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Lectures

Topic: Patterns, Entropy and Gibbs measures

Instructor: Nishant Chandgotia, TIFR CAM, Bangalore

Statistical physics attempts to explain natural physical phenomena like freezing and magnetism by means of simplistic probabilistic models. In this series of lectures, after building on prerequisites like ergodic theory, probability and information theory, we will introduce several models of statistical physics like the Ising model, hardcore model and the dimer model explaining how they arise and what we know about them. In between I also hope to introduce some open questions which have fascinated me for quite a few years like that of the Arctic circle phenomena, formation of quasicrystals and finiteness of phases of a simplistic class of models arising from graph homomorphisms.

References:

  1. An introduction to information theory by Cover and Thomas
  2. Equilibrium States in Ergodic theory by Keller
  3. Thermodynamic Formalism by David Ruelle

Topic: Geometric Measure Theory

Instructor: Debabrata Karmakar, TIFR CAM, Bangalore

Review of measures: The definition of Lebesgue and Hausdorff measure, Caratheodory measurability, covering theorems, Hardy-Littlewood maximal function and differentiation theorems. 

Hausdorff measures: The length of a curve and one dimensional Hausdorff measure, The Lipschitz functions, Radamacher theorem, Lipschitz extension and Kirszbraun’s theorem, The isodiametric inequality and H^n = L^n, Hausdorff dimension of Lipschitz Graph.

The area formula: The polar decomposition of matrices, Jacobian determinant, The proof of area formula, Applications: length of a curve, surface area of a Lipschitz graph,  volume element of embedded submanifolds, The Gauss-Green formula for C^1-domain.

The coarea formula: The proof of co-area formula, Applications: Polar coordinates, integration over level sets.

Topic: Microlocal Analysis and Inverse Problems 

Instructor: Venky Krishnan, TIFR CAM, Bangalore

We will review the basics of microlocal analysis and give an introduction to microlocal analysis in imaging and tomography problems. Topics covered will include :

  1. Introduction of microlocal analysis
  2. Applications of microlocal analysis in tensor tomography problems and image reconstruction problems
  3. Applications of microlocal analysis in Calderón inverse problems 

References:

  1. The Analysis of Linear Partial Differential Operators by Lars Hormander Vol 1
  2. Microlocal Analysis for Differential Operators, By Grigis and Sjostrand

Topic: Introduction to Harmonic Measure and Geometric Function Theory

Instructor: Atul Shekhar, TIFR CAM, Bangalore

List of topics to be covered include,  Jordan Domains and their conformal equivalence, non-tangential limit, Stolz angle and prime ends, Dirichlet problem and harmonic measure, Caratheodory theorem, some results about univalent functions, Green's functions and Poisson kernels, boundary smoothness, extremal distance, Teichmuller Modulsatz, Bloch Functions and quasicircles. 

References:

  1. Harmonic Measure  by Garnett and Marshall
  2. Boundary Behaviour of Conformal Maps by Pommerenke

Topic: Reproducing kernel Hilbert space (RKHS): from theory to applications

Instructor: Sreekar Vadlamani, TIFR CAM, Bangalore

Abstract: Reproducing kernel Hilbert spaces have become a vital tool in numerous disciplines particularly in statistics/machine learning. It not only provides an alternate perspective of modelling problems in machine learning, but also provides a framework to answer several fundamental questions. In these lectures, we shall introduce the theory of RKHS through basic (finite dimensional) examples, and graduate to more intricate settings to unravel the full potential of RKHS. We shall also go through some pedagogical, and some realistic, applications of the RKHS tool.

References: (more references will be shared during the lectures)

  1. An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, by V. I. Paulsen and M. Raghupathi

  2. Kernel Mean Embedding of Distributions: A Review and Beyond, by K. Muandet, K. Fukumizu, B. Sriperumbudur and B. Scholkopf

Topic: Multiscale Problems in PDEs

Instructor: M Vanninathan 

Abstract : Starting with Continuum Mechanics Modelling, we see how multiscale problems arise in PDE in simple situations. Main feature of such problems is the  small scale variations of their solutions. Due to this, classical ideas of solving them fail, both theoretically and numerically. To overcome this difficulty, one of the ideas developed is homogenization.  A surprising conclusion is that the original model PDE is not ideal in the presence of these small scales and a new approximate model is needed. I intend to demonstrate this by giving an introduction to some developments of Homogenization Theory.  I would like to cover the following chapters: Examples of Microstructures, Elliptic Homogenization, Effective coefficients associated with periodic structures and general structures,  Compensated Compactness, Effective set, Two-scale convergence, Bloch wave method etc.  Some possible applications will also be mentioned. 

References: (other references will be given during lectures): 

  1. A Bensoussan, J-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures 1978.

  2. L.Tartar, The General Theory of Homogenization, 2009.

  3. G. Allaire, Shape Optimization by the Homogenization Method, 2002.