Past seminar colloquium

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: We will describe: (i) What is Controllability problem (ii) Examples and known results: ODE (finite dim), transport equation, Heat equation (infinite dim ). Then we consider compressible Navier-Stokes equations in one dimension, linearized around a constant steady state $(Q_0,V_0)$, with $Q_0 > 0,V_0\geq 0$. It is a coupled system involving both transport and parabolic effects. We study the controllability of this linearized system in bounded interval $(0,L)$. We find that the properties of the two semigroups $(e^{tA})_{t\geq0} $ (the one when $V_0 = 0$ and the one when $V_0> 0$) and the spectrum of $A$ are completely different where $A$ is the corresponding linearized operator. We obtain several interesting positive and negative results for the null controllability and approximate controllability of the system using interior or boundary control in both the cases $V_0 =0$ and $V_0>0$.

Randomly evolving systems with components that interact with each other have been of great interest in research. In particular, in this talk, we will discuss interacting urn processes with reinforcement. We consider N interacting two-colour Friedman urns and discuss the synchronization limit and fluctuation theorems. Theinteraction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fractionof balls of that colour in all urns combined. Further extensions and applications to understanding opinion dynamics on networks etc. will also be discussed.

Abstract: In this talk, our focus will be on certain classical results due to Ingham, Levinson and Paley-Wiener which find optimal decay of the Fourier transform of nonzero functions vanishing on `large sets'. We will talk about these theorems in details and their generalizations on the $n$- dimensional Euclidean space, the $n$-dimensional torus and certain non-commutative Lie groups. 

In this talk, we study the configuration of systoles (minimum length geodesics) on closed hyperbolic surfaces. The set of all systoles forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surfaces, we call these admissible. There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first result characterizes admissibility. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, to the classical result that there are only two minimal non-planar graphs).

I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories. 

 In Quantum Information Theory, quantum states and quantum channels are central objects of study. Mathematically, a quantum state is a positive semidefinite matrix of unit trace, and a quantum channel is formalized by a linear map which is completely positive and trace preserving (CPTP).  

In the first part of this talk, we describe a rank function, that we call the ``entanglement breaking rank", of a special class of quantum channels called ``entanglement breaking channels". We show how this rank parameter for a particular channel links to one of the most celebrated problems in frame theory, commonly referred to as Zauner's conjecture. This helps us present an analytic, perturbative approach to the conjecture rather than algebraic. This part of the talk is based on a joint work with Vern Paulsen, Jitendra Prakash (NISER alumnus), and Mizanur Rahaman.

In the second part of this talk, we discuss an ordering of quantum states referred to as ``quantum majorization'', which is a natural generalization of the concept of matrix majorization in the quantum mechanical setting. We shall briefly revisit the work by Gour et al. (Nature Communications, 2018); they established a characterization for majorization of quantum states in the finite-dimensional setting via the notion of conditional min-entropy. We then outline our work where we extend the characterization by Gour et al. to the context of semifinite von Neumann algebras. Our method relies on a connection between the conditional min-entropy and the operator space projective tensor norm for injective von Neumann algebras. This part of the talk is based on a joint work with Priyanga Ganesan (NISER alumnus), Li Gao, and Sarah Plosker.

This is the third lecture on this topic with the following:

Abstact: Radon-Nikodym theorem of measure theory was proved in 1930 and A. N. Kolmogorov laid the foundation of modern probability in 1933 where the formal played a crucial role. In this Short Lecture Series in Mathematics (SLSM), we discuss the conditioning of probability theory in the light of Radon-Nikodym theorem. Upon the interest of the audience, we may introduce the notion of martingals and their convergence theorems. We have scheduled these lectures in the F-slot of Thursday and planned to go in accordance with the pace of the audience. Basic understanding of measure theory and functional analysis will be the prerequisite.

TBA

Abstract: River basin geomorphology is a very old subject of study initiated by Horton (1945). Various statistical models of drainage networks have been proposed. Each such model is a random directed graph with its own nuances. In recent years physicists have been interested in these models because of the commonality of such branching networks in many areas of statistical physics (see Rodríguez-Iturbe  and Rinaldo (1997) for a detailed survey). We discuss the geometric features of one such model and also its scaling limit. The scaling limit of this model is the Brownian web, which has lately been the focus of extensive study among probabilists. Most of this talk will be accessible to a general audience.

For a nonnegative potential function q and a given locally finite graph G, we study thecombinatorial Schr¨odinger operator Lq(G) = ∆G + q with Dirichlet boundary condition on aproper finite subset S of the vertex set of G such that the induced subgraph on S is connected.Let Υp = {q ∈ Lp(S) : q(x) ≥ 0, Px ∈Sqp(x) ≤ 1}, for 1 ≤ p < ∞. We prove the existenceand uniqueness of the maximizer of the smallest Dirichlet eigenvalue of Lq(G), whenever thepotential function q ∈ Υp. Furthermore, we also establish the analogue of the Euler-Lagrangeequation on graphs.

Abstract  In the 18th century, while dealing with astronomical and geodesic measurements, the scientists were confronted with a statistical problem, which in those days was described as "the problem of combining inconsistent equations". People who worked on this problem and contributed towards its solutions include Euler,  Laplace, Gauss and Legendre among many others. I shall discuss the history of the problem and how it eventually led to the invention of the method of least squares.

