Abstract: In a celebrated paper in 1971, Serrin studied the overdetermined problem for the Laplacian operator. The proof relied on a powerful technique, now known as the method of moving planes, a refinement of a reflection principle conceived by Alexandrov. Later this idea of reflection principle was picked-up by Gidas, Ni and Nirenberg in 1979 in their landmark work on the symmetry of positive solutions of elliptic problems. The overdetermined problems were then extended for annulus as well as for unbounded domains. These problems are also related to the parallel surface problems. Recently, similar problems have been studied for nonlocal operators. In this talk, we shall give an overview of these problems and its extension.
Abstract: We consider the action of the semigroup of$n\times n$ nonsingular integral matrices, on the space of $p$-tuples of $n$-vectors, where $p < n$. Then the orbit of a $p$-tuple is dense whenever the coordinates are not all contained in a proper rational subspace. We discuss the aspects of effectiveness of approaching a general target point, in terms of the sizes of the matrices involved. The critical exponents of the behaviour will be discussed.
Abstract: Group actions on bundles have interesting consequences. I will introduce the notion of connection on principal bundles both in the analytic and algebraic settings. I will try to explain how certain types of connection are related to the existence of group actions on a bundle.
Abstract: Studying excursion sets of random fields has a long history with motivations coming from statistical hypothesis testing, cosmology, stereology, and integral geometry, to name a few. Questions pertaining to the characterisation of excursion sets are often considered very relevant, and very challenging. For instance, one is often interested in obtaining precise statistical description of the volume, (cumulative) surface area, perimeter of excursion sets of random fields. In this talk, we shall present an asymptotic description of (some) such geometric functionals of excursion sets of random fields in different scenarios.
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Abstract: Just as is the case for a stretched guitar string, for which any deformation can be viewed as a combination of sines and cosines of multitude of amplitudes and frequencies (that is, it can be expressed as a combination of "pure oscillations", or "pure vibrations", or "eigenfunctions"), physics seems determined that the same should happen with absolutely everything: from sound and light, to matter and time, and including laws that govern them, our understanding of the physical world is based, at a fundamental level, on consideration of eigenfunctions. With this motivation in mind we explore some areas in the mathematics of the world of vibrations, and we consider, in particular, a certain special combination of eigenfunctions: the Green function. After reviewing basic ideas concerning eigenfunctions, we will mention recent numerical methods in which the eigenfunctions are computed precisely using the aforementioned Green function. Based on Green's functions, we will build, study and use eigenfunctions related to various physical problems, including, for example, oscillations describing disturbances on the surface of water in a glass (which are given by Steklov eigenfunctions), the interaction of light with nano-optical devices and the design of such devices (Maxwell eigenfunctions), quantum mechanics and sound propagation (Schrodinger and Helmholtz eigenfunctions), problems related to probability theory (eigenfunctions of Laplace and fractional Laplacian operators), and, notably, the temporal sinusoidal vibrations themselves.We will also briefly consider certain geometric characteristics of the eigenfunctions, such as the distribution of the corresponding eigenvalues (oscillation frequencies) and their "nodal curves"--that is, the points in space where the eigenfunctions vanish--including a discussion, with some details, of the intriguing nodal lines of the Steklov eigenfunctions related to the aforementioned glass of water.
Abstract: This will be a survey talk on the subconvexity problem. After a brief historical introduction, we will discuss the several aspects of the problem and the different techniques that are used to tackle them. At the end we will mention some recent results.
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Abstract: Starting with the definition of conditional probability we shall discuss some examples of Markov Chains. Shall explain how they model practical phenomena. Briefly discuss how Markov chains can be used for simulation.
Abstract: Classical permutations are symmetries of sets with finitely many points. In the matrix representation, classical permutations are specific square matrices with entries of either 0 or 1. This perspective was generalised by Shuzhou Wang in 1998, and he introduced the notion of quantum permutation group. This talk primarily aims to motivate Wang's pioneering idea and discuss a few properties of the quantum permutation group.
Abstract: In this talk, we shall provide a broad overview of the models used in India and their capabilities. We will examine instances of successes, failures, and the reasons for these outcomes. Then we will focus on some of the work done for Government of Karnataka. We shall conclude with a road map of how generated data could be stored, presented, and shared so that the expertise of the broader modelling community can be fully tapped for better pandemic preparedness.