Past seminar colloquium

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.

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: 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.

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.

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.

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: 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 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. 

It is well known that the Orlicz space is a natural generalization of the Lebesgue spaces. A vector measure is a countably additive Banach space valued measure. We discuss the Banach algebra structure on the Orlicz spaces associated to a vector measure over compact groups. We also discuss the Fourier transform of functions that are integrable with respect to vector measures over compact groups. We also introduce and study the convolution of functions from Lp-spaces associated to a vector measure.

Abstract: First I shall introduce differential of forms and integration over chains and finally I give proof of Stokes theorem. 

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.

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.

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.

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

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.

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.

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.

I will survey some recent results on actions of groups on von Neumann algebras, the emphasis being on joinings and spectral properties. 

The first talk will be a leisurely introduction to dynamics of group actions on homogeneous spaces of Lie groups.

We show that $\mathcal H$, with an $\mathfrak S_n$ invariant reproducing kernel $K$on an $\mathfrak S_n$ domain in $\C^n$, splits into reducing submodules $\mathbb P_{\bl p} \m H$, over the invariant ring $\C[\boldsymbol z]^{\mathfrak S_n}$, indexed by the partitions $\bl p$ of $n$. We then discuss the problem of minimality, inequivalence and realization of the submodules $\mathbb P_{\bl p} \m H$, particularly in the case when $\mathcal H$ is the weighted Bergman space $\mb A^{(\lambda)}(\mb D^n)$, for $\lambda>0$. One way to deal with the equivalence problem is through the realization and for which an analogue of Chevalley-Shephard-Todd Theorem for $\mathfrak S_n$ in the analytic setup seems quintessential. In fact, we show that the analytic version do exist for the most general version, that is, for finite pseudo-reflection groups. These results are from the joint works with Swarnendu Datta, Gargi Ghosh, Gadadhar Misra and Subrata Shyam Roy. 

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

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).

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. 

In this talk we explore the problem of determining the dimension of the L^2 [0, ∞)span of n linearly independent L^2_ loc [0, ∞) functions and the fascinating geometry associatedwith it. This problem appears in the context of Weyl’s limit classification problems of formallysymmetric differential expressions on [0, ∞).

In this talk, we will report on recent work (with J. Hilgert and S. Hansen, Paderborn) relating resonances and scattering poles on Riemannian symmetric spaces of rank one. We use boundary values in the sense of Kashiwara and Oshima to show that resonances and scattering poles coincide, along with their residues. Our methods also enable us to give a new and simple proof of the Helgason's conjecture in the rank one case. Time permitting, we'll mention progress made for symmetric spaces of higher rank. 

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: 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.

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.

First I shall introduce the moduli space of curves over complex numbers.This has been a central object of study in Algebraic Geometry with connections in mathematical physics, number theory and complex dynamics. This is also the primary object of study in my research. I shall then talk about a stratification of the moduli space by affine varieties and mention some partial results in this direction. Changing topics I shall then talk about more recent work on some moduli spaces of elliptic curves with applications to number theory.

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.

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.

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.

This will be the last lecture of a series of five on Finite Fields.

TBA

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.

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: 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: 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.

We first recall the fundamental work of Barthel, Livne and Breuil on the mod p representation theory of GL2(Qp) and then present a variant of their approach in which a supersingular representation of GL2(Qp) is realized as a quotient of a representation induced from the Iwahori subgroup, instead of the maximal compact subgroup. Certain computations appear to be easier to execute via this model. We briefly illustrate an instance of this by obtaining the K-socle filtration of a supersingular representation. This latter result is originally due to Stefano Morra. We will also quickly see some results on self-extensions of supersingular representation in Iwahori-Hecke model. This is a joint work with U. K. Anandavardhanan.

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.

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).

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. 

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. 

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. 

TBA

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: 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 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.

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.

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.

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.

In this talk we will be discussing about vanishing pressure limit for the equation

u_t + (u^2/2)_x = 0

ρ_t + (ρu)_x = 0,

with initial data

u(x, 0) = u_0(x),

ρ(x, 0) = ρ_0(x),

where u is the velocity component and ρ is the density component. This equation is considered as one of the model for the large scale structure formation of universe. In this direction we find some partial results for Reimann type initial data.

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:

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.

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.

In the first part of the talk, we will discuss briefly the theory of Equidistribution. In the second part of the talk we will see the distribution of gaps between eigenvalues of Hecke operators in both horizontal and vertical settings. Moreover, we will deduce a stronger version of multiplicity one theorem for the space of cusp forms of weight k and level N from the joint Sato-Tate conjecture. This is a joint work with M. Ram Murty. Finally, using recent developments in the theory of l-adic Galois representations we will study the normal number of prime factors of sums of Fourier coefficients of eigenforms. The final part is a joint work with M. Ram Murty and V. Kumar Murty.

