Past seminar colloquium

Surrogate end-points are used when the true end-points are costly or time consuming.In a typical set up we observe a fixed proportion of true-and-surrogateresponses, and the remaining proportion are only-surrogate responses. It is obviousthat the inclusion of such only-surrogate end-points increase the efficiency ofassociated estimation. In this present paper we want to quantify the gain in efficiencyas a function of the proportion of available true responses. Also we obtain theexpression of the gain in true sample size at the expense of surrogates to achieve afixed power, as a function of the proportion of true responses. We present ourdiscussion in the two-treatment set up in the context of odds ratio.

Abstract: Queueing theory is vastly used to study service systems arising in management problems, where customers are considered to be indifferent in the sense that the decisions to control a system, are only made by the management and the users are impelled to follow the decisions. However, for a service system to be more realistic, it is essential to consider customers’ decisions about their actions (join or balk, wait or abandon, buy priority or not) which depends on the information provided to them at their arrival epochs. The study of queuing systems with strategic customers was initiated by Naor in 1969. After Naor’s work, an emerging tendency to study customers’ behavior imposing a reward-cost structure on the system took place. In this scenario, customers want to maximize their net benefit against others who have the same objective, which can be viewed as a symmetric game among them. When the system is empty, the server goes on vacation and returns after a random time to serve waiting customers if any.During the server’s vacation, customers continue to arrive at the system and if the present customers did not receive their service in due time, they became impatient and decide sequentially whether they will abandon the system or not upon the availability of a secondary transportation facility. In this presentation, the focus is to study customers’ equilibrium and socially optimal behavior in queuing systems with server vacation. Numerical experiments showing the dependence of performance measures on system parameters is demonstrated via several figures. Finally, a potential application of the model is suggested in managing a perishable inventory store.

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).

Noncommutative generalisation of complex and Kähler manifolds was proposed and studied by Fröhlich-Grandjean-Recknagel. In this talk, I shall discuss a class of examples of such objects coming from C*-dynamical systems with invariant trace.

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. 

In the second talk, I will describe some recent developments in and around the question of “effectivity”.

A partition of a square matrix A is said to be equitable if all the block of the partitioned matrix have constant row sums and each of the diagonal block is of square order. A quotient matrix Q of a square matrix A corresponding to an equitable partition is a matrix whose entries are the constat row sums of the corresponding blocks of A. A quotient matrix is an useful tool to find some eigenvalues of the matrix A. I will discuss some matrices whose eigenvalues are the eigenvalues of A and which are not the eigenvalues of a quotient matrix. Using this result we find eigenvalue localization theorems for matrices having an equitable partition. Finally, I will discuss some problems related to distance regular graph, Gersgorin disk theorem and distance matrix of graphs.

We will explain what is a differential k-form and define the De Rham Cohomology of a smooth manifold.

Abstract: Stream ciphers are designed to provide randomness. In 2008, eSTREAM standardize some stream ciphers, Grain-v1 was among them. In this talk, I will talk about the existence of non-randomness of Grain-v1 and its variants for different rounds.

Absract: In this talk, we will introduce Sobolev spaces and discuss some of their properties like approximations, extensions and existence of traces. 

Hyperbolic systems of conservation laws appear in many different areain Physics and has a history of more than 200 years. It became part of the morderntheory of analysis of PDES just 65 years ago. Now this theory is a well developedbranch of Analysis. Aim of this talk is to tell this story. The first part of thetalk is concerned with the Lax- Glimm theory of classical shocks and wellposednessof initial value problem. The second part is on initial boundary value problemfor systems and different formulation based on entropy inequalities and boundarylayers. We touch up on the recent work in the subject with Anupam Pal Chaudhuryand P.G.LeFloch

Positive approximation processes play an important role in Approximation Theory and appear in a very natural way dealing with approximation of continuous functions, especially one, which requires further qualitative properties such as monotonicity, convexity and shape preservation and so on. In this talk, we discuss the degree of approximation of signals (functions) using various types of summability transforms methods in different spaces. Approximation of functions by positive linear operators using Quantum calculus (q-calculus) will also be highlighted. During this talk, few applications of approximations of functions will also be highlighted.

