surprising irregularities popping up alongside expected results. A survey of several recent and important results would be presented in a way accessible to non-experts.
Abstract: In the first part of the talk we discuss a different formulation for describing maximal surfaces in Lorentz-Minkowski space $ \mathbb{L}^3:=(\mathbb{R}^3, dx^2+dy^2-dz^2) $ using the identification of $ \mathbb{R}^3 $ with $ \mathbb{C}\times \mathbb{R} $. This description of maximal surfaces help us to give a different proof of the singular Bj\"orling problem for the case of closed real analytic null curve. As an application, we show the existence of maximal surfaces which contain a given closed real analytic spacelike curve and has a special singularity. In the next part we make an observation that the maximal surface equation and Born-Infeld equation (which arises in physics in the context of nonlinear electrodynamics) are related by a Wick rotation. We shall also show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. Finally in the last part of the talk we show the connection of maximal surfaces to analytic number theory through certain Ramanujan’s identities.
In this talk we demonstrate some relations in degenerate Bernoulli polynomials , which may be expressed as a general convolution identity. We also show some properties of hypergeometric degenerate Bernoulli polynomials and numbers.
Abstract:- There are various graphs (e.g., intersection graph, commuting graph, prime graph, etc.) constructed from groups to study various properties of groups via graphs and vice versa. The power graph of a group $G$ is a graph with vertex set $G$ and two distinct vertices are adjacent if one is a power of the other. We begin this talk with a study on connectivity of power graphs of groups. Using the concept of quotient power graphs, we first present some characterizations of minimal separating sets of power graphs. Then we obtain some minimal separating sets of power graphs of finite cyclic groups. Consequently, we determine two upper bounds of vertex connectivity of power graphs of finite cyclic groups along with their exact value for some orders. We further provide some structural properties of power graphs of $p$-groups. Then we determine minimum degree of power graphs of finite cyclic groups (partially), dihedral groups, dicyclic groups and abelian $p$-groups. Moreover, we ascertain the equality of minimum degree with edge connectivity for power graphs of all finite groups, and then characterize its equality with vertex connectivity for power graphs of aforementioned groups. Proceeding further, we investigate Laplacian spectra of power graphs of finite cyclic groups, dicyclic groups and $p$-groups. We supply some characterizations for the equality of vertex connectivity and algebraic connectivity of power graphs of the above groups. Finally, we conclude with some unsolved problems concerning power graphs for future research.
Abstract: Gauge theories, which have their origin in physics, have had a profound effect on mathematics and nowhere is it more strikingly evident than in dimension four. Seiberg-Witten equations and Seiberg-Witten invariants have proven to be powerful instruments for the study of smooth structures on four-manifolds.
In my talk, I will introduce the equations and talk about a generalisation of the same. Almost all the well-known gauge theories on four-manifolds can be treated as special case of this generalisation. I will then discuss some of my latest results that show an interesting pattern, in which a class of such gauge theories on four-manifolds can be thought of as statements about de-generate metrics. I will then discuss some consequences of this observation and end the talk with some research directions I plan to pursue.
In this talk, we shall discuss the theta series associated with positive definite integral quadratic forms and some of its applications in getting formulas for the number of representations of positive integers by quadratic forms, using the theory of modular forms. The second part of this talk is about the construction of the Shimura and Shintani mappings between certain subspaces of modular forms of half-integral weights and integral weight respectively.
In my talk, I will address several mathematical questions about constrained Stochastic Partial Differential Equations (SPDEs) arising from the dynamics of (I) ferromagnetism, (II) viscoelastic fluid models, and (III) mechanism for spatial pattern formation analysing chemotaxis patterns under random perturbation. Standard or recent techniques for the analysis of parabolic, semilinear and non-constrained SPDEs do not apply in a straightforward way to the model problem. Here, we take this opportunity to study the solvability of such problems. Due to the environmental structure of such models, it is natural to follow that incorporation of noise in the usual Itˆo (or Itˆo-L ́evy) sense does not work in this context.
To start with, we analyse the theory of magnetisation of ferromagnetic materials below a certain temperature. Since the magnetisation has a unit length at initial time, therefore the question of how to incorporate a suitable perturbation modelling the constrained structure without destroying its invariance property is a delicate one; [4, 5, 9, 10].
In the later part of the talk, we will focus on the viscoelastic fluid model where the stress tensor is invariant under coordinate transformation. One needs to take into account this property in order to have a full understanding of the effect of fluctuating forcing field; [1, 2, 3, 6, 7].
Finally, to study the oriented movement of cells (or an organism) in response to a chemical gradient, we are careful with the environmental structure and the fluctuation of parameters. An appropriate mathematical approach to establish such realistic models with the incorporation of randomness in the system is exquisite; [8, 11, 12, 13].
References
[1] U. Manna and D. Mukherjee, ‘Strong Solutions of Stochastic Models for Viscoelastic Flows of Oldroyd Type,’ Nonlinear Analysis. 165, 198—242 (2017).
[2] P. Agarwal, U. Manna and D. Mukherjee, ‘Stochastic Control of Tidal Dynamics Equation with L ́evy Noise,’ Applied Mathematics and Optimization. 76(2), 1–70 (2017).
[3] U. Manna and D. Mukherjee, ‘Optimal relaxed control of stochastic hereditary evolution equations with L ́evy noise,’ ESAIM: Control, Optimisation and Calculus of Variations. 25(61), 48 pp. (2019).
[4] Z. Brze ́zniak, U. Manna and D. Mukherjee, ‘Existence of solution for Stochastic Landau-Lifshitz-Gilbert equation via Wong-Zakai approximation,’ Journal of Differential Equations. 267(2), 776-825 (2019).
[5] Akash A. Panda, U. Manna and D. Mukherjee, ‘Wong-Zakai approximation for the stochastic Landau- Lifshitz-Gilbert equations with anisotropy energy.’ J. Math. Anal. Appl. 480(1), 13 pp. (2019).
[6] U. Manna and D. Mukherjee, ‘Weak solutions and invariant measures of stochastic Oldroyd-B type model driven by jump noise.’ Journal of Differential Equations. 272, 760–818 (2021).
[7] U. Manna and D. Mukherjee, ‘Weak martingale solution of stochastic critical Oldroyd-B type models perturbed by pure jump noise.’ Stochastic Analysis and Applications, DOI:10.1080/07362994.2021.1947855; (2021).
[8] E. Hausenblas, D. Mukherjee, and T. Tran, ‘The one-dimensional stochastic Keller–Segel model with time- homogeneous spatial Wiener processes,’ to appear in Journal of Differential Equations, arXiv:2009.13789v1.
[9] E. Hausenblas, D. Mukherjee and Kitsolil Fahim, Wong–Zakai approximation for Landau–Lifshitz-–Gilbert
equations driven by geometric rough paths. Applied Mathematics and Optimization, DOI:10.1007/s 00245-021-
09808-1; (2021).
[10] M. Biswas, E. Hausenblas, and D. Mukherjee, Landau–Lifshitz–Gilbert Equation: Controllability by low
modes forcing for deterministic version and Support Theorems for stochastic version, (Preprint available),
(2021).
[11] E. Hausenblas, D. Mukherjee and Johannes Lankeit, ‘Existence of a local solution to the two dimensional
stochastic Keller Segel Model.’ (Submitted, preprint available), (2020).
