Abstract: How would you strategize in a game with say, 50 players ? If the game were repeated many times, and you saw that your neighbour was doing well in several previous rounds, would you be tempted to imitate the neighbour in the next round ? Would that be {\em rational} on your part ? In games with a large number of players, outcomes are associated not with the actual tuple of strategies chosen by players but with the distribution of what fraction of players choose which move. The pattern of reasoning in such games is different from those in which all players know each others' types. We discuss Nash equilibria, and some logical / automata theoretic formulations of stability in such games.
About the speaker: Professor R. Ramanujam from The Institute of Mathematical Sciences, Chennai.
Abstract: Urn models are one of the basic models studied in stochastic processes. It has nearly a century long history dating back to the works of Polya and Eggenberger. It has useful applications in modelling spread of contagious diseases, prey-predator analysis, diffusion of gases, reinforced learning, database management, analysis of algorithms - to name a few only. It is also useful in clinical trials, but in that case it is important to assume the replacement matrix to be random. Bai and Hu (2005) provided a detailed analysis of such models using martingale techniques under the assumption of existence of $(2+\epsilon)$ moments with $\epsilon>0$. Laruelle and Pages (2013) improved upon the results using stochastic approximation by relaxing the moment assumptions to existence of second moment alone. We relax the moment conditions further by careful application of stochastic approximation technique. The presentation will provide a brief overview of stochastic approximation technique in the current context. This is a joint work with Ujan Gangopadhyay.
About the speaker: Professor Krishanu Maulik from Indian Statistical Institute, Kolkata.
Abstract: The classical Wierstrass approximation theorem of a continuous function from [0,1] to R by polynomials in uniform norm will be proved using the weak law of large numbers for coin tossing. This will be generalised to continuous functions on the unit simplex and the unit cube in higher dimensions. The approximation by convolution to produce smooth functions will also be discussed. Finally, Stirlings formula will be proved using Fourier analysis and the local clt for the Poisson distribution.
About the speaker: Professor Athreya obtained his Phd. in mathematics from Stanford University, California USA in 1967. He returned to India in 1971 and was a professor of mathematics at the Indian Institute of Science from 1971 to 1980. He was a professor at Iowa State University from 1980 till 2013. He is currently a professor emeritus at Iowa State University. He was given the title of professor in the College of Liberal arts and Sciences, Iowa State University, in 1998.
His fields of interest are Probability Theory, Mathematical Statistics, Mathematical Analysis and Mathematical Modelling. He is a Fellow of the Indian Academy of Sciences. He is a Fellow of the Institute of Mathematical Statistics, USA.
He is an elected member of the International Statistics.
He has authored two books published by Springer Verlag and written a number of research papers in mathematics and statistics. He is deeply interested in teaching mathematics at all levels. His hobbies include listening to Indian classical and folk music.
Abstract: Among the many contributions of George Polya in Mathematics, "Polya's Theorem on Polynomials", stands out for being a surprising result and for its beauty of proof. So much so that it has been favorite to Paul Erdos. We present this wonderful result and its astoundingly beautiful and simple proof.
About the speaker: Professor Harish Chandra is from the Department Of Mathematics Institute of Science, Banaras Hindu University.
Abstract: Riemann surfaces or complex structures on surfaces are ubiquitous in mathematics. I shall introduce some other associated geometric structures like hyperbolic and projective structures, and talk of their deformations. The study of their moduli spaces involves an interplay of topology, geometry and analysis, and I will highlight some of that interaction in the context of my own work.
Abstract: We explore the entropy solution framework for scalar conservation laws that are perturbed by multiplicative Levy noise. The primary focus of this talk is to establish existence and uniqueness of entropy solutions of conservation laws with multiple spatial dimensions that are driven by jump processes. The entropy inequalities are formally obtained by Ito-L´evy chain rule. The issue of multidimensionality requires generalized ´ interpretation of entropy inequalities to accommodate Young measure valued solution. We first establish the existence of entropy solution in the generalized sense via vanishing viscosity approximation, and then establish the L1-contraction principle which also requires vanishing viscosity regularization. Finally, the L1 contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.
Based on joint works with K. H. Karlsen, U. Koley and A. K. Majee.