Given a differentiable manifold $M$, understanding 'topology of $M$' means solving the Vector Field Problem on $M$, analyzing $K$ rings of $M$, immersion problem, etc. It is considered as a first step while analyzing the space completely. We will begin by explaining terms and an overview of the concept of topology of a manifold. We will consider actions of a finite cyclic group of order $m$ and the circle on the complex Stiefel manifold. Manifolds obtained as orbit spaces of these actions are called $m$-projective Stiefel manifold and right generalized complex projective Stiefel manifold respectively. We will discuss topology of these manifolds. (This is part of joint work with P. Sankaran and B. Subhash.)

There is a deep interplay between geometry and holomorphic function theory on domains in $\mathbb{C}^n$, $n>1$, which is quite unlike the situation in $\mathbb{C}$. This leads to striking new phenomena in higher dimensions that are absent in the one dimensional setting. We will begin by exploring classical results of this nature - Hartogs extension phenomenon and biholomorphic inequivalence of the ball and polydisc. We will then proceed to discuss more recent results concerning the tangential Lipschitz gain of holomorphic functions and a smoothing property of the Bergman projection.

This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This is the second 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: Let P ⊂ R ^d be a closed convex cone. Assume that P is pointed, i.e. the intersectionP ∩ −P = {0} and P is spanning, i.e. P − P = R ^d . Denote the interior of P by Ω. Let Ebe a product system over Ω. We show that there exists an infinite-dimensional separable Hilbert space H and a semigroup α := {α _x }_ x∈P of unital normal ∗-endomorphisms ofB(H) such that E is isomorphic to the product system associated with α. 

In this talk we will briefly talk about the resonance method and some recent progress in the theory of the Riemann zeta function. We will also see some new results on Hardy's Z function, Argument of the Riemann Zeta function, Dirichlet L- functions, Divisor problem and Circle problem as applications of the resonance method. These results have appeared (and some works are under progress) in different manuscripts due to the speaker and coauthors. 

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.

Confidence intervals constructed from a fixed-sample size procedure may sometimes be too wide to be of any practical use. Hence, to avoid such a problem, it is desirable to construct confidence intervals with fixed-width. G. B. Dantzig (1940) showed that there did not exist any fixed sample size procedure which produces fixed-width confidence interval with a prescribed confidence level. Charles Stein (1945, 1949) introduced the groundbreaking idea of sampling in two stages to construct fixed-width confidence intervals with a prescribed confidence level. In this talk, we consider the problem of fixed-width interval estimation of normal mean with unknown variance. In doing so, we develop Stein’s two-stage procedure and successively go through its modifications leading to modified two stage and purely sequential procedures. We also discuss properties associated with these procedures and how they compare with each other. If time permits we will also consider the problem of fixed-width interval estimation of a p dimensional normal mean vector with a certain structure for the dispersion matrix.

Abstract: We will characterize Toeplitz operators on the Hardy space using Berezin transform and then we will derive the necessary and sufficient condition under which two Toeplitz operators commute.

Abstract:

We undertake journey, starting with `pollen  particles' of
  Robert Brown to `stock prices' of Louis Bachelier to
   `suspended particles' of Albert Einstein to `mathematics'
of  Norbert Wiener to `random walk' view of Monroe Donsker
and enter the  `stochastic calculus' Kiyosi Ito, a garden
with beautiful flowers.

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
$$
0<|\delta qu+\beta-p|<\frac{1}{H^\varepsilon(u)q^{d+\varepsilon}},
$$
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.

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.

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 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. 

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.

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.

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 Seminar, School of Mathematical Sciences, NISER-Bhubaneswar.

Abstract: I shall talk about the Krein-Milman theorem and its application and the Krein-Smulian theorem and its application.

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.

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:

1. Review of local and global root numbers
2. To explain the computations of the local root numbers for a particular local Galois representations (e.g. Heisenberg representations)
3. And the computation of the lambda function \lambda_{K/F} except when K/F is quadratic wildly ramified extensions
4. If time permits, then I will explain some of my ongoing projects regarding root numbers and some open problems.

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: 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

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

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.