[12] E. Hausenblas, D. Mukherjee and Ali Zakaria, ‘Strong solution to a stochastic chemotaxis system with
porous medium diffusion.’ (Submitted, preprint available), (2020).
[13] E. Hausenblas, and D. Mukherjee, ‘Pathwise uniqueness to the stochastic Keller-Segel systems.’ (Submitted,
preprint available), (2020).
Abstract: Modular forms are important objects in number theory and it has a wide range of applications in all other branches of Mathematics as well as in Physics. A modular for has Fourier expansion and the Fourier coefficients determine the modular form. Dirichlet series (e.g.,Riemann Zeta function) are important objects in number theory, used to study the distribution and properties of primes. Certain special values of Dirichlet series appears as Fourier coefficients of modular form. Kohnen [1991] constructed certain cusp forms whose Fourier coefficients involve special values of certain Dirichlet series of Rankin type by computing the adjoint map w.r.t. the Petersson scalar product of the product map by a fixed cusp form. Using differential operators one can define certain bilinear operators called the Rankin-Cohen brackets which is generalization of product. Recently the work of Kohnen has been generalized by Herrero [2015], where the author computed the adjoint of the map constructed using Rankin-Cohen brackets instead of product by a fixed cusp form. Fourier coefficients of the image of a cusp form under the adjoint map involves special values of certain Dirichlet series of Rankin-Selberg type similar to the one which appeared in the product case with certain twisting arising from binomial coefficients appearing in the Rankin-Cohen bracket. The work of Kohnen has been generalised to other automorphic forms (e.g., Jacobi forms, Siegel modular forms, Hilbert modular forms etc.,). Rankin-Cohen brackets for Jacobi forms and Siegel modular forms of genus two were studied by Choie explicitly using certain differential operators. Therefore it is natural to ask, how one can extend the work of Herrero to the case of Jacobi forms and Siegel modular forms of genus 2. A part of the thesis discuss about these generalizations. We shall also discuss the similar generaliza- tion for the case of half integral weight modular forms developed by Shimura [1973] and as a consequence, we get non-vanishing of certain Rankin- Selberg type Dirichlet series associated with modular forms. We shall also see how our method can be used to give a different proof of Rankin’s method in case of certain automorphic forms.
NB: This is a Thesis Colloquium/Open
Abstract: I shall talk about the Krein-Milman theorem and its application and the Krein-Smulian theorem and its application.
Abstract: Normal distribution has been used quite extensively in different areas of science and technology both in theory and practice. Although, it has several desirable properties, it has its own limitations also. Recently, various non-normal distributions (both univariate as well as multivariate) have been proposed in the literature for data analysis purposes. In this talk, we will consider different non-normal distributions and discuss their properties and provide applications in different areas.
This talk consists of two parts. In the first part, we discuss the convergence of finite volume method for solving non-linear aggregation-breakage equation. The pro of re lie s on showing the consistency of the scheme and Lipschitz continuity of numerical fluxes. It is investigated that the technique is second order convergent independently of the meshes for pure breakage problem while for aggregation and coupled problems, it depends on the type of grids chose n for the computations. Next, we show the efficient representation of d-point correlation functions for a Gaussian random field. To avoid the curse of dimensionality for d > 2, a truncated KarhunenLo´eve expansion of the random field is used together with the low rank Tensor Train decomposition. The target application of this work is the computation of statistics ofthe solution of linear PDEs with random Gaussian forcing terms.
This is a topic in classical algebraic K-Theory. I will recall definitions of elementary linear group, elementary symplectic group, linear transvection group, symplectic transvection group and symplectic group w.r.t. any alternating form. These groups have natural action on the set of unimodular elements. I will briefly discuss how bijections between orbit spaces of unimodular elements under different group actions are established. Finally, I will talk about an application of these results, namely improving injective stability bound for K1 group.
This will be the last lecture of a series of five on Finite Fields.
We will consider the basic material on the geometric approach to partial differential equations and symmetries, including an introductory part on the geometry of jet spaces.
C*-tensor categories are important descriptors of generalized symmetries appearing in non-commutative analysis and mathematical physics. An important algebra associated to a rigid semisimple C*-tensor category $ \mathcal{C} $ is the tube algebra $ \mathcal{A}\mathcal{C} $. The tube algebra admits a universal C*-algebra, hence has a well behaved representation category. Further, this representation category provides a useful way to describe the analytic properties of initial C*-tensor categories, such as amenability, the Haagerup property, and property (T).With a brief motivation from different directions, in this talk, I will move on to describing the annular algebra $\mathcal{A}\Lambda$ associated to a rigid C*-tensor category $ \mathcal{C} $. The annular representation category of $ \mathcal{C} $ is the category of $*$-representations of the annular algebra $\mathcal{A}\Lambda$. I will then present a description of the annular representation category of free product of two categories with an application to the Fuss-Catalan subfactor planar algebra.We then move onto oriented extensions of subfactor planar algebras (or equivalently singly generated C*-2-categories), which are a class of singly generated C*-tensor categories (or equivalently oriented factor planar algebras). I will end the talk with few problems which could extend this work.
In 1980s Goldman introduced various Lie algebra structures on the free vector space generated by the free homotopy classes of closed curves in any orientable surface F. Naturally the universal enveloping algebra and the symmetric algebra of these Lie algebras admit a Poisson algebra structure. In this talk I will define and discuss some properties of these Poisson algebras. I will briefly explain their connections with symplectic structure of moduli space and the skein algebras of $F\times [0,1]$. This is an ongoing work with Prof. Moira Chas.
Abstract: Horrocks' Theorem is an indispensable tool to prove Quillen-Suslin Theorem, which gives a sufficient condition for finitely generated projective modules to be free. In this talk, we will talk about a version of Horrocks' Theorem which uses concepts of completable unimodular rows over a ring. First, we will prove the theorem for local rings, then generalize it using the Local-Global Principle.
Abstract: All the statistical analysis assumes that all the measurements are obtained correctly. In practice, the observations can be correctly obtained only in controlled environment and this assumption is violated in many real life data. Instead of the true value of the observations, they are obtained with some error 'measurement error' and the true value of the observation is unknown. The usual statistical tools become invalid when the data has measurement error. What are the consequences of presence of measurement error on the linear regression analysis and how to obtain the correct statistical inferences by using the erroneous data in linear regression analysis are the issues to be addressed in this talk.
Abstract: Enumerative geometry is a branch of mathematics concerned with the following question: "How many geometric objects are there that satisfy certain constraints?" The simplest example of such a question is: "How many lines pass through two distinct points?'' A more interesting example is: "How many lines are there in three dimensional space that intersect four generic lines?'' In this talk we will describe a topological method to approach enumerative questions. We will use this method to count how many degree d curves are there in CP^2 that pass through certain number of generic points and have certain singularities.
It is conjectured that all separable polynomials with integers coefficients, under some local conditions, take infinitely many (in fact positive density of) square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. We report on some progress towards showing that ``cyclotomic polynomial$\Phi_{\ell}(X)$ take square free values with positive proportion".
This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This is the final lecture in this series.
Abstract: The first half of the mini course will be an introducing to the two classical models of random graphs (a.k.a. Erdős-Rényi random graphs) and discuss the phenomenon of phase transition. We will also discuss thresholds for monotonic properties with examples including connectivity threshold and sub-graph containment threshold.