Let M be a compact manifold without boundary. One can define a smooth real valued function of the space of Riemannian metrics of M by taking Lp-norm of Riemannian curvature for p ≥ 2. Compact irreducible locally symmetric spaces are critical metrics for this functional. I will show that rank 1 symmetric spaces are local minima for this functional by studying stability of the same at those metrics. I will also exhibit examples of symmetric metrics which are not local minima for it. In the 2nd part of my talk I will talk about Wilking’s criterion for Ricci Flow. B Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators C(S), which are nonnegative in a suitable sense, to every AdSO(n,C) invariant subset S ⊂ so(n,C). We show that if S is an AdSO(n,C) invariant subset of so(n, C) such that S ∪ {0} is closed and C+(S) ⊂ C(S) denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in C+(S) also admits a metric with curvature operator in C+(S) (b) The normalized Ricci flow on any compact Riemannian manifold M with curvature operator in C+(S) converges to a metric of constant positive sectional curvature.

Lecture series abstract: We will survey the von Neumann algebraic approach to locally compact quantum groups in the sense of Kustermans and Vaes, roughly following the treatment by Van Daele. We will begin with an intro to the theory of weights of von Neumann algebras. We will then proceed to describe locally compact quantum groups. If there is any time left, we will present some recent progress in the field. The prerequisite for attending the talks is some basic knowledge of C*-algebras (and perhaps also von Neumann algebras, depending on the audience).

The Schur multiplier of a finite group is the second cohomology group with complex coefficients. It was introduced in the beginning of 20th century by Issai Schur in his work on projective representations. Since then they have been proved to be a powerful tool in group theory. To determine the Schur multiplier of a given finite group is often a difficult task. Therefore it is of interest to provide bounds for the numerical qualities such as the order, exponent and the rank of the Schur multiplier. In this talk we shall provide sharp lower and upper bounds for the order of some special p-groups and classify their covering groups.

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

The E_2-term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for other Bousfield-Kan types spectral sequence. In this talk, I'll explain how the higher terms of such spectral sequences are determined by truncations of functors, defined in terms of certain (spectrally) enriched functor called mapping algebras. This is joint work with David Blanc.

We will continue with the discussion on (co)tangent bundle of a manifold. We will define the differential of a smooth map and state the implicit function theorem. We will also work out some concrete examples. 

Abstract: Simply put, a graph is nothing but a set of points ("vertices"), along with edges which connect them. A planar graph is a special type of graph- one where the edges do not cross each other ( i.e., meet only at the vertices). In this talk, we will discuss an interesting problem in planar graph theory- that of trying to colour the vertices with as few colours as possible, subject to the condition that no two adjacent vertices have the same colour. We will go on to prove the Five Colour Theorem, which says that for any planar graph, there exists such a colouring utilising only five colours. We will further discuss the Four Colour "Conjecture".

The only prerequisites are curiosity and an eagerness to learn!

Regression analysis is a branch of statistics that provides us with tools to model a relationship between variables. In this presentation, we will discuss two types of regression models: the classical linear regression and non-parametric regression. Linear regression analysis aims to fit a linear model to data with the help of Least Square Estimates. For this model, we will discuss the techniques for checking the appropriateness of the model, the tests for determining the significance of regressor variables, the methods for dealing with multi-collinearity, and the procedure for incorporating categorical (or qualitative) variables into our regression model. We will also talk about a few topics from non-parametric regression such as density estimation and smoothing. Most of the concepts will be illustrated in statistical software R using real-life data-sets.

Abstract: WSNs finds wide applications in both military and civilian arenas of life. They consists of numerous low cost wireless devices, called sensors or nodes that deals with a lot of sensitive data. Due to resource constraints in these sensors, symmetric key systems are preferred over a public key setup in such networks; hence, both the sender and the receiver need to possess the same key prior to message exchange. This can be achieved by various techniques; the most preferred being Key Pre–distribution Schemes (KPS) for resource constraint WSNs. Such schemes involves pre–loading the symmetric cryptographic keys before deployment followed by establishing afterwards. They are refreshed periodically during successive deployment operations. My Ph.D. thesis is directed towards the analysis of existing these KPSs related to the security of WSN. This led to the observation of two major weaknesses in several KPS, namely the 'selective node attack' and the 'lack of full and direct inter-nodal communication' between nodes.  For the first time in the literature of WSNs, we propose the idea of treating communication and connectivity as two separate aspects of any WSN. One may model these two aspects independently and then secure them using two different cryptographic frameworks in order to prevent the prevailing problem of 'selective node attack'. Though the connectivity design is hierarchical, our research outlined its application to distributed networks. Further, this is the first hierarchical WSN structure with the provision for (unlimited) addition and/or replacement of Cluster Heads (CH) , which results in significant improvement in network scalability. This automatically addressed the problems of jamming or capture of CH (or GN). Application of this generic idea to any deterministic KPS yields improvement in resiliency (a standard measure of network security), scalability and communication probability of the combined network.  To address the issue of 'lack of full and direct inter–nodal communication' between nodes in some prominent existing designs, we proposed three unique deterministic merging techniques for those WSNs. The techniques are design­ specific and each assures full communication in the merged network. Critical analysis and comparative study establishes the strength of the designs.  Some of the standard weaknesses present in most existing KPSs motivated us to investigate for systems that have small key ring, and yet are capable of supporting large number of nodes with appreciable resiliency, scalability and communication probability. Our research in this direction led to the proposal of a new KPS based on Unique Factorization of Polynomial Rings over Finite Fields. Barring the resiliency issue, this KPS satisfies all the above properties; and a combination of this KPS with the generic design is well equipped to meet all the desired results.

Multiple zeta values are the real numbers defined by the convergent series $\displaystyle\zeta(a_1,\ldots,a_r):=\sum_{n_1>n_2>\cdots>n_r\geq1}\frac{1}{n_1^{a_1}\ldots n_r^{a_r}}$, where $a_1,\ldots,a_r$ are positive integers with $a_1>1.$ They are generalization of classical Riemann zeta values to more variables. In this talk, we shall classify all the convergent sequences of multiple zeta values which do not converge to $0.$Using this classification, we will completely describe the iterated derived sets of the set of multiple zeta values in $\mathbb{R}.$Our results will imply that the set of multiple zeta values, ordered by $\geq,$ is a well-ordered set.We will also determine its type of order, which is $\omega^3,$ where $\omega$ is the order type of the set $\mathbb{N}$ of positive integers. Finally, using these results we shall show that there are only finitely many multiple zeta values which all represent the same value. It is in fact expected that there are at most two multiple zeta values that represents the same value!