• Title: The sign of the Gauss sum  
• Abstract: One way to understand the connections between two different structures is to mix the two and to observe the resulting object! This is what Gauss did when he started his study of what is now called a Gauss sum. A Gauss sum is a mix of an additive map with a multiplicative map, where both the operations are done modulo a fixed prime number. It is not hard to see that the value of the most interesting case of the Gauss sum, appropriately normalized, is either $\pm 1$ or $\pm \imath$. However, Gauss proved in 1805, apparently it took four years for him to do this (!), that it is always $1$ or $\imath$. We'll discuss this result of Gauss and some of its generalizations.

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 .

Fractal Interpolation Function is a novel method to construct irregular functions frominterpolation data. The talk will begin with a brief introduction of Fractals and differentkinds of Fractal Interpolation Functions. Then, I will intro duce a special kind of FractalInterpolation Function called “Super Fractal Interpolation Function” (SFIF) for finersimulation of the objects of the nature or outcomes of scientific experiments that revealone or more structures embedded into another (i.e. hybrid structures). We will lookat construction of SFIF wherein, an Iterated Function System (IFS) is chosen from apool of several IFS at each level of iteration leading to implementation of the desiredrandomness and variability in fractal interpolation of the given data. Further, I willdiscuss properties of integrability and differentiability of an SFIF. Finally, I will showa result on convergence of a Cubic Spline SFIF.

Abstract: Independence is a fundamental idea in probability theory. This is the same as product measures with total measure 1.
However, for non-commutative structures, the classical definition of independence does not work. This gave rise to the concept
of free independence as initiated by Voiculescu. We shall present an easy introduction to this idea using partition theory, Mobius function, moments and free cumulants. We shall also explain the deep and interesting connection between free independence and large dimensional random matrices. If time permits, we shall also see how it becomes useful in statistical analysis of high dimensional time series.  

We will discuss jet spaces and symmetries of partial differential equations.

Let $Z(t):=\zeta\left(\half+it\right)\chi^{-\half}\left(\half+it\right)$ be Hardy's function,where the Riemann zeta function $\zeta(s)$ has the functional equation $\zeta(s)=\chi(s)\zeta(1-s)$.Hardy's function has been used to compute the zeros of $\zeta\left(\half+it\right)$ and plays a crucial role in the theory of the Riemann zeta function.In this talk we will compute the large values of $Z(t)$ and $-Z(t)$ using the resonance method. 

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. 

This talk consists of two parts; the first one dealing with a study of certain positive curvatures under the Ricci flow and the second one dealing with a rigidity question in geodesic flows.In 2010 Buchard Wilking proved that for every Ad_SO(n;C)-invariant subset S of the Lie algebraso(n;C) one can attach a notion positive curvature, which we call positive S-curvature, which is preserved by the Ricci flow. We study the properties of positive S-curvatures in reference to the Ricci flow. This part of work is in collaboration with Harish Seshadri and Soma Maity. We shall also discuss a problem motivated by blow-up considerations coming from a conjecture of Richard Schoen.In the second part of the talk we will study a rigidity question in Riemannian geometry, viz, when isRiemannian manifold determined by its geodesic flow? After a brief overview of this problem, We will present our work on the rigidity of the at cylinder. This is a joint work with C. S. Aravinda.

The well-known Pythagorean theorem which relates the lengths of sides of a right triangle to the length of the hypotenuse dates as far back in time as the Babylonians. Later in school when one gets a flavour of trigonometry, this result is neatly packaged in the equation $cos^2 \theta + sin^2 \theta = 1$. With further mathematical sophistication in terms of analytic geometry, one starts talking about $cos \theta, sin \theta$ as representing projections of a unit vector onto the x-axis and y-axis. In this talk, we will discuss the manifestations of the Pythagorean theorem and its converse in higher dimensions. In $n$ dimensions, we will see how this can be viewed as the problem of characterizing diagonal entries of a $n \times n$ projection matrix. Somewhat surprisingly, the set of vectors representing diagonal entries of rank $k$ projections in $M_n(\mathbb{C})$ forms a convex set with a simple description of its extreme points. In more generality, these results reveal connections between various parts of mathematics like combinatorics, representation theory, symplectic geometry which we will briefly touch upon.

We will discuss the general framework of quantum probability as the non-commutative generalization of the classical probability, operations on that and some of the quantum impossibilities. 

Abstract: In this talk, we define normal number and state Borel conjecture. In 1973, Mahler first proved a result towards Borel conjecture. Here, we discuss a conditional quantitative version of Mahler result.

