We shall describe recent progress on the question of writing down the reductions of certain local Galois representations. We shall focus on the case of half integral slopes (especially slope 3/2) where the behaviour of the reduction is both more complicated and more interesting. Our proof uses the mod p Local Langlands Correspondence to reduce the problem to computing the reductions of certain locally algebraic representations of GL_2 of the p-adics on certain functions on the underlying tree

We shall describe recent progress on the question of writing down the reductions of certain local Galois representations. We shall focus on the case of half integral slopes (especially slope 3/2) where the behaviour of the reduction is both more complicated and more interesting. Our proof uses the mod p Local Langlands Correspondence to reduce the problem to computing the reductions of certain locally algebraic representations of GL_2 of the p-adics on certain functions on the underlying tree.

This talk is based on the recently concluded 19-lecture course by Xavier Viennot at IMSc, and is meant as publicity for the videos (freely and perpetually accessible) of those lectures on the Matscience Youtube channel. The lectures are jam-packed with new and elegant proofs of well known results, myriad applications--- from graph theory to Lie algebras and their representations to statistical physics and even quantum gravity---and open problems of varying difficulty. We will take a tour through the basic definition, the main technical results, and some applications.

It is known that contractive homomorphisms of the disc and the bi-disc algebra to the space of bounded linear operators on a Hilbert space are completely contractive, thanks to the dilation theorems of B. Sz.-Nagy and Ando respectively. Examples of contractive homomorphisms of the (Euclidean) ball algebra which are not completely contractive was given by G. Misra. From the work of V. Paulsen and E. Ricard, it follows that if m >= 3 and B is any ball in C^m with respect to some norm, then there exists a contractive linear map which is not complete contractive. The characterization of those balls in C^2 for which contractive linear maps are always completely contractive remained open. In this talk, we intend to answer this question for balls in C^2 which are of the form {z= (z1, z2) :||zA||=||z1A1+z2A2||op<=1} for some choice of an 2-tuple of 2x2 linearly independent matrices A = ( A1, A2)

We observe periodic phenomena everyday in our lives. The daily temperature of Delhi or the number of tourists visiting the famous Taj Mahal or the ECG data of a normal human being, clearly follow periodic nature. Sometimes, the observations may not be exactly periodic due to different reasons, but they may be nearly periodic. The received data is usually disturbed by various factors. Due to random nature of the data, statistical techniques play important roles in analyzing the data. Statistics is also used in the formulation of appropriate models to describe the behavior of the system, development of an appropriate technique for estimation of model parameters, and the assessment of model performances. In this talk we will discuss different techniques which we have developed for the last twenty five years for analyzing periodic data, other than the standard Fourier analysis.

Consider a sample of size N from a linear process of dimension p where n, p are tending to infinity, p/n is tending to a finite non-negative number. We prove, in a unified way, that the limiting spectral distribution (LSD) of any symmetric polynomial in these matrices exist. Our approach is through the intuitive algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. We suggest statistical uses of these LSD and related results in problems such as order determination and white noise testing. The ideas extend to several independent processes and is useful for statistical tests in to sample problems.

The talk will outline the development of the PascGalois Project. Its origins are in an exercise using Pascal's Triangle and modular arithmetic. Colors are assigned to the numbers 0, 1, ..., n-1, and Pascal's Triangle modulo n is drawn. The patterns in the triangle are then related to the properties of the cyclic group Zn. The process of drawing the triangles is then generalized to non-cyclic and non-abelian groups and the new patterns are examined in light of the properties of these groups. The images can help develop visual and intuitive understanding of concepts such as subgroup closure and quotient groups. They can also be used to discuss the relationship between mathematical properties and visual aesthetics. Finally we view Pascal's Triangle as a one-dimensional cellular automata and generalize to more general initial conditions and two dimensional automata. Many of the investigations in this project have been undertaken with students in undergraduate research projects and one outgrowth of the project has been the development of a set of visualization exercises to supplement the standard undergraduate course in abstract algebra. The web site for the project is at www.pascgalois.org.

An important class of problems in mathematics deals with the reconstruction of a structure from the eigenvalues of an associated matrix. The most famous such problem is: Can one hear the shape of a drum? Here we deal with the question: Which graphs are determined by the spectrum (eigenvalues) of its adjacency matrix? More in particular we ask ourselves whether this is the case for almost all graphs. There is no consensus on what the answer should be, although there is a growing number of experts that expect it to be affirmative. In this talk we will present several results related to this question. This includes constructions of cospectral graphs and characterizations of graphs by their spectrum. Some of these results support an affirmative answer, some support the contrary. It will be explained why the speaker believes that it is true.

Since time immemorial, people have been trying to understand the rational number solutions of systems of homogeneous polynomial equations with integer coefficients (called a Diophantine system). It is more convenient to think of them as rational points on associated projective varieties X, which we wll take to be smooth. This talk will introduce the various questions of this topic, and briefly review the reasonably well understood one-dimensional situation. But then the focus will be on dimension 2, and some progress for those covered by the unit ball will be discussed. The talk will end with a program (joint with Mladen Dimitrov) to establish an analogue of a result of Mazur.

We consider a nonlinear age structured McKendrick-von Foerster population model with diffusion term (MV-D). We prove the existence and uniqueness of solution of the MV-D equation. We also prove the convergence of the solution to its steady state as time tends to infinity using the generalized relative entropy inequality and Poincare Writinger type inequality. We propose a numerical scheme for the linear MV-D equation. We discretize the time variable to get a system of second order ordinary differential equations. Convergence of the scheme is established using the stability estimates by introducing Rothe function.