 For a number field K over Q, there is an associated invariant called the class number, which captures how far the ring of integers of K, is from being a principal ideal domain (PID). The study of class numbers is an important theme in algebraic number theory. In order to understand how the class number varies on varying the number field, Siegel showed that the class number times the regulator approaches infinity for a sequence of quadratic number fields. Later, Brauer extended this to a sequence of Galois extensions over Q, with some additional hypothesis. This is called the famous Brauer-Siegel Theorem. Recently, Tsfasman and Vladut conjectured a generalized Brauer-Siegel statement for sequence of number fields. In this talk, we prove the classical Brauer-Siegel and the generalized version in several unknown cases.

We discuss factoriality and structural properties of the q-deformed algebras of Hiai. These are type III cousins of the free group factors, extremely complicated objects and even very basic questions about them remain open. Using maximal abelian subalgebras we demonstrate that they are factors except few cases - the problem being open since 2000.

Abstract
In Combinatorial Matrix Theory, the study of graph structures via dif- ferent properties of matrices associated with it is an interesting and popular topic. Among the various matrices associated with a graph, the adjacency matrix is probably the most popular and widely investigated one.
A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G, when it exists, is the unique weighted graph whose adjacency matrix is signature similar to A(G)−1. The inverse graph of an invertible graph G is denoted by G+. A nonsingular graph G satisfies reciprocal eigenvalue property or property R if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property or property SR if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. In many ways these two concepts are related to each other. Both of these play important roles in Quantum Chemistry. In this talk, we will study the graph structure with regard to the concepts inverse graph and reciprocal eigenvalue properties.
 

In this talk, I shall discuss some basic definitions and examples of Finsler geometry. Afterthat I shall talk about some special Finsler metrics such as Kropina change of m-th rootmetrics, Finsler spaces with rational spray coeffiecients, the general (α, β) Finsler metricsand some of their non-Riemannian curvature properties.

The study of projective representations has a long history starting with the pioneering work of Schur for finite groups (1904). It involves understanding the homomorphisms from a group into the projective general linear groups. Two essential ingredients to study the group’s projective representations are describing its Schur multiplier and representation group. In the first half of my talk, l shall start with a brief introduction to this topic. Then we shall discuss about the Schur multiplier of p-groups and give a characterization of non-abelian p-groups having Schur multiplier of maximum order. In the second half, we shall study the projective representations of discrete Heisenberg groups by describing its Schur multiplier and representation group. I shall finish my talk by discussing some future problems in this direction.

Abstract:
Statistical classification techniques attempt to predict the true class label of a new observation (i.e. an observation not used in the construction of the classifier) based on a set of observations whose class labels are known beforehand. During the presentation, we will discuss the following classification methods - linear discriminant analysis, quadratic discriminant analysis, Fisher discriminant analysis, decision tree classifier, and artificial neural network classifier. Underlying assumptions of each of these techniques along with how each of these methods work will be discussed.

Google Meet Link: https://meet.google.com/gce-weco-wai

Abstract: 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 .

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

Abstract:Message Authentication Code  (MAC) is a symmetric key cryptographic primitive to ensure data authentication. Traditionally, MACs are constructed using block ciphers. In this talk, we review how MACs can be constructed using Algebraic Structures and analyze their strength against related-key attacks. The talk will be self explanatory and no formal cryptographic background will be assumed.

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

Local Index formula is at the heart of the so called hard Riemannian aspects of Noncommutative Geometry. We will try to see why thisis so important. However so far we have only one computation of the local index formula. We will discuss two more computations. 

In this talk, we will see how the notion of graph homomorphism generalizes the concept of coloring and help defining parameters such as clique number, independence number, chromatic number etc. for different types of graphs, such as, oriented graphs, signed graphs, colored mixed graphs etc.More importantly, we will see how these new concepts and works relate itself to the popularly known theories.

Abstract: In this talk, we try to understand elements of chaos theory in dynamical systems numerically and graphically. We also try to understand the the various properties like period doubling and chaos of the famous Logistic Map. Along with it some other less known maps, the Tent Map and the Gauss Map will also be analyzed. In particular Gauss Map shows very interesting properties like coexisting attractors and reverse period doubling.The principal aim is to explore the deep relationship among dynamical systems, chaos and fractals, and to uncover structure even when order seems to be absent.Prerequisite :Mathematical Background till class 12.

Google Meet link: meet.google.com/bvx-swhw-epg

Abstract: 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.

Abstract : I shall discuss how Monte Carlo methods can be used to solve complex optimization problems. In particular, I shall discuss the simulated annealing algorithm.

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. 

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. 

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: 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.

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.

We will define what is a Laplacian on a smooth Riemannian Manifold. We will then define what are harmonic forms and state the Hodge Decomposition Theorem. In particular, we will explain how it implies Poincare Duality. We will then go on to define the notion of a weak solution and explain how elliptic regularity of the Laplacian, implies the Hodge Decomposition Theorem. We will review the basic notions of smooth manifolds and keep prerequisites to a minimum. Those who are interested in Analysis and PDE are encouraged to attend.  

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 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.