In the second half of the course we will consider other kind of random graphs. In particular, we will discuss various models for complex networks, including Albert-Barabási preferential attachment models. We will discuss "scale-freeness", asymptotic degree distribution and "small-world phenomenon". Properties of super and sub-linear preferential attachment models and some recent developments in de-preferential attachment models will also be discussed.
If time permits we will also introduce the random geometric graphs and discuss asymptotic of the connectivity threshold.
Abstract: In this talk, our focus will be on certain classical results due to Ingham, Levinson and Paley-Wiener which find optimal decay of the Fourier transform of nonzero functions vanishing on `large sets'. We will talk about these theorems in details and their generalizations on the $n$- dimensional Euclidean space, the $n$-dimensional torus and certain non-commutative Lie groups.
I will survey some recent results on actions of groups on von Neumann algebras, the emphasis being on joinings and spectral properties.
I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories.
This talk broadly has two parts. The first one is about the signs of Hecke eigenvalues of modular forms and the second is about some results on the magnitude of Hecke eigenvalues of modular forms, especially some Omega results.
We will discuss some methods to compute the De-Rham Cohomology of smooth manifolds. We will discuss how to apply Mayer-Vietoris (which is the analogue of Van-Kampen Theorem for (co)homology).
In this talk we will first discuss about the non-commutative Lp spaces and recall the construction of the mixed q-deformed Araki-Woods von Neumann algebras. Then we will define the Ornstein-Uhlenbeck semigroups for this von Neumann algebra and study their ultracontractivity.
Abstract: We will talk about states and state spaces of partially ordered abelian groups with an order unit. A state on a partially ordered abelian group with order unit (G,u) is a positive homomorphism (a group homomorphism that takes positive elements to positive elements) from G to the real numbers with natural ordering such that u is mapped to 1. We will also see state spaces in general, i.e., we will talk about the collection of all states on (G,u) and will try to see the effects on the state space of changing the order unit.
Friday, November 26 · 4:00 – 5:00pm
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Abstract: Lattice points contained in regions of the plane, and higher dimensional Euclidean spaces, have been a topic of much study, going back at least to the century-old work of Minkowski. In recent decades there have been many developments in the general area, with many new insights brought in. The talk will aim at giving a flavour of the ideas involved, at a rudimentary level.
Tea: 3.30 p.m
TBA
Abstract: Two dimensional polynomial phase signal has uses in modeling black and white texture. We will discuss how to estimate the parameters of the model from observed data set and about the large sample properties of these estimators.
Lecture series abstract: We will survey the von Neumann algebraic approach to locally compact quantum groups in the sense of Kustermans and Vaes, roughly following the treatment by Van Daele. We will begin with an intro to the theory of weights of von Neumann algebras. We will then proceed to describe locally compact quantum groups. If there is any time left, we will present some recent progress in the field. The prerequisite for attending the talks is some basic knowledge of C*-algebras (and perhaps also von Neumann algebras, depending on the audience).
In 1931 Plancherel and Polya observed the following amusing fact: If the averages of a locally integrable function on \mathbb R over balls of radius R with center x converges to f(x) for all x as R goes to infinity then f(x)=ax+b for some real numbers a and b. They went on to show that in higher dimension the limit of ball averages is a harmonic function. We shall talk about a generalization of this result for Riemannian symmetric spaces of noncompact type with rank one.
Abstract: In this talk, we shall introduce the notion of a "quantum channel" and then define a special class of quantum channels called "entanglement breaking channels". For a map in this special class, we shall define what we mean by its entanglement breaking rank. We shall show how this rank parameter for a certain map links to an open problem in linear algebra: Zauner's conjecture. The prerequisites for this talk will be minimal with some background in linear algebra, and will be aimed towards students.
Abstract: We discuss approximate Birkhoff-James orthogonality of bounded linear operators defined between normed linear spaces X and Y. As an application of the results obtained, we characterize smoothness of a bounded linear operator T under the condition that K(X, Y), the space of compact linear operators is an M−ideal in L(X,Y), the space of bounded linear operators.
In his Ph.D thesis, John Tate attached local root numbers with characters of a non-Archimedean local field of characteristic zero. Robert Langlands (later P. Deligne ) proved the existence theorem of non-abelian local root numbers of higher dimensional complex local Galois representations. The local Langlands correspondence preserves thisroot numbers and the global root number is a product of local root numbers. So the explicit computation of the local root numbersis an integral part of the Langlands programs. But for arbitrary higher dimensional Galois representations we do not have any explicit formula for the local root numbers. To give an explicit formula of the local root number of an induced representation of a local Galois group of a non-Archimedean local field F ofcharacteristic zero, first we have to compute the Langlands' lambda function \lambda_{K/F} for a finite extension K/F . The plan of the talk is as follows:
We will continue with the discussion on (co)tangent space and smooth maps. We will define what is the differential of a smooth map and state the implicit function theorem. We will also work out some concrete examples.
We consider certain rings of smooth functions on a nice space, like a circle, and try to explore its algebraic nature using the smoothness of these functions. As a motivation for this, recall the ring C[0,1] of continuous real valued function on [0,1]. Recall how we prove that its maximal ideals (algebraic structure) are points using the compactness (topological structure) of [0,1].Only basic knowledge of ring theory and analytic functions is required, however, even these 2 topics will be revised in the beginning of the talk.
Abstract: The goal of the talk is to study complex tori and see some of its properties .We will begin with defining what a complex tori is and show that it is a complex manifold. Then we will classify complex tori of dimension one and show that each such tori is an elliptic curve.
Abstract: I will introduce to the basics of the machinery of motivic homotopy theory and explain some of the foundational aspects of the theory in the study of algebraic varieties.
Let F be a non-Archimedean local field with ring of integers O and finite residue field k of odd characteristic. In contrast to the well-understood representation theory of the finite groups of Lie type GL_n(k) or of the locally compact groups GL_n(F), representations of groups GL_n(O) are considerably less understood. In this talk we construct a family of continuous complex-valued irreducible representations of the groups GL_n(O) called regular representations. This talk is based on a joint work with Roi Krakowski and Uri Onn.
An automorphism T of a locally compact group is said to be distal if the closure of the T-orbit of any nontrivial element stays away from the identity. We discuss some properties of distal actions on groups. We will also relate any distal group with the behaviour of convolution powers of probability measures on it. (This would be a survey talk and most of it would be accessible to most people.)
The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966) studied similar versions of this notion in the context of non separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators via interspersing subspaces in a purely Banach space theoretic set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kania, Kochanek, Kozmider and Laustsen(2013), and also Kania and Laustsen (2017), we give complete answers for reflexive Banach spaces and the non-reflexive space constructed by Kania and Laustsen from the celebrated Argyros-Haydon's space with few operators.
TBA
The critical group of a graph is an interesting isomorphic invariant. It is a finite abelian group whose order is equal to the number of spanning forests in the graph. The Smith normal form of the graph's Laplacian determines the structure of its critical group. In this presentation, we will consider a family of strongly regular graphs. We will apply representation theory of groups of automorphisms to determine the critical groups of graphs in this family.
Abstract: Hardy - Littlewood maximal function is one of the most fundamental operators in real analysis with many applications in different areas. In the seminar we will discuss about the Hardy- Littlewood function on sphere and its properties. Also the maximal function on sphere for weighted integral in case of Dunkl operator will be discussed.