 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.

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 $V_0>0$.

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.)

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: 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!

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.

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.

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. 

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. 

 Rational matrices G(x) arise in many applications such as in vibration analysis of machines, buildings, and vehicles, in control theory and linear systems theory and as approximate solutions of other nonlinear eigenvalue problems. The spectral data (poles, zeros, eigenvalues, eigenvectors, minimal bases, and minimal indices) of G(x) play a vital role in many applications. In this talk, we propose the definition of Rosenbrock strong linearization of rational matrices: by a Rosenbrock strong linearization of a rational matrix G(x) we mean a matrix pencil L(x) preferably of smallest dimension that reveals the pole-zero structure of G(x). Then we construct a family of pencils (which we refer to as GFPRs) of G(x) and show that GFPRs are Rosenbrock strong linearizations of G(x). Moreover, we show that GFPRs of G(x) is a rich source of structure-preserving linearizations of G(x) and utilize these pencils to construct structure-preserving Rosenbrock strong linearizations of structured (symmetric, skew-symmetric, even and odd) rational matrices G(x). Finally, we describe the recovery of eigenvectors, minimal bases, and minimal indices of G(x) from those of the GFPRs.

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: 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.

Abstract:  In this talk, we aim to introduce the process of mathematical modeling using differential equations. Basic compartmental models for spread of infectious diseases (mathematical epidemiology) will be described. In particular, models on the impact of awareness on epidemic outbreak will be studied. Also, methods to analyze the local stability of the equilibrium states of the system will be discussed.

Prerequisite: High school level knowledge of Ordinary Differential Equations

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.

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).

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.

This will be the first lecture of a series of five on Finite Fields.

This is a Short Lecture Series in Mathematics (SLSM) consisting of 4 lectures of 90 minutes each. This lecture schedule in this series is the following:

1st Lecture: Monday, October 09, 2017 - 11:30 to 13:00

2nd Lecture: Tuesday, October 10, 2017 - 11:30 to 13:00

3rd Lecture: Wednesday, October 11, 2017 - 16:30 to 18:00

4th Lecture: Thursday, October 12, 2017 - 15:30 to 17:00

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.

References:

   1. Random Graphs by Svante Janson, Tomasz Łuczak and Andrzej Rucinski;
   2. Random Graphs by Béla Bollobás; 
   3. Random Graphs and Complex Networks by Remco van den Hofstad;
   4. Random Geometric Graphs by Mathew Penrose. 

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. 

The talk contains a type of differential cryptnalysis on a popular stream cipher (Grain v1) with some conditions on IV bits.

TBA

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:
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: We show that the solution  of the equation   in  and  in  is radial by using maximum principle

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.

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.

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.

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.)

For simple versus simple hypothesis testing, the famous Neyman-Pearson lemma (1933) provides the best fixed sample size test procedure that has minimum type II error among all the tests that control type I error at some prescribed level. However, it is well-known that even such a best test cannot guarantee control of type II error while retaining the type I error control as long as the sample size is fixed. Abraham Wald (1943) extended the Neyman-Pearson lemma for sequentially observed data by introducing the Sequential Probability Ratio Test (SPRT) that simultaneously controls type I and type II error probabilities at some prefixed levels. In this talk, I will be discussing this test procedure along with boundary conditions, average sample size, the optimality property of SPRT and truncated SPRT. If time permits, I will discuss how to extend this idea for testing k (>2) simple-vs-simple hypotheses with the goal of selecting the correct hypothesis. This problem is known as Multi-hypothesis sequential probability ratio test (MSPRT).

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).

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

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.

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: 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.

The aim of this talk is to describe some conjectures which are part of the Langlands program.The talk is aimed at non-experts and should be accessible to a broad audience. 

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.

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.

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.  

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.

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.

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.

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.

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. 

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.

 We study the limiting behavior of the solutions of Euler equations of one-dimensional compressible fluid flow as the pressure like term vanishes. This system can be thought of as an approximation for the one dimensional model for large scale structure formation of universe. We show that the solutions of former equation converges to the solution of later in the sense of distribution and agrees with the vanishing viscosity limit when the initial data is of Riemann type. 

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. 

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.

Digital signature is a fundamental cryptographic primitive. It enables an entity to validate the authenticity and integrity of a digital document. In this talk, I will be discussing Group signature, Nominative Signature and Multi-signature. Group signature scheme enables any group member to produce a signature anonymously on behalf of the group and in case of misbehavior, the signer can be traced out. In a nominative signature scheme, a nominator and a nominee jointly produce a signature such that only the nominee can verify the signature. Multisignature is a powerful cryptographic primitive that helps to reduce the bandwidth taken by N signatures from O(N ) to O(1). It provides a group of signers the ability to sign collaboratively a common message in such a way that the size of the multisignature remains the same as that of a single signature and the verifier gets convinced that the message has been signed by all the signers.