A rigidity conjecture by Goswami states that existence of a smooth and faithful action of a compact quantum group G on a compact connected Riemannian manifold M forces G to be compact group. In particular, whenever the action is isometric, or G is finite dimensional, Goswami and Joardar have proved that the conjecture is true. The first step in the investigation of a non-compact version of this rigidity conjecture demands a correct notion of faithful actions of locally compact quantum groups on classical spaces. In this talk, we show that bicrossed product construction for locally compact groups provides a large class of examples of non-Kac locally compact quantum groups acting faithfully and ergodically on classical (non-compact) spaces. However, none of these actions can be isometric, leading to the aforementioned rigidity conjecture may hold in the non-compact case as well. This is based on the joint work in progress with Debashish Goswami.

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.

Abstract:- Suppose $X$ is a minimal surface, which is a ramified double covering $\pi:X\to S$, of a rational surface $S$, with dim $|-K_S|\geq 1$. And suppose $L$ is a divisor on $S$, such that $L^2\geq 7$ and $L\cdot C\geq 3$ for any curve $C$ on $S$. Then the divisor $K_X+\pi^*L$ on $X$, is base-point free and the multiplication map in it's section ring : $Sym^r(H^0(K_X+\pi^*L))\to H^0(r(K_X+\pi^*L))$, is surjective for all $r\geq 1$. In particular this implies, when $S$ is also smooth and $L$ is an ample line bundle on $S$, that $K_X+n\pi^*L$ embeds $X$ as a projectively normal variety for all $n\geq 3$. In this talk we will present this result and various things associated to it.

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 $C$ be a smooth, projective, irreducible curve over the
field of complex numbers, and $C^n$ denotes the $n$-fold Cartesian
product of $C$. The symmetric group of $n$ elements acts on $C^n$ and
let $S^n(C)$ be the quotient. This is a smooth, irreducible, projective
variety of dimension $n$, called the $n$-th symmetric power of $C$.
Given a vector bundle $E$ of rank $r$ on $C$, one can naturally
associate a rank $nr$ vector bundle on $S^n(C)$, called the $n$-th
secant bundle of $E$. In this talk, we will discuss stability conditions
of secant bundles on $S^n(C)$. 

 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.

Attached file

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. 

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

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

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

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

Abstract: Employing solution of heat equation, we prove Taylor's theorem with Peano form of the remainder. In addition, we derive the Taylor series of an infinitely differentiable function under the additional assumption that the n'th derivative does not grow faster than the n'th power of some fixed positive constant.

Googlemeets link: https://meet.google.com/pdg-ymra-wkv

All are cordially invited. 

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.

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. 

A major theme of interest in mathematics is the study of Geometric structures on manifolds. Geometric structures are modelled on nice topological spaces with a group of automorphisms acting transitively on them. The spaces of negative curvature are widely studied from this viewpoint. A special class of these being real and complex hyperbolic spaces. In this talk, we will first focus on the geometry of complex hyperbolic spaces. In this pursuit, we will provide algebraic classification of the isometries of these spaces, and surface group representations into their isometry groups. This is achieved by means of algebraic data given by the traces of these isometries, and some other conjugacy invariants.In the second half of this talk, we will focus on hyperbolic structures on closed oriented surfaces of negative Euler characteristic. To this end, we enter into the realm of mapping class group which serves as a significant tool in connecting the geometry and topology of surfaces. In particular, we will give a method to decompose a periodic mapping class into irreducible components, and provide some interesting algebraic and geometric applications. We will conclude this talk by connecting the study of these mapping classes to the theory of “ribbon” graphs.

I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories. 

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.

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, ∞).

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).

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

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

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

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

Lecture series abstract: We will survey the von Neumann algebraic approach to locally compact quantum groups in the sense of Kustermans and Vaes, roughly following the treatment by Van Daele. We will begin with an intro to the theory of weights of von Neumann algebras. We will then proceed to describe locally compact quantum groups. If there is any time left, we will present some recent progress in the field. The prerequisite for attending the talks is some basic knowledge of C*-algebras (and perhaps also von Neumann algebras, depending on the audience).

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

Recent decades have witnessed significant growth and progress in spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. This series of lectures will focus on basic theory and computations of spatial statistics. Topics will include conditional and intrinsic autoregressions, connections between Markov random fields and geostatistics, variogram calculations, h-likelihood methods and matrix-free computations. Applications from agricultural variety trials, environmental sciences and geographical epidemiology will be discussed. 

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

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

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

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

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

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

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

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

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

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

In this talk, we shall discuss the theta series associated with positive definite integral quadratic forms and some of its applications in getting formulas for the number of representations of positive integers by quadratic forms, using the theory of modular forms. The second part of this talk is about the construction of the Shimura and Shintani mappings between certain subspaces of modular forms of half-integral weights and integral weight respectively. 

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

Abstract : I shall begin with the description of the lattice points counting problem in Euclidean spaces. Namely, establishing an (asymptotic) error estimate for the number of points that the lattice of integral points has in a Euclidean ball of large radius, as the radius goes to infinity. 
This problem has a very long history and vast literature is available in obtaining error estimates for Euclidean dilates of the unit ball as well as various other convex bodies. 
In the first half of the talk, I shall sketch the Fourier spectral method which originated with Minkowski for Euclidean balls. This classical method does not give the best possible error estimates for the balls, but it gives optimal error in the class of Euclidean dilates of convex bodies with surfaces having non-vanishing Gaussian curvature at all points.

In the last half of the talk, I shall discuss the lattice points counting problem in the context of the Heisenberg groups, for families of balls corresponding to certain radial and homogeneous norms, including the canonical Cygan-Koranyi norm. 

In the end, I shall briefly mention the scope of this method discussing more general nilpotent Lie groups. 

This talk is based on my joint work with Amos Nevo and Krystal Taylor which is available on arXiv. Most of this talk should be accessible to those who are familiar with basic functional analysis. 

• Title: Gaussian Elimination in classical groups
• Abstract: The classical row-column operations provide an efficient algorithm to solve word problem in the matrix group GL(n,k). We will see an analogues result for orthogonal and symplectic groups too.