TBA
Abstract: Employing solution of heat equation, we prove Taylor's theorem with Peano form of the remainder. In addition, we derive the Taylor series of an infinitely differentiable function under the additional assumption that the n'th derivative does not grow faster than the n'th power of some fixed positive constant.
Googlemeets link: https://meet.google.com/pdg-ymra-wkv
All are cordially invited.
Abstract: The theory of coloring of random graphs has now been studied for several years and there are several very exciting results in this area. We shall see some of the known results in this area, before turning to a variant on graph coloring (distinguishing coloring) and discuss results on the distinguishing chromatic number of random cayley graphs.
This will be the first lecture of a series of five on Finite Fields.
In this talk, we will discuss some recent results on the existence and uniqueness of strong solutions of certain classes of stochastic PDEs in the space of Tempered distributions. We show that these solutions can be constructed from the solutions of "related" finite dimensional stochastic differential equations driven by the same Brownian motion. We will also discuss a criterion, called the Monotonicity inequality, which implies the uniqueness of strong solutions.
Suppose $\mathcal{H}$ is a Hilbert space. A bounded linear operator $T$ on $\mathcal{H}$ is said to be norm attaining if there exists a non-zero vector $x_0\in \mathcal{H}$ such that $\|Tx_0\|=\|T\|\|x_0\|$. It is natural to study operators that attain the norm. In this talk, we shall discuss on the norm attainment of three classical operators, namely, idempotent operators, model operators and Toeplitz operators. This is based on a joint work with Neeru Bala, Aryaman Sensarma and Jaydeb Sarkar.
Google Meet Link: meet.google.com/ajp-wiyo-vwm
Abstract: In this talk, using the basics of the Arithmetic that was learned in school, some of the elementary theorems of Number theory will be proven and also a flavor of RSA encryption will be given using the theory discussed.
The prerequisite to attend this talk is just to have a basic high school (till 10th) background in Mathematics.
All are cordially invited
The talk contains a type of differential cryptnalysis on a popular stream cipher (Grain v1) with some conditions on IV bits.
Given a holomorphic map between compact Riemann surfaces, Hurwitz's formula relates the genera of the domain and range with the degree and ramification of the map. Hence, given two compact Riemann surfaces we get a restriction on the possible types of holomorphic maps between them. In particular, we also get the possible types of meromorphic functions on a compact Riemann surface.
Abstract: The study of moduli spaces of stable maps and quantum cohomology theory plays a prominent role in modern enumerative geometry. A landmark result in this area is Kontsevich's recursion formula to enumerate rational curves in projective space. In the first part of this talk, we shall study a fiber bundle version of the above problem. We will consider the problem of enumerating rational curves in CP^3 whose image lies inside a CP^2 (which is also called a planar curve). We will show how Kontsevich's idea can be extended to the setting of fiber bundles.
In the second part of this talk, we will turn to classical enumerative geometry. We will study singular curves in a linear system that are tangent to a
given divisor. When the singularities are nodes, the question has been extensively studied by Caporaso and Harris. In this talk, we will give an approach to solve this question when the curve has more degenerate singularities. The method we will discuss comprises an explicit computation of the Euler class of an appropriate bundle. We then use excess intersection theory to compute the degenerate contribution to the Euler class.
Systems of Conservation laws which are not strictly hyperbolic appear in many physical applications. Generally for these systems the solution spaceis larger than the usual BVl oc space and classical Glimm-Lax Theory does not apply. We start with the non-strictly hyperbolic system(uj) t +j Xi=1( uiuj −i +12) x = 0, j = 1, 2...n.For n = 1, the above system is the celebrated Bugers equation which is well studiedby E. Hopf. For n = 2, the above system describes one dimensional model for largescale structure formation of universe. We study (n = 4) case of the above system,using vanishing viscosity approach for Riemann type initial and boundary data andpossible integral formulation, when the solution has nice structure. For certain classof general initial data we construct weak asymptotic solution developed by Panovand Shelkovich. As an application we study zero pressure gas dynamics system, namely,ut + (u. ∇)u = 0, ρt + ∇.(ρu) = 0,where ρ and u are density and velocity components respectively
We give an explicit formula for the integral kernel of the twistedKoecher-Maass series associated to a degree two Siegel cusp form F, where the twist is realized by any Maass waveform whose eigenvalue is in the continuum spectrum. From such a kernel we deduce the analytic properties of those twisted Koecher-Maassseries, and show how the later can be expressed in terms of Dirichlet series associated to the Fourier-Jacobi coefficients of $F$.
Economic inequality arises due to the inequality in the distribution of income and assets among individuals or groups within a society, or region or even between countries. For continuous evaluation of different economic policies taken by the government, computation of Gini index periodically for the whole country or state or region is very important. But not all countries can afford or do not collect data from households in a relatively large scale periodically.In order to compute a confidence interval of Gini index for a particular country or a region at given time, there exist fixed-sample size methods. However, for achieving a level of accuracy of estimation within some pre-specified error bound i.e. for constructing a fixed-width confidence intervals for Gini Index, no fixed sample size methodology can be used. This problem falls in the domain of sequential methodology. To date there does not exist any multi-stage or sequential procedure for constructing fixed-width confidence intervals for Gini Index.In this presentation, a fixed-width confidence interval estimation procedure of Gini index will be presented under simple random sampling scenario along with several asymptotic properties like convergence results on final sample size and also the coverage probability which are proved without any specific distributional assumption.A discussion will be made on use of other sampling schemes as well.
We will describe the structure of isometries on Hilbert spaces, following H. Wold and J. von Neumann. Then we will proceed to represent (concrete and not so concrete) tuples of commuting isometries. We will draw a list of observations on subtlety of this object and link it up with some deep problems in operator theory and function theory. We will also report on some recent results concerning invariant subspaces of the Hardy space over the unit polydisc.
Abstract : Here we study the Marshall-Olkin formulation of bivariate Pareto distribution, which includes both location and scale parameters and find efficient estimation techniques of the parameters of corresponding distribution. We use Maximum likelihood Estimation through the EM algorithm for the parameter estimation. Pareto distribution is heavy-tail in nature. It plays an important role in the Extreme Value Theory. So these distributions can be very useful in modeling the data related to finance, insurance, climate and network-security etc. These distributions can be used to analyze data related to any bivariate-component systems, e.g. axial length of two eyes of a diabetic patient. A numerical simulation is performed to verify the performance of different proposed algorithms
We shall discuss covering spaces of graphs, leading to facts about subgroups of free groups.We shall then construct geometric models, inspired by these graphs,and analyze some concrete examples.
The study of of degree d curves in $\mathbb{CP}^2$ is one of the interesting topic which arises in the context of Enumerative Geometry. The specific problem, which I will talk about, can be stated as follows:
Let $\mathbb{CP}^2$ be a compact complex surface and $L \rightarrow \mathbb{CP}^2$ a holomorphic line bundle that is sufficiently ample. Let $\mathcal{D} := \mathbb{P}^\frac{d(d+3)}{2}$ be the space of all degree d complex curves in $\mathbb{CP}^2$ - What is $\mathcal{N}(\mathfrak{X}_{k})$, the number of curves in $\mathbb{CP}^2$, passing through $\frac{d(d+3)}{2} -k$ generic points, having singularity of type $\mathfrak{X}_{k}$, where $k$ is the codimension of the singularity $\mathfrak{X}_{k}$?