In the talk, we discuss certain lifting between ellipticcusp forms and Hilbert cusp forms over real quadratic field which isknown as Doi-Naganuma lifting. We also discuss its close relationship to theShimura lifting which is a lifting from elliptic modular forms tomodular forms of half-integral weight.

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. 

Abstract: In this talk we will discuss the w*-complete metric approximation property for  mixed q-Araki-Woods von Neumann algebras for any Q=(q_i,j)_i,j  where q_i,j real numbers with max |q_i,j|<1. By adapting an ultraproduct technique of Junge and Zeng, we prove that the mixed q-Araki-Woods von Neumann algebras is isomorphic to ultraproducts of some mixed q-Gaussian and some q-Araki-Woods von Neumann algebras. As a consequence, we discuss its applicability for  the completely bounded norm of radial multipliers on mixed q-Gaussian von Neumann algebras to mixed q-Araki-Woods von Nuemann algebras, hence establishing the w*-cmap for mixed q-Araki-Woods von Neumann algebras.

Abstract: Hilbert noted that the polynomial x^4 - 10 x^2 + 1 is irreducible over the integers whereas it is reducible modulo all primes. What is behind this? If two polynomials f,g with integer coefficients take the same set of values modulo all primes, what is the relation between f and g? What proportion of primes divide numbers of the form 2^n + 1? How many of these are of the form 4m+3? What about primes dividing 7^n + 12^n in some arithmetic progression? Are there infinitely many prime numbers such that the decimal expansion of 1/p recurs with period p-1? Given an integer a, if every prime dividing a^n-1 for some n also divides b^n-1, is b necessarily a power of a? We discuss the interesting mathematics behind such questions.

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. 

Generalized Order Statistics(GOS) models and Non-homogeneous PoissonProcess (NHPP) models form a significant subclass of the many softwarereliability models proposed in the literature. First, we discuss themodels followed by some logical implications of NHPP models andestimability of the underlying parameters. We prove an importantlimitation of NHPP models for which the limit of the expected numberof failures $m(t)$ as the testing time $t\to\infty$ is finite.Specifically, the parameters of those models cannot be estimatedconsistently as the testing time approaches infinity. Then, we presenta nonparametric method for estimating $\nu$, the number of bugs present in acode, and investigate its properties. Our results show that theproposed estimator performs well in terms of bias and asymptoticnormality, while the MLE of $\nu$ derived assuming that the commonrenewal distribution is exponential may be seriously biased if thatassumption does not hold. We present a new parametric approach aswell.

Abstract:Starting with the problem of heat equation witha potential and using Lie-Trotter product formula, Feynmanhas a heuristic way of explaining Brownian Motion.We shall discuss this. If time permits we shall discussWiener and Ito integrals.

Abstract: In the first part of the talk I will talk about nef cones of divisors and pseudo-effective cones of k-cycles on products of projective bundles over curves. In the 2nd part, I will present some results about the Seshadri constants of ample line bundles on various blow-ups of projective spaces.

The concept of Operator Ideals will be explained through examples and several properties. These objects first occurred in the famous work ``The Memoirs'' of A. Grothendieck published in 1955. Since then, it has taken form of a vast theory of Operator Ideals.

Abstract: In this talk we will introduce graph colouring and chromatic number of a graph. We will discuss some application of graph colouring. Further we will present the proof of 'Perfect Graph Theorem' by Lovász.

Classical distribution theory in higher dimensions is largely focused on the Gaussian distribution. However, for the Gaussian distribution it is well known that every marginal distribution, every conditional distribution and all linear transformations are also Gaussian. Besides, it is also obvious that these properties chosen individually are not sufficient conditions to characterize the Gaussian distribution. There are many non-Gaussian distributions which share some of these features with the Gaussian distribution. In the present talk we introduce couple of new theorems which characterizes the Gaussian distribution. Besides, one of the extensive application of the Gaussian distribution will be in classical classification theory. In the current note we also discusses the extensions of classical classification results to the theory of extreme value analysis. Problem in action: Let P and Q be a two sets of probability measures defined on the same sigma field. Let T be a transformation which maps every measure from P to one and only measure in Q. Now, given Q and T can we characterize the set P ? 

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.

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.

 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 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.

TBA

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.

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. 

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.