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 third lecture of a series of five on Finite Fields.

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: This will be an introductory talk towards the theory of finite simple groups and their corresponding geometries. We shall discuss some examples which will be accessible to master level students.

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.

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.

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

 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. 

Abstract: In the first half of the talk we define unimodular rows over a ring R and see some of its basic properties. The completability of such unimodular rows can be viewed in terms of some special multiplicative subgroups of the matrix ring over R. In the second half, we define projective modules. Free modules are projective. In 1976 Quillen-Suslin gave a sufficient condition under which the converse is true. The completability of unimodular rows over certain polynomial rings plays an important role to show the fact that finitely generated projective modules over such ring are free. 

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

Abstract:

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

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

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

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

Abstract: The first half of the mini course will be an introducing to the two classical models of random graphs (a.k.a. Erdős-Rényi random graphs) and discuss the phenomenon of phase transition. We will also discuss thresholds for monotonic properties with examples including connectivity threshold and sub-graph containment threshold.

In the second half of the course we will consider other kind of random graphs. In particular, we will discuss various models for complex networks, including Albert-Barabási preferential attachment models. We will discuss "scale-freeness", asymptotic degree distribution and "small-world phenomenon". Properties of super and sub-linear preferential attachment models and some recent developments in de-preferential attachment models will also be discussed.

If time permits we will also introduce the random geometric graphs and discuss asymptotic of the connectivity threshold.

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

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

Abstact: Radon-Nikodym theorem of measure theory was proved in 1930 and A. N. Kolmogorov laid the foundation of modern probability in 1933 where the formal played a crucial role. In this Short Lecture Series in Mathematics (SLSM), we discuss the conditioning of probability theory in the light of Radon-Nikodym theorem. Upon the interest of the audience, we may introduce the notion of martingals and their convergence theorems. We have scheduled these lectures in the F-slot of Thursday and planned to go in accordance with the pace of the audience. Basic understanding of measure theory and functional analysis will be the prerequisite.

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

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 ? 

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

Functional Data Analysis is one of the frontline areas of research in statistics. The field has grown considerably mainly due to the plethora of data types that cannot be handled and analyzed by using conventional multivariate statistical techniques. Such data are very common in areas of meteorology, chemometrics, biomedical sciences, linguistics, finance etc .The lecture series will primarily aim at introducing the field of functional data analysis. Since functional data analysis is broadly defined as the statistical analysis of data, which are in the form of curves or functions, we will start with probability distributions and random elements in infinite dimensional Hilbert spaces, concepts of mean and covariance kernel/operator, the associated Karhunen-Loeve expansion and some standard limit theorems in Hilbert spaces. We will then discuss some selected statistical inference problems involving functional data like inference for mean and covariance operators, functional principal component analysis, functional linear models, classification problem with functional data, robust inference techniques for functional data etc. We will recall some results from functional analysis as and when required during the lectures. 

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

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

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

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

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

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.

In this talk we will see the well posedness results for the nonlinear Schr\"{o}dinger equation for the magnetic Laplacian on $\R^{2n}$, corresponding to constant magnetic field, namely the twisted Laplacian on $\C^n$ with power type nonlinearity $\lambda |u|^{\alpha} u$. We establish the well posedness in certain first order Sobolev spaces associated to the twisted Laplacian. The approach is via the spectral theory of the Schr\"{o}dinger propagator for the twisted Laplacian, and local existence is proved using Strichartz estimates established for the same.Using blowup analysis and conservation laws, we conclude global well posedness in the defocussing case (\lambda>0) with $0\leq \alpha< 2/(n-1)$ and also in the focussing case (\lambda<0) with $0\leq \alpha< 2/n$.We also prove finite time blow up in the focussing case (\lambda<0) with $2/n\leq \alpha< 2/(n-1)$. In this talk we also see a Hardy-Sobolev inequality for the twisted Laplacian on $\C^n$. We also show that the inequality is optimal in the sense that weight can not be improved.

Abstract:
Traditionally, Poisson process with constant rate have been extensively used to model arrival and potential service processes arising in queueing and queueing networks models under the Markov regime.
Generalization to non Markovian models, usually has taken the route of Renewal Processes for the arrival and the services schemes. This is neither natural nor realistic in the Modelling sense. In this talk, I will provide a brief history of queues and queueing networks, and emphasize the need for time variability in parameters, and hence the use of non Homogenous Poisson process as an alternative renewal process.

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. 

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: Dattarya Ramchandra Kaprekar was an Indian recreational mathematician who described several classes of natural numbers. The motive of this talk is to give a flavor of Elementary Number Theory and Iterations, by discussing the contributions of D. R. Kaprekar. I will discuss Kaprekar Numbers, Kaprekar Routines and Kaprekar Sequences. Nothing more than class 10 mathematics is needed to understand this talk.

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: An equivalent formulation of Ando's (generalization of Nagy's theorem) theorem providing a unitary dilation for a pair of commuting contractions is the statement: every contractive homomorphism of the bi-disc algebra is completely contractive. This is not true for any function algebra in three variables leaving the question of function algebras in two variables other than the bi-disc algebra open. The answer to this question for a large class will be discussed.  

We shall discuss a new method of computing (integral) homotopy groups of certain manifolds in terms of the homotopy groups of spheres. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we verify a conjecture of J. C. Moore. This is joint work with Somnath Basu.

Chirp signals are frequently used in different areas of science and engineering. MCMCbased Bayesian inference is done here for purpose of one step and multiple step prediction incase of one dimensional single chirp signal with i.i.d. error structure as well as dependenterror structure with exponentially decaying covariances. We use Gibbs sampling techniqueand random walk MCMC to update the parameters. We perform five simulationstudies for illustration purpose. We also do some real data analysis to show how the methodis working in practice.