In this talk we will describe what is a zero set of vector bundle over $\mathbb{CP}^2$ and then calculate $\mathcal{N}({A_1})$ for degree d complex curves in $\mathbb{CP}^2$. Here
$\mathcal{N}({A_1})$ represents number of degree d curves in $\mathbb{CP}^2$ passing through $\frac{d(d+3)}{2} -1$ points having $A_1$ singularity.
Abstract: We show that the solution of the equation in and in is radial by using maximum principle
Let S ⊂ CP 3 be ’almost any’ projective surface of degree ≥ 4. ClassicalNoether-Lefschetz theorem states that any curve C ⊂ S can be written as an intersectionC = S∩ S 0 where S 0 is some surface in CP 3. Grothendieck-Lefschetz theorem generalizesthis result for higher dimension.In this survey talk we will discuss these theorems and the corresponding results forvector bundles. We will study related aspects of vector bundles over hypersurfaces andcomplex projective spaces. Our emphasis will be on extendibility theorems and varioussplitting criterion for vector bundles.We will also mention some recent results and open problems. The talk should beaccessible to a graduate student.
In this lectures, we would discuss probabilistic model of primes leading to heuristics about their distribution. We would see many surprising irregularities popping up alongside expected results. A survey of several recent and important results would be presented in a way accessible to non-experts.
Functional Data Analysis is one of the frontline areas of research in statistics. The field has grown considerably mainly due to the plethora of data types that cannot be handled and analyzed by using conventional multivariate statistical techniques. Such data are very common in areas of meteorology, chemometrics, biomedical sciences, linguistics, finance etc .The lecture series will primarily aim at introducing the field of functional data analysis. Since functional data analysis is broadly defined as the statistical analysis of data, which are in the form of curves or functions, we will start with probability distributions and random elements in infinite dimensional Hilbert spaces, concepts of mean and covariance kernel/operator, the associated Karhunen-Loeve expansion and some standard limit theorems in Hilbert spaces. We will then discuss some selected statistical inference problems involving functional data like inference for mean and covariance operators, functional principal component analysis, functional linear models, classification problem with functional data, robust inference techniques for functional data etc. We will recall some results from functional analysis as and when required during the lectures.
Abstract: This is an informal talk on Random graphs and its relation to complex analysis. NB: This talk is organized as a part of ongoing "Advanced Instructional School on Stochastic Processes". Please note the unusual timing.
Abstrat: We prove an h-principle for poisson structures on closed manifolds. Equivalently
we prove h-principle for symplectic foliation (singular) on closed manifolds. On open
manifolds however the singularities could be avoided and it is a known result by Fernandes
and Frejlich [1].
References
[1] Fernandes, Rui Loja; Frejlich, Pedro An h-principle for symplectic foliations. Int. Math. Res. Not. IMRN
2012, no. 7, 1505–1518. (Reviewer: David Iglesias Ponte)
Presidency University, Kolkata, India., e-mail:mukherjeesauvik@yahoo.com,
In this talk, I will present Lax-Oleinik type explicit formulafor the solution of some balance laws and discuss its Structure Theory.
I will discuss about the nonlocal operators, in particular, the fractional Laplacian, and investigate the positivity properties of nonlocal Schrödinger type operators, driven by the fractional Laplacian by developing a criterion that links the positivity of the spectrum of such operators with the existence of certain positive supersolutions, thereby establishing necessary and sufficient conditions for the existence of a configuration of poles that ensures the positivity of the corresponding Schrödinger operator.
For a given nonempty subset L of the line set of the projective plane PG(2,q), a blocking set with respect to L (or simply, an L-blocking set) is a subset B of the point set of PG(2,q) such that every line of L contains at least one point of B. Let E (respectively; T, S) denote the set of all lines which are external (respectively; tangent, secant) to an irreducible conic in PG(2,q). We shall discuss minimum size L-blocking sets of PG(2,q) for L = E, S, T, SUT, EUT, SUE.
Abstract. Bozejko and Speicher in 1994 defined a von-Neumann algebra corresponding to the qij -deformed commutation relation. In this talk we will define the von-Neumann algebra and study its factoriality under some restriction.
Abstract: I shall talk about the Gelfand-Naimark-Segal construction and prove that any general C*-algebra is isometrically ∗-isomorphic to a norm closed subalgebra of B(H) for some Hilbert space H.
Abstract: We observe periodic phenomena everyday in our lives. The daily temperature of Delhi or the number of tourists visiting the famous Taj Mahal or the ECG data of a normal human being, clearly follow periodic nature. Sometimes, the observations may not be exactly periodic due to different reasons, but they may be nearly periodic. The received data is usually disturbed by various factors. Due to random nature of the data, statistical techniques play important roles in analyzing the data. Statistics is also used in the formulation of appropriate models to describe the behavior of the system, development of an appropriate technique for estimation of model parameters, and the assessment of model performances. In this talk we will discuss different techniques which have developed for the last twenty five years for analyzing periodic data, other than the standard Fourier analysis.
In this talk, I will discuss an infinite family of numbers involving Briggs-Euler- Lehmer constants, Euler’s constant and linear forms of logarithms of non-zero algebraic numbers. I will show that these infinite family of numbers are transcendental with at most one exception. This result generalizes a recent result of Murty and Zaytseva.
Conservation laws with discontinuous flux appears in the models of two phase flow in porous media, traffic flow with discontinuous road surface, clarifier thickener models of continuous sedimentation, enhanced oil recovery process etc. In this talk we begin with an introduction to both theoretical and numerical aspects of scalar conservation laws with discontinuous flux (CL-DF)[1, 2, 4, 6]. Apart from the basic difficulties for the mathematical analysis, this discussion include the convergence analysis of a second order scheme to the physically relevant (entropy) solution[3]. We continue the discussion with the applications of CL-DF to the system of non strictly hyperbolic partial differential equations, where we propose an efficient numerical method which overcomes the difficulties in the discretization [7]. Together with the stability analysis, this method is applied to a system of equations which models the multicomponent polymer flooding problem of enhanced oil recovery process. In the latter half we discuss a high order numerical method of discontinuous Galerkin scheme applied to a coupled two phase flow-transport problem in the context of discontinuous flux [5]. Apart from this, prior to the summary and future work we discuss about the instability issue which arises in the Buckley-Leverett problem[8].References[1] Adimurthi, J. Jaffre, G. D. Veerappa Gowda, Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal. 42(2004) 179-208.[2] Adimurthi, G. D. Veerappa Gowda, Conservation laws with discontinuous flux, J.Math. Kyoto Univ. 43 (1) (2003) 27-70.[3] Adimurthi, K. Sudarshan Kumar, G. D. Veerappa Gowda, Second order schemefor scalar conservation laws with discontinuous flux, App. Numer. Math. 80 (2014)46-64.[4] R. B¨urger, K.H. Karlsen, J.D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer.Anal. 47 (2009), 1684-1712.[5] R. B¨urger, S. Kumar, K. Sudarshan Kumar, R. Ruiz-Baier, Discontinuous approximation of viscous two phase flow in heterogeneous porous media, J. Comput.Phys. 321 (2016), 126-150.[6] T. Gimse, N. H. Risebro, Solution of the Cauchy Problem for a conservation lawwith discontinuous flux function, SIAM J. Math. Anal. 23 (1992), 635-648.[7] K. Sudarshan Kumar, C. Praveen, G.D. Veerappa Gowda, A finite volume methodfor a two-phase multicomponent polymer flooding, J. Comput. Phys. 275 (2014)667–695.[8] H. P. Langtangen, A. Tveito, R. Winther, Instability of Buckley-Leverett Flow ina Heterogeneous Medium, Transp. Porous Media, 9 (1992) 165-185.