Abstract: In 1927, Van der Waerden proved that if the set of natural numbers is partitioned into two sets, one of them will have arbitrarily long arithmetic progressions. Generalizing, Erdos conjectured that one could partition any sufficiently dense infinite set S of numbers into two, and one of the parts will contain arbitrarily long arithmetic progressions. This was proved in a celebrated result by E. Szemeredi in 1975, using combinatorial techniques.In 1978, H. Furstenberg provided an ergodic theoretic proof of Szemeredi's result. The talk will cover the topological dynamical approach to questions on arithmetic progressions on integer sets using Furstenberg's approach. We will briefly mention some effective versions of Furstenberg's result, obtained in a recent joint work of the speaker with Rod Downey and Andre Nies.As a minor digression, we will also mention a "proof from the book" by H.
Furstenberg, which sheds some light on the connection between topology and number theory.
About the speaker: Satyadev Nandakumar is an assistant professor in computer science and engineering at IIT Kanpur. He received his Ph. D from Iowa State University in 2009.
Abstract: In this talk we discuss a proof of the result that the surfaces of revolution that are geodesically conjugate to the flat cylinder are isometric to it. This is a joint work with H. A. Gururaja.
About The Speaker: Professor C.S. Aravinda finished M.Sc. in 1985 from Central College, Bangalore and Ph.D. in Mathematics in 1995 from TIFR, University of Mumbai. His research interests are in Geometry, Topology and Dynamics in negative curvature.He was a faculty in Chennai Mathematical Institute from 1997-2007, and has been at TIFR-CAM since 2007.Professor Aravinda has translated the biography "The Man Who Knew Infinity" of Ramanujan from English to Kannada. Also he has been conducting workshops for college and university teachers and has been in the Editorial board of RMS newsletter, Mathematics Student, Hardy-Ramanujan Journal.
Abstract: Embeddings of Sobolev spaces play an important role in the analysis of partial differential equations. We will discuss some of these sharp embeddings known as Moser-Trudinger and Adams Inequalities and present some of the recent results obtained.
About The Speaker: Prof. K. Sandeep is a faculty at TIFR-CAM, Bangalore. His research interests lie in Variational Methods of PDEs, including PDE posed on hyperbolic spaces, and nonlinear Functional Analysis. He completed his Ph.D. from TIFR, Bangalore centre in 2002. He received the Shanti Swarup Bhatnagar Prize for Mathematical Sciences in 2015.
Abstract: Every one knows Fermat's Last Theorem was proved by Andrew Wiles in 1994. But not many are aware that some 10 years before that Gerd Faltings had made substantial progress towards proving the theorem. In fact Faltings proved a very general result (known as Mordell's conjecture) which asserts in particular that for a large class of homogeneous polynomials with coefficients in the rational number field $\Q$, the set of zeros in $\Q^3$ is finite upto scaling by $\Q^x$. This class includes the Fermat polynomials $x^n+ y^n + z^n$ for $n > 4$. In this talk I will formulate and explain the statement of Falting's theorem and add some general comments. I will say nothing about the proof (which I am not adequately familiar with).
Abstract: We consider certain lower order perturbations of polyharmonic operators and prove uniqueness of recovery of the perturbations from the knowledge of full and partial boundary Neumann data. Time permitting, we will prove stability estimates for the recovery of zeroth order perturbation of the biharmonic operator from full and partial boundary Neumann data as well.
Abstract: Width (resp. height) for a subgroup of a group roughly measures the pairwise (resp. total) intersection of conjugates of the subgroup. In this talk, after going through some basics of Hyperbolic Geometry, I will prove width and height of quasiconvex subgroups of closed hyperbolic surface groups to be finite. This proof is due to R.Gitik, M.Mitra, E.Rips and M.Sageev. They have proved it in more general case: quasiconvex subgroups of hyperbolic groups have finite height and width..
Abstract: One simple approach to visualize and summarize complex curve/image data is to extend the classical boxplot to the functional setting. This necessitates to develop a ranking of the functions. A first possibility is to use the notion of band depth that produces an ordering from the center outward. A second possibility is to use a tilting approach to assessing the influence that functional data have on the value of a statistic, and to rank the data in terms of that influence. We describe the computational aspects of those two approaches, explore their properties by simulations, and illustrate their application to data from climate science and brain imaging.