Abstract: In this talk I will briefly introduce the study of random polynomials. I will provide a brief survey of some results. Further we will focus on the study critical points (zeros of derivative) in relation to the zeros of random polynomials, initiated by Pemantle and Rivin. It can be seen from the simulations that the zeros closely pair with the critical points and we investigate this phenomenon. We also establish similar phenomenon for zeros of higher derivatives. Further we attempt to compute the pairing distance between zeros and critical points of random polynomials. We will discuss the problem in the case when all the zeros of the polynomial are real. I will also pose some questions of interest and connections to other well known problems.

Abstract: The 'Thresholding Greedy Algorithm' was introduced by Konyagin and Temlyakov in order to study some special bases in Banach spaces. This basis is known as greedy basis and is characterized by unconditional and democratic properties of the basis. Later, Albiac, Dilworth, Kalton, Wojtaszczyk and few other author's studied the weaker versions of greedy basis, namely, almost greedy basis and partially greedy basis. Almost greedy basis can be characterized by quasi-greediness and democratic property and partially greedy basis can be characterized by quasi-greediness and conservative property. In this talk we will discuss some new characterizations of almost greedy and partially greedy basis. We will also discuss generalizations of greedy basis and its relatives. First part of this talk is joint work with Stephen J. Dilworth.

I will present interactions among 2-dimensional conformal field theory, which is a kind of quantum field theory in physics, theory of operator algebras and the Moonshine conjecture which predicted mysterious relations between the finite simple group Monster and the elliptic modular function. I will emphasize representation theoretic aspects and do not assume anyknowledge of these theories. 

Let $\Gamma\subset \overline{\mathbb Q}^{\times}$ be a  
finitely generated multiplicative group of algebraic numbers.  Let  
$\delta, \beta\in\overline{\mathbb Q}^\times$  be  algebraic numbers  
with $\beta$ irrational.  In this talk,  I will prove  that  there  
exist only finitely many triples $(u, q,  p)\in\Gamma\times\mathbb{Z}^2$
with $d = [\mathbb{Q}(u):\mathbb{Q}]$  such that
$$
0<|\delta qu+\beta-p|<\frac{1}{H^\varepsilon(u)q^{d+\varepsilon}},
$$
where $H(u)$ denotes  the absolute Weil height.  As an application of  
this result, we also prove a transcendence result, which states as  
follows:  Let $\alpha>1$ be a real number. Let $\beta$ be an algebraic  
irrational and  $\lambda$ be a non-zero real algebraic number.   For a  
given real number $\varepsilon >0$, if there are infinitely many  
natural numbers $n$ for which  $||\lambda\alpha^n+\beta|| < 2^{-  
\varepsilon n}$ holds true, then  $\alpha$ is transcendental, where  
$||x||$ denotes the distance from its nearest integer.

Google Meet Link: meet.google.com/rpj-qpwn-ows

We shall begin with a characterization of algebraic orthogonality in C^*-algebras in terms of order and norm. We apply this chacterization to propose a model of ``Absolutely ordered spaces'' which turns out to be a vector lattice under `commutative' conditions. This model suits to operator algebras and has potential to be applicable in quantum theory.

This talk will be in elementary level and first year Integrated MSc students should attained it.

Centerless Lie tori play an important role in explicitly constructing the extended affine Lie algebras; they play similar role as derived algebras modulo center play in the realization of affine Kac-Moody algebras. In this talk we consider the universal central extension of a centerless Lie torus and classify its irreducible integrable modules when the center acts non-trivially. They turn out to be highest weight modules for the direct sum of finitely many affine Lie algebras upto an automorphism.

An $m$-order $n$-dimensional square real tensor $\mathcal{A}$ is a multidimensional array of $n^m$ elements of the form$\mathcal{A} = (A_{i_1\dots i_m})$, $A_{i_1\dots i_m} \in \mathbb{R}$, $1 \leq i_1, \dots , i_m \leq n.$ (A square matrix of order $n$ is a $2$-order $n$-dimensional square tensor.) An $m$-order $n$-dimensional square real tensor is said to be a nonnegative (positive) tensor if all its entries are nonnegative (positive). We shall discuss the Perron-Frobenius theory for nonnegative tensors. Using these results we establish a sufficient condition for the positive semidefiniteness of homogenous multivariable polynomials.

The topology of real algebraic varieties is the study of the topology of objects that can be defined real algebraically, particularly the restrictions that the real algebraic structure imposes on the topology. The main problem that motivated interest in this area is the so called Hilbert's sixteenth problem that was suggested by Hilbert in his famous address: to classify, up to isotopy, non-singular planar real algebraic curves of a given degree. Topologically, a non-singular real algebraic curve is merely the union of circles (in fact, the fixed point set a complex conjugation on a Riemann surface), and its isotopy class is simply determined by the arrangement of these circles with respect to each other. Nevertheless, a complete classification, which would mean identifying which of these arrangements can be realized as the zero set of a real polynomial in two variables of a given degree, has as yet only been successful up to degree 7. However, it has led to several tangential questions and generalizations, which we will discuss after a brief overview of the motivating problem.

I shall discuss on control problems governed by the partial differential equations-mainly compressible Navier-Stokes equations,visco-elastic flows. I shall mention some of the basic tools applicable to study the control problems. We mainly use spectral characterization of the operator associated to the linearized PDE and Fourier series techniques to prove controllability and stabilizability results. I shall also indicate how the hyperbolic and parabolic nature of equations affects their main controllability results. Then some of our recent results obtained in this direction will be discussed.  

TBA

Recent decades have witnessed significant growth and progress in spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science. In recent years, new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations. This series of lectures will focus on basic theory and computations of spatial statistics. Topics will include conditional and intrinsic autoregressions, connections between Markov random fields and geostatistics, variogram calculations, h-likelihood methods and matrix-free computations. Applications from agricultural variety trials, environmental sciences and geographical epidemiology will be discussed.