One of the objectives of Noncommutative Geometry is to study operator algebras through differential geometric techniques. The geometric data associated to an operator algebra is a spectral triple. We will describe the construction of the space of forms coming from the spectral triple. Then we will discuss the question of existence and uniqueness of Levi-Civita connections for a spectral triple. Based on a joint work with D. Goswami and S. Joardar.
C*-tensor categories are important descriptors of generalized symmetries appearing in non-commutative analysis and mathematical physics. An important algebra associated to a rigid semisimple C*-tensor category $ \mathcal{C} $ is the tube algebra $ \mathcal{A}\mathcal{C} $. The tube algebra admits a universal C*-algebra, hence has a well behaved representation category. Further, this representation category provides a useful way to describe the analytic properties of initial C*-tensor categories, such as amenability, the Haagerup property, and property (T).With a brief motivation from different directions, in this talk, I will move on to describing the annular algebra $\mathcal{A}\Lambda$ associated to a rigid C*-tensor category $ \mathcal{C} $. The annular representation category of $ \mathcal{C} $ is the category of $*$-representations of the annular algebra $\mathcal{A}\Lambda$. I will then present a description of the annular representation category of free product of two categories with an application to the Fuss-Catalan subfactor planar algebra.We then move onto oriented extensions of subfactor planar algebras (or equivalently singly generated C*-2-categories), which are a class of singly generated C*-tensor categories (or equivalently oriented factor planar algebras). I will end the talk with few problems which could extend this work.
In this talk I will discuss mixed multiplicities of (not necessarily Noetherian) filtrations of m-primary ideals in a Noetherian local ring (R, m). I will give the construction of a real polynomial whose coefficients give the mixed multiplicities generalizing the classical theory for m-primary ideals. As a consequence, I will show that a classical theorem due to Rees, holds true for (not necessarily Noetherian) filtrations. I will also discuss a result which deals with non m-primary and non-Noetherian filtrations of ideals and partially generalizes another theorem of Rees.
Let $G$ be a simple finite graph with vertex set $V(G)=\{x_1,x_2,x_3,\dots,x_n\}$ and edge set $E(G)=\{e_1,e_2,e_3,\dots,e_q\}$. Also suppose that $I(G)$ is the edge ideal of $G$, where $I(G)=\langle x_{i}x_{j}~|~\{x_i,x_j\}\in E(G)\rangle$ $\subset R=K[x_1,x_2,\dots,x_n].$ We assume that $R(I(G))$ and $K[G]$ are the Rees algebra and toric algebra of $I(G)$ respectively.
In this talk we give a new upper bound for the regularity of edge ideals of gap-free graphs, in terms of their minimal triangulation. We also provide a new class of gap-free graphs such that $I(G)^s$ has linear resolution for $s\geq 3.$
We also show that if $G$ is connected and $R(I(G))$ is normal, then $\reg(R(I(G)))\leq \alpha_0(G)$, where $\alpha_0(G)$ is the vertex cover number of $G$. As a consequence, every normal K\"onig connected graph $G$, $reg(R(I(G))) = \mat(G)$, the matching number of $G$.
For a gap-free graph $G$, we give various combinatorial upper bounds for $\reg(R(I(G)))$. As a consequence we give various sufficient conditions for the equality of $\reg(R(I(G)))$ and $\mat(G)$. Finally we show that if $G$ is a chordal graph such that the toric algebra $K[G]$ has $q$-linear resolution$(q\geq 4)$, then $K[G]$ is a hypersurface, that is the defining ideal $I_G$ of $K[G]$ is generated by a single element, which proves the conjecture of Hibi-Matsuda-Tsuchiya affirmatively for chordal graphs.
Abstract: Plücker Formula gives the genus of a smooth projective plane curve in terms of its degree, specifically, a curve of degree d has genus (d-1)(d-2)/2. We will first prove Bezout's theorem and as an application look at Plücker formula.
Abstract: We will describe: (i) What is Controllability problem (ii) Examples and known results: ODE (finite dim), transport equation, Heat equation (infinite dim ). Then we consider compressible Navier-Stokes equations in one dimension, linearized around a constant steady state $(Q_0,V_0)$, with $Q_0 > 0,V_0\geq 0$. It is a coupled system involving both transport and parabolic effects. We study the controllability of this linearized system in bounded interval $(0,L)$. We find that the properties of the two semigroups $(e^{tA})_{t\geq0} $ (the one when $V_0 = 0$ and the one when $V_0> 0$) and the spectrum of $A$ are completely different where $A$ is the corresponding linearized operator. We obtain several interesting positive and negative results for the null controllability and approximate controllability of the system using interior or boundary control in both the cases $V_0 =0$ and $V_0>0$.
Randomly evolving systems with components that interact with each other have been of great interest in research. In particular, in this talk, we will discuss interacting urn processes with reinforcement. We consider N interacting two-colour Friedman urns and discuss the synchronization limit and fluctuation theorems. Theinteraction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fractionof balls of that colour in all urns combined. Further extensions and applications to understanding opinion dynamics on networks etc. will also be discussed.
Abstract: In this talk, our focus will be on certain classical results due to Ingham, Levinson and Paley-Wiener which find optimal decay of the Fourier transform of nonzero functions vanishing on `large sets'. We will talk about these theorems in details and their generalizations on the $n$- dimensional Euclidean space, the $n$-dimensional torus and certain non-commutative Lie groups.
In this talk, we study the configuration of systoles (minimum length geodesics) on closed hyperbolic surfaces. The set of all systoles forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surfaces, we call these admissible. There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first result characterizes admissibility. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, to the classical result that there are only two minimal non-planar graphs).
I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories.
In Quantum Information Theory, quantum states and quantum channels are central objects of study. Mathematically, a quantum state is a positive semidefinite matrix of unit trace, and a quantum channel is formalized by a linear map which is completely positive and trace preserving (CPTP).
In the first part of this talk, we describe a rank function, that we call the ``entanglement breaking rank", of a special class of quantum channels called ``entanglement breaking channels". We show how this rank parameter for a particular channel links to one of the most celebrated problems in frame theory, commonly referred to as Zauner's conjecture. This helps us present an analytic, perturbative approach to the conjecture rather than algebraic. This part of the talk is based on a joint work with Vern Paulsen, Jitendra Prakash (NISER alumnus), and Mizanur Rahaman.
In the second part of this talk, we discuss an ordering of quantum states referred to as ``quantum majorization'', which is a natural generalization of the concept of matrix majorization in the quantum mechanical setting. We shall briefly revisit the work by Gour et al. (Nature Communications, 2018); they established a characterization for majorization of quantum states in the finite-dimensional setting via the notion of conditional min-entropy. We then outline our work where we extend the characterization by Gour et al. to the context of semifinite von Neumann algebras. Our method relies on a connection between the conditional min-entropy and the operator space projective tensor norm for injective von Neumann algebras. This part of the talk is based on a joint work with Priyanga Ganesan (NISER alumnus), Li Gao, and Sarah Plosker.