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: 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 

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 ? 

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.  

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.

Abstract: In this talk, we will discuss some arithmetic and distribution type results on various partition functions, namely, Andrews’ singular overpartitions, cubic and overcubic partition pairs, and Andrews’ integer partitions with even parts below odd parts. We use some dissection formulas of Ramanujan’s theta functions, and arithmetic properties of modular forms and eta-quotients to study the distributions and to find some infinite family congruences of these partition functions. Also, using Radu’s algorithm on modular forms, we prove a conjecture on cubic partition pairs.

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: 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: 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.

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.

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.

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. 

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.

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.

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. 

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Abstract: Tensor is a multidimensional array (for example matrix is a tensor of order 2). Tensors often arises from the discretizations of multidimensional functions that are involved in the numerical treatment of complex problems in many different areas of natural, financial or social sciences. The direct numerical treatment of these arrays leads to serious problems like memory requirements and the complexity of basic operations (they grow exponentially in d). In the last decade the approximation of multidimensional arrays has become a central issue in approximation theory and numerial analysis. The main idea of the approximation of a tensor is decomposing the given tensor as sums of outer products of vectors. In the language of functions, it is an approximation of multivariable functions by sums of products of univariate functions. Tensor decompositions has lot of applications in image processing, quantum chemistry, data mining, machine learning stochastic partial differential equations etc.In the matrix case (i.e tensor of order 2), the singular value decomposition (SVD) represents a matrix as sum of outer product of vectors. SVD algorithm requires O(n3) arithmeticoperations (if the matrix is of size n × n). So it is very expensive when the matrix dimensions are large. Various inexpensive techniques of low rank approximation based on skeleton/cross approximation are available in the literature. SVD and its applications, other low rank approximation techniques like RRQR, Interpolative decomposition, randomized algorithms, skeleton/cross approximation techniques will be discussed in the talk. Canonical,Tucker, Tensor Chain and Tensor Train formats for higher order tensors will be introduced.

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.

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. 

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). 

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, 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. 

TBA

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: 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: 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

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.

TBA

This talk basically consists of two parts. First is, attacks on the stream cipher 'Sprout' and the second is, constructions of T-function, which carries good cryptographic properties.The design Specification of Sprout was proposed at FSE, 2015. Firstly, I will discuss about fault attack on Sprout then I will talk about the weak key-IV pairs of the same cipher. The second part of my talk deals with a new and interesting cryptographic primitive known as 'T-function', which was introduced by Klimov and Shamir in 2003. I will discuss its construction along with its various properties.

This will be the second lecture of a series of five on Finite Fields.

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.

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: 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 $\GL_2(\Q_p)$. 

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

Abstract: In this talk, we will prove a fundamental theorem on differential geometry, namely the embedding theorem due to Hassler Whitney, which shows that every smooth manifold can be embedded into a Euclidean space.

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!

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).

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.

I will discuss the distribution of some arithmetic and geometric objects and describe the basic tools used to study questions of equidistribution.

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: In this talk we will describe Random Graph through two models, 'Binomial Random Graph' and 'Erdos-Rényi' random graph. We will describe some of their properties and the relation between them. Further we will discuss threshold function of a graph property and how graph properties vary as the number of edges of a random graph increases. 

Abstract:

The Lie groups were discovered by Sophus Lie around 1880 while searching for a framework to analyze the continuous symmetries of differential equations in much the same way as permutation groups are used in Galois theory for analylyzing the discrete symmetries of algebraic equations. One of the important idea in the theory of Lie groups is to replace the gobal object, the group, with its local or linearized version which Lie called its “infinitesimal group”, now known as its Lie algebra. Around 1940, after Elie Cartan’s beautiful classification of finite dimensional semisimple Lie algebras, Lie algebras emerged as an independent branch of Algebra. An important class of infinite dimensional Lie algebras generalizing the finite dimensional semisimple Lie algebras were discovered independently by Victor Kac and Robert Moody in 1968.  An important family of these infinite Lie algebras is known as affine Lie algebras. These Lie algebras have proved to be very important with interactions in many areas of mathematics and physics. One such interaction is with number theory, particularly combinatorial identities.  In this talk I will give an overview of some applications in this direction.

• 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.

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.

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. 

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. 

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 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.

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.

 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 ). 

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

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.

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.

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. 

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. 

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 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.

See the attachement. 

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.

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.

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.

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. 

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.  

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: 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 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: 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. 

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

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.

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.