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

A basis of the centralizer algebra for the action of the complex reflection group G(r, p, n) on the tensor product of its reflection representa- tion was given by Tanabe, and for p = 1, the corresponding partition algebra was studied by Orellana. In this talk, we first define the partition algebra for G(r, p, n) and call it Tanabe algebra. Along with the corresponding Schur–Weyl duality, using a confluence of ideas from Okounkov–Vershik approach, Clifford theory and higher Specht polynomials, we give a parametrization of the irreducible modules of Tanabe algebras and construct the Bratteli dia- gram. Furthermore, we give Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin basis of irreducible modules of Tanabe algebras. In the process, we also obtain some new results in the representation theory of complex reflection groups. The results presented in this talk form a part of a joint work with Dr. Shraddha Srivastava.

We will explain what it means to integrate differential forms on a manifold and state Stokes Theorem. We will see how it generalizes the fundamental theorem of calculus and Green's theorem. 

The Riemann-Roch Theorem is the foundation of the theory of algebraic curves. It gives a precise answer for the dimension of the space L(D) for a divisor D of an algebraic curve. The qualitative information that a Riemann surface is an algebraic curve is seen to be equivalent to more quantitative statement of Riemann-Roch.As an application of Riemann-Roch Theorem, it can be shown that any Algebraic curve can be holomorphically embedded into projective space. Any genus zero algebraic curve is isomorphic to the Riemann Sphere(C∞). Any non-degenerate smooth projective curve X in Pn of minimal degree n is a rational normal curve.

Abstract: In this talk, we will discuss the existence of weak solutions for some elliptic boundary value problems. In particular, we will discuss the weak solutions corresponding to the Laplace operator and the Biharmonic operator.

Abstract : We discuss about the large values of the Riemann zeta-function on the line $1$ and it's consequence to the error term of the summatory function of $d^2(n)$ for $1 \le n \le x$ for large $x$ considered by Srinivasa Ramanujan where $d(n)$ denotes the sum of the positive divisors of the positive integer $n$.

The realization of the Total Character (or Gel’fand Character) τG of a finite group G, i.e. the sum of all ordinary irreducible characters of G is an old problem in character theory of finite groups. One possible approach is to try to realize τG as a polynomial in some irreducible character of G. In this vein, K. W. Johnson has asked whether it is possible to express the total character of G as a polynomial, with integer coefficients, in a single irreducible character of G. We study for several classes of finite nilpotent groups, the problem of existence of a polynomial f(x) ∈ Q[x] such that f(χ) = τG for some irreducible character χ of G. As a consequence, we completely determine the p-groups of order at most p5 (with p odd) which admit such a polynomial. Indeed, we prove that: If G is a non-abelian p-group of order p5, then G has such a polynomial if and only if Z(G) is cyclic and (G,Z(G)) is generalized Camina pair and, we conjecture that this holds good for p-groups of any order. In the talk, we also discuss about the nonlinear irreducible characters of p-groups of order at most p5.

Abstract:  A well known result on translates of a function $\varphi$ in $L^{2}(\mathbb{R})$ states that the collection $\{\tau_{k}\varphi: k\in\mathbb {Z}\}$ forms an orthonormal system in $L^{2}(\mathbb{R})$ iff $p_{\varphi}(\xi)=\sum\limits_{k\in\mathbb {Z}}|\widehat{\varphi}(\xi+k)|^{2}= 1~ a.e.~ \xi\in\mathbb {T}$. Similarly in the literature there are interesting characterizations of Bessel sequence, frame, Riesz basis of a system of translates in $L^{2}(\mathbb{R})$ in terms of Fourier transform.In this talk, we aim to study frames in twisted shift-invariant spaces in $L^{2}(\mathbb{R}^{2n})$ and shift-invariant spaces on the Heisenberg group $\mathbb{H}^{n}$. First we shall define a twisted shift-invariant space $V^{t}(\varphi)$ in $L^{2}(\mathbb{R}^{2n})$ as the closed linear span of the twisted translates of $\varphi$. We shall obtain characterizations of orthonormal system, Bessel sequence, frame and Riesz basis consisting of twisted translates $\{T_{(k,l)}^{t}\varphi: k,l\in\mathbb{Z}^{n}\}$ of $\varphi\in L^{2}(\mathbb{R}^{2n})$ in terms of the kernel $K_{\varphi}$ of the Weyl transform of $\varphi$. In particular, we shall prove that if $\{T_{(k,l)}^{t}\varphi:k,l\in\mathbb{Z}^{n}\}$ is an orthonormal system in $L^{2}(\mathbb{R}^{2n})$, then $w_{\varphi}(\xi)=1$ a.e. $\xi\in\mathbb{T}^{n}$, where $w_{\varphi}(\xi)=\sum\limits_{m\in\mathbb{Z}^{n}}\int\limits_{\mathbb{R}^{n}}|K_{\varphi}(\xi+m,\eta)|^{2}d\eta,~\xi~\in\mathbb{T}^{n}$. Unlike the classical case on $\mathbb{R}^{n}$, it turns out to be a surprising fact that the converse of the above theorem need not be true. The converse is true with an additional condition, which we call "condition C". In fact, we shall see that $\{T_{(k,l)}^{t}\varphi:k,l\in\mathbb{Z}^{n}\}$ is an orthonormal system in $L^{2}(\mathbb{R}^{2n})$ if and only if $w_{\varphi}(\xi)=1$ a.e. $\xi\in\mathbb{T}^{n}$ and $\varphi$ satisfies condition C. Next we shall study shift-invariant spaces associated with countably many mutually orthogonal generators $\mathscr{A}$ on the Heisenberg group. We shall conclude the talk by providing a sampling formula on a subspace of a twisted shift-invariant space as an application of our results.