This is the third lecture on this topic with the following:
Abstact: Radon-Nikodym theorem of measure theory was proved in 1930 and A. N. Kolmogorov laid the foundation of modern probability in 1933 where the formal played a crucial role. In this Short Lecture Series in Mathematics (SLSM), we discuss the conditioning of probability theory in the light of Radon-Nikodym theorem. Upon the interest of the audience, we may introduce the notion of martingals and their convergence theorems. We have scheduled these lectures in the F-slot of Thursday and planned to go in accordance with the pace of the audience. Basic understanding of measure theory and functional analysis will be the prerequisite.
TBA
In this talk, we discuss our attempts to solve an important open problem regarding Terrain Guarding: is the state-of-the-art polynomial-time approximation scheme for the problem [Gibson2014], a simple local search construct, optimal? In this regard, we first sketch a proof of a recent advancement in this domain by Mustafa and Jartoux [Jartoux2018]. This work proves, for geometric hitting set problems which give rise to arbitrarily large grid-like exchange graphs, that the local search algorithm is indeed optimal. In light of this result, we question if the Terrain Guarding problem can give rise to such grids. We will conclude the talk by laying out a possible plan for future work on this problem.
Abstract: River basin geomorphology is a very old subject of study initiated by Horton (1945). Various statistical models of drainage networks have been proposed. Each such model is a random directed graph with its own nuances. In recent years physicists have been interested in these models because of the commonality of such branching networks in many areas of statistical physics (see Rodríguez-Iturbe and Rinaldo (1997) for a detailed survey). We discuss the geometric features of one such model and also its scaling limit. The scaling limit of this model is the Brownian web, which has lately been the focus of extensive study among probabilists. Most of this talk will be accessible to a general audience.
For a nonnegative potential function q and a given locally finite graph G, we study thecombinatorial Schr¨odinger operator Lq(G) = ∆G + q with Dirichlet boundary condition on aproper finite subset S of the vertex set of G such that the induced subgraph on S is connected.Let Υp = {q ∈ Lp(S) : q(x) ≥ 0, Px ∈Sqp(x) ≤ 1}, for 1 ≤ p < ∞. We prove the existenceand uniqueness of the maximizer of the smallest Dirichlet eigenvalue of Lq(G), whenever thepotential function q ∈ Υp. Furthermore, we also establish the analogue of the Euler-Lagrangeequation on graphs.
Abstract In the 18th century, while dealing with astronomical and geodesic measurements, the scientists were confronted with a statistical problem, which in those days was described as "the problem of combining inconsistent equations". People who worked on this problem and contributed towards its solutions include Euler, Laplace, Gauss and Legendre among many others. I shall discuss the history of the problem and how it eventually led to the invention of the method of least squares.
Let M be a compact manifold without boundary. One can define a smooth real valued function of the space of Riemannian metrics of M by taking Lp-norm of Riemannian curvature for p ≥ 2. Compact irreducible locally symmetric spaces are critical metrics for this functional. I will show that rank 1 symmetric spaces are local minima for this functional by studying stability of the same at those metrics. I will also exhibit examples of symmetric metrics which are not local minima for it. In the 2nd part of my talk I will talk about Wilking’s criterion for Ricci Flow. B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators C(S), which are nonnegative in a suitable sense, to every AdSO(n,C) invariant subset S ⊂ so(n,C). We show that if S is an AdSO(n,C) invariant subset of so(n, C) such that S ∪ {0} is closed and C+(S) ⊂ C(S) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in C+(S) also admits a metric with curvature operator in C+(S) (b) The normalized Ricci flow on any compact Riemannian manifold M with curvature operator in C+(S) converges to a metric of constant positive sectional curvature.
Lecture series abstract: We will survey the von Neumann algebraic approach to locally compact quantum groups in the sense of Kustermans and Vaes, roughly following the treatment by Van Daele. We will begin with an intro to the theory of weights of von Neumann algebras. We will then proceed to describe locally compact quantum groups. If there is any time left, we will present some recent progress in the field. The prerequisite for attending the talks is some basic knowledge of C*-algebras (and perhaps also von Neumann algebras, depending on the audience).
The Schur multiplier of a finite group is the second cohomology group with complex coefficients. It was introduced in the beginning of 20th century by Issai Schur in his work on projective representations. Since then they have been proved to be a powerful tool in group theory. To determine the Schur multiplier of a given finite group is often a difficult task. Therefore it is of interest to provide bounds for the numerical qualities such as the order, exponent and the rank of the Schur multiplier. In this talk we shall provide sharp lower and upper bounds for the order of some special p-groups and classify their covering groups.
The first talk will be a leisurely introduction to dynamics of group actions on homogeneous spaces of Lie groups.
The E_2-term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for other Bousfield-Kan types spectral sequence. In this talk, I'll explain how the higher terms of such spectral sequences are determined by truncations of functors, defined in terms of certain (spectrally) enriched functor called mapping algebras. This is joint work with David Blanc.
We will continue with the discussion on (co)tangent bundle of a manifold. We will define the differential of a smooth map and state the implicit function theorem. We will also work out some concrete examples.
Abstract: Simply put, a graph is nothing but a set of points ("vertices"), along with edges which connect them. A planar graph is a special type of graph- one where the edges do not cross each other ( i.e., meet only at the vertices). In this talk, we will discuss an interesting problem in planar graph theory- that of trying to colour the vertices with as few colours as possible, subject to the condition that no two adjacent vertices have the same colour. We will go on to prove the Five Colour Theorem, which says that for any planar graph, there exists such a colouring utilising only five colours. We will further discuss the Four Colour "Conjecture".
The only prerequisites are curiosity and an eagerness to learn!
Regression analysis is a branch of statistics that provides us with tools to model a relationship between variables. In this presentation, we will discuss two types of regression models: the classical linear regression and non-parametric regression. Linear regression analysis aims to fit a linear model to data with the help of Least Square Estimates. For this model, we will discuss the techniques for checking the appropriateness of the model, the tests for determining the significance of regressor variables, the methods for dealing with multi-collinearity, and the procedure for incorporating categorical (or qualitative) variables into our regression model. We will also talk about a few topics from non-parametric regression such as density estimation and smoothing. Most of the concepts will be illustrated in statistical software R using real-life data-sets.