It is known that there are only finitely many perfect powers in non degenerate binary recurrencesequences. However explicitly finding them is an interesting and a difficult problem for binary recurrencesequences. A breakthrough result of Bugeaud, Mignotte and Siksek states that Fibonacci sequences$(F_n)_{n\geq 0}$ given by $F_0=0, F_1=1$ and $F_{n+2}=F_n+F_{n+1}$ for $n\geq 0$ are perfect powersonly for $F_0=0, F_1=1, F_2=1, F_6=8$ and $F_{12}=144$.In this talk, we the problem of finding perfect powers in products of terms of Recurrence Sequences. We show thatthere are only perfect powers and also give an explicit method to find them. We explicitly find the perfect powers in products of terms of some well known recurrence sequence including Fibonacci, Pell, Jacobsthal and Mersenne sequences and associated Lucas sequences. 

Abstract: Let $p$ be a prime number. The two-dimensional crystalline representations of the local Galois group $\mathrm{Gal}(\bar\Q_p|\Q_p)$ are parametrized by the pairs $(k,a)$ up to twists, where $k\geq 2$ is an integer and $a\in m_{\bar\Z_p}$, the maximal ideal in the ring of integers of $\bar\Q_p$. We are interested in studying the map$(k,a)\mapsto \bar V_{k,a}$, where $\bar V_{k,a}$ denotes the semisimplified mod $p$ reduction of a typical crystalline representation $V_{k,a}$. These reductions have been computed when $k\leq 2p+1$ or when $a$ has a small $p$-adic valuation. Using the theory of Wach modules, Laurent Berger has also shown that the map displayed above is locally constant with respect to both the variables and found an explicit bound on the radius of local constancy when %$k$ is fixed and$a$ varies (2012). However, if $a$ is fixed and $k$ varies, nothing more than the existence is known about the radius of local constancy. So we ask the following simple question: for any given $p$-adic integer $a$, how close do $k$ and $k'$ need to be in the weight space to ensure that $\bar V_{k,a}\cong\bar V_{k',a}$? We give a partial answer to this question using some explicit computations in the automorphic side of the $p$-adic and mod $p$ Local Langlands Correspondences for $\GL_2(\Q_p)$. 

Abstract: We shall begin with the homotopy invariance property of K-theory.After reviewing monoids and monoid algebras, we present some results which aremonoid version of the homotopy invariance property in K-theory. This answers a question ofGubeladze. Next, we will discuss the monoid version of Weibel's vanishing conjectureand some results in this direction. Finally, we will talk about the homology stability for groups.Here we present a result which improves homology stability for symplectic groups.If the time permits, some application of the homology stability will be given to the hermitian K-theory.  

Cartan subgroups of Lie groups play a crucial role in the study of the structure of Lie groups, the behavior of the power maps, and also in determining whether the exponential map is surjective or has a dense image. Here we present some recent progress on `the density/surjectivity of the power maps of Lie groups ', and `the structural aspects of Cartan subgroups' and its applications on the characterization of the density of the power map in Lie groups.

Abstract: In this talk we will review on the minimal blocking sets of external, tangent and secant lines to an irreducible conic in PG(2,q).

Abstract: The theory of $M$-ideals in Banach spaces as well as in operator spaces is one of the important areas of research. It is well known from the literature that order structures and compact convex sets play a significant role in this area. Especially, the characterization of $M$-ideals in $A(K)$-spaces in terms of split face of the compact convex set $K$, established by Anderson and Alfsen, is a classical theorem of great importance.In this thesis, we have characterized $M$-ideal in order smooth $\infty$-normed spaces in terms of split faces of the quasi-state spaces. Also, we discuss the complete $M$-ideals in matricially order smooth $\infty$-normed spaces. For $p\neq \infty$, we introduce the notion of ideals, smooth $p$-order ideals to initiate the study of ideals in order smooth $p$-normedspaces.We introduce the notion of an $L^{1}$-matrix convex set in $*$-locally convex space. We show that $\{A_{0}(K_{n}, M_{n}(E))\}$(the `quantized functional space’) is a $\mathrm{C}^*$-ordered operator space. Conversely, every $\mathrm{C}^*$-ordered operator space is complete isometrically, completely isomorphic to $\{A_{0}(Q_{n}(V) M_{n}(V))\}$, where $Q_{n}(V)$ is the quasi-state space of $M_{n}(V)$ (in the matrix duality).

We start with defining binary quadratic forms and discussing some properties of them.  Given an integer, we look for the forms representing that. We define a composition rule for forms and define the class group. Universal forms represent every integer. We present a theorem on the classification of universal binary quadratic forms. Finally, we discuss Ramanujan's universal quaternary diagonal quadratic forms and give formulas for the number of representations of an integer by some of those forms.

Every probability distribution has some uncertainty associated with it. The concept of Shannon’s entropy provide a quantitative measure of this uncertainty. But for some probability distributions the Shannon’s entropy measure may be negative and then it is no longer an uncertainty measure. To overcome this problem the concept of generalized entropy has been introduced in literature. In many practical situations it has been seen that when an investigator collect a sample of observations produced by nature, according to a certain model, the original distribution may not be reproduced due to various reason. For this reason it is important to consider the concept of weighted distribution. Motivated with the usefulness of the generalized entropy and the weighted lifetime distributions, we introduce the concept of weighted generalized entropy and discuss several properties of this model.

The Petersson scalar product split the space of modular forms for congruence subgroup into the space of cusp forms and the space of Eisenstein series.In this talk we show how one gets the Hecke theory  for the space of Eisentein series for weight is either integer or half-integer. We also show their connection through Hecke equivalent maps.