Abstract: WSNs finds wide applications in both military and civilian arenas of life. They consists of numerous low cost wireless devices, called sensors or nodes that deals with a lot of sensitive data. Due to resource constraints in these sensors, symmetric key systems are preferred over a public key setup in such networks; hence, both the sender and the receiver need to possess the same key prior to message exchange. This can be achieved by various techniques; the most preferred being Key Pre–distribution Schemes (KPS) for resource constraint WSNs. Such schemes involves pre–loading the symmetric cryptographic keys before deployment followed by establishing afterwards. They are refreshed periodically during successive deployment operations. My Ph.D. thesis is directed towards the analysis of existing these KPSs related to the security of WSN. This led to the observation of two major weaknesses in several KPS, namely the 'selective node attack' and the 'lack of full and direct inter-nodal communication' between nodes. For the first time in the literature of WSNs, we propose the idea of treating communication and connectivity as two separate aspects of any WSN. One may model these two aspects independently and then secure them using two different cryptographic frameworks in order to prevent the prevailing problem of 'selective node attack'. Though the connectivity design is hierarchical, our research outlined its application to distributed networks. Further, this is the first hierarchical WSN structure with the provision for (unlimited) addition and/or replacement of Cluster Heads (CH) , which results in significant improvement in network scalability. This automatically addressed the problems of jamming or capture of CH (or GN). Application of this generic idea to any deterministic KPS yields improvement in resiliency (a standard measure of network security), scalability and communication probability of the combined network. To address the issue of 'lack of full and direct inter–nodal communication' between nodes in some prominent existing designs, we proposed three unique deterministic merging techniques for those WSNs. The techniques are design specific and each assures full communication in the merged network. Critical analysis and comparative study establishes the strength of the designs. Some of the standard weaknesses present in most existing KPSs motivated us to investigate for systems that have small key ring, and yet are capable of supporting large number of nodes with appreciable resiliency, scalability and communication probability. Our research in this direction led to the proposal of a new KPS based on Unique Factorization of Polynomial Rings over Finite Fields. Barring the resiliency issue, this KPS satisfies all the above properties; and a combination of this KPS with the generic design is well equipped to meet all the desired results.
Multiple zeta values are the real numbers defined by the convergent series $\displaystyle\zeta(a_1,\ldots,a_r):=\sum_{n_1>n_2>\cdots>n_r\geq1}\frac{1}{n_1^{a_1}\ldots n_r^{a_r}}$, where $a_1,\ldots,a_r$ are positive integers with $a_1>1.$ They are generalization of classical Riemann zeta values to more variables. In this talk, we shall classify all the convergent sequences of multiple zeta values which do not converge to $0.$Using this classification, we will completely describe the iterated derived sets of the set of multiple zeta values in $\mathbb{R}.$Our results will imply that the set of multiple zeta values, ordered by $\geq,$ is a well-ordered set.We will also determine its type of order, which is $\omega^3,$ where $\omega$ is the order type of the set $\mathbb{N}$ of positive integers. Finally, using these results we shall show that there are only finitely many multiple zeta values which all represent the same value. It is in fact expected that there are at most two multiple zeta values that represents the same value!
For a number field K over Q, there is an associated invariant called the class number, which captures how far the ring of integers of K, is from being a principal ideal domain (PID). The study of class numbers is an important theme in algebraic number theory. In order to understand how the class number varies on varying the number field, Siegel showed that the class number times the regulator approaches infinity for a sequence of quadratic number fields. Later, Brauer extended this to a sequence of Galois extensions over Q, with some additional hypothesis. This is called the famous Brauer-Siegel Theorem. Recently, Tsfasman and Vladut conjectured a generalized Brauer-Siegel statement for sequence of number fields. In this talk, we prove the classical Brauer-Siegel and the generalized version in several unknown cases.
We discuss factoriality and structural properties of the q-deformed algebras of Hiai. These are type III cousins of the free group factors, extremely complicated objects and even very basic questions about them remain open. Using maximal abelian subalgebras we demonstrate that they are factors except few cases - the problem being open since 2000.
Abstract
In Combinatorial Matrix Theory, the study of graph structures via dif- ferent properties of matrices associated with it is an interesting and popular topic. Among the various matrices associated with a graph, the adjacency matrix is probably the most popular and widely investigated one.
A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G, when it exists, is the unique weighted graph whose adjacency matrix is signature similar to A(G)−1. The inverse graph of an invertible graph G is denoted by G+. A nonsingular graph G satisfies reciprocal eigenvalue property or property R if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property or property SR if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. In many ways these two concepts are related to each other. Both of these play important roles in Quantum Chemistry. In this talk, we will study the graph structure with regard to the concepts inverse graph and reciprocal eigenvalue properties.
In this talk, I shall discuss some basic definitions and examples of Finsler geometry. Afterthat I shall talk about some special Finsler metrics such as Kropina change of m-th rootmetrics, Finsler spaces with rational spray coeffiecients, the general (α, β) Finsler metricsand some of their non-Riemannian curvature properties.
The study of projective representations has a long history starting with the pioneering work of Schur for finite groups (1904). It involves understanding the homomorphisms from a group into the projective general linear groups. Two essential ingredients to study the group’s projective representations are describing its Schur multiplier and representation group. In the first half of my talk, l shall start with a brief introduction to this topic. Then we shall discuss about the Schur multiplier of p-groups and give a characterization of non-abelian p-groups having Schur multiplier of maximum order. In the second half, we shall study the projective representations of discrete Heisenberg groups by describing its Schur multiplier and representation group. I shall finish my talk by discussing some future problems in this direction.
Abstract:
Statistical classification techniques attempt to predict the true class label of a new observation (i.e. an observation not used in the construction of the classifier) based on a set of observations whose class labels are known beforehand. During the presentation, we will discuss the following classification methods - linear discriminant analysis, quadratic discriminant analysis, Fisher discriminant analysis, decision tree classifier, and artificial neural network classifier. Underlying assumptions of each of these techniques along with how each of these methods work will be discussed.
Google Meet Link: https://meet.google.com/gce-weco-wai
Abstract:Message Authentication Code (MAC) is a symmetric key cryptographic primitive to ensure data authentication. Traditionally, MACs are constructed using block ciphers. In this talk, we review how MACs can be constructed using Algebraic Structures and analyze their strength against related-key attacks. The talk will be self explanatory and no formal cryptographic background will be assumed.
This will be the second lecture of a series of five on Finite Fields.
Local Index formula is at the heart of the so called hard Riemannian aspects of Noncommutative Geometry. We will try to see why thisis so important. However so far we have only one computation of the local index formula. We will discuss two more computations.
In this talk, we will see how the notion of graph homomorphism generalizes the concept of coloring and help defining parameters such as clique number, independence number, chromatic number etc. for different types of graphs, such as, oriented graphs, signed graphs, colored mixed graphs etc.More importantly, we will see how these new concepts and works relate itself to the popularly known theories.
Abstract: In this talk, we try to understand elements of chaos theory in dynamical systems numerically and graphically. We also try to understand the the various properties like period doubling and chaos of the famous Logistic Map. Along with it some other less known maps, the Tent Map and the Gauss Map will also be analyzed. In particular Gauss Map shows very interesting properties like coexisting attractors and reverse period doubling.The principal aim is to explore the deep relationship among dynamical systems, chaos and fractals, and to uncover structure even when order seems to be absent.Prerequisite :Mathematical Background till class 12.
Google Meet link: meet.google.com/bvx-swhw-epg
Abstract: Enumerative geometry of curves in the complex projective plane is very classical. In this talk, we consider two generalisations of classical enumerative geometric problems of curves in the plane. We consider curves in that lie inside a in ; such curves are called planar curves. We discuss the enumeration of degree planar curves in having nodes and another singularity of codimension , and that intersect generic lines and pass through generic points in , where with . We also count the number of elliptic () curves of degree in a del Pezzo surface that pass through generic points of .
Abstract : I shall discuss how Monte Carlo methods can be used to solve complex optimization problems. In particular, I shall discuss the simulated annealing algorithm.