Summer Programme 2025

TIFR Centre for Applicable Mathematics is organizing a Summer Programme mainly targeted toward Master’s and entry-level PhD students focusing on Analysis and related topics during the period June 16 - July 11, 2025. The programme will consist of ten 90-minute lectures in each of the following topics:

Topics

• Patterns, Entropy and Gibbs measures by Nishant Chandgotia
  
• Geometric Measure Theory by Debabrata Karmakar
• Microlocal Analysis and Inverse Problems by Venky Krishnan
• Introduction to Harmonic Measure and Geometric Function Theory by Atul Shekhar
• Reproducing kernel Hilbert space (RKHS): from theory to applications By Sreekar Vadlamani

• Multiscale Problems in PDEs by M Vanninathan

The detailed list of topics that will be covered is given below. This set of courses will be offered only in offline format. 

Important Information : Due to budget and accommodation constraints, we cannot offer a fellowship and accommodation to attend this programme. If you are an out of town person interested in attending this programme, please bear this in mind before applying. There are several paying guest accommodations in the vicinity of TIFR CAM and the monthly rent for this is in the range of Rs. 10000. You are free to use the institute cafeteria during your stay in Bangalore where food is available at reasonable prices.

At the end of the program, if you have attended all the lectures, you will be given a participation certificate.

How to Apply

If you are interested in attending this program, please send an email to summerprog@tifrbng.res.in with the following information.

(please cut and paste the following into your email and fill in the details in your email):

Name:

Current programme of study:

Institute/University:

Topic of specialization (if you are a PhD student):

Why are you interested in attending this program? (Give a brief description of up to 500 words):

Important Dates

• Summer Programme Duration : June 16 - July 11, 2025
• Deadline: April 30, 2025
• The list of selected participants will be announced by May 2, 2025 on TIFR CAM website

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.

Special Lectures

Title : Gram matrices: connecting GPS triangulation, Euclidean metric embeddings, and Heron’s formula

Title :  Sobolev type inequalities and related problems.

Title : Distance between walks on graphs

Title: The probabilistic method in analysis

Speaker : Koushik Ramachandran, TIFR CAM, Bangalore

Abstract : Paul Erdos invented the probabilistic method to answer some questions in graph theory and combinatorics. In the last 50 years it has become clear that the probabilistic method is very general and is an useful tool in analysis, algebraic geometry, computer science, etc. In this talk I will introduce the probabilistic method as envisioned by Erdos but quickly move to some applications in harmonic and complex analysis. The talk should be accessible to students with a background in basic graduate level analysis, and discrete probability.

Title: Ballot problem, Young tableaux and Non-intersecting paths

Speaker : Manjunath Krishnapur

Abstract: In the ballot problem, there are k candidates who get votes a_1\ge a_2\ge ...\ge a_k. All the votes are placed in a box and drawn one after another at random while counting. What is the probability that at all times during the counting process, the first candidate is leading over the second candidate who is leading over the third and so on?

It turns out that this problem is equivalent to that of counting the number of standard or semi-standard Young tableau of a given shape (the terms will be defined in the lecture). And both problems are solved by a lemma of Karlin--Mcgregor and Lindstrom-Gessel-Viennot that gives a determinantal formula for the number of non-intersecting collections of paths with given starting and ending points in certain directed graphs.  Only very basic probability and linear algebra knowledge will be assumed. 

 Download Notes(pdf)

Title: Exact solvability of the dimer model 

Speaker : Terrence George

Abstract: The dimer model—originally a toy model for the adsorption of diatomic molecules on surfaces—has grown into a central object in statistical mechanics, combinatorics, discrete geometry, etc. In this talk, I will introduce the model from first principles and explore its exact solvability on planar graphs via Kasteleyn's remarkable determinant method.

 Download Notes (pdf)

Title: On the depth query problem for convex bodies: a geometric approach

Title : Protecting quantum states against environmental perturbations by dynamical decoupling using inverting pulses

Speaker : Deepak Dhar, ICTS Bangalore

Abstract : One important difficulty in developing powerful quantum computers is the fact that information stored in qubits is susceptible to small changes in environment, and suffers degradation and loss of fidelity with time. One way to protect quantum states against this to subject the quantum system to a series of magnetic field pulses. In this, a particularly promising candidate are “inverting pulses”, which are very short duration magnetic field pulses, approximated as delta functions in time, that are designed so that one subspace of the Hilbert space picks up a phase π compared to perpendicular subspace.

I will discuss the problem of designing the optimal sequence of interval between pulses to to supress the transition rate between these two subspaces to be of Ο(λ
2N+2), where λ is the strength of pertubation. It was shown by Uhrig that one can achieve this by only using N pulses. I will discuss the problem in a general setting, and show that optimization problem involves solution of order 2N coupled polynomial equations in M variables. One might expect that M will have to be
of order 2N. Miraculously, it turns out there is consistent solution exists with only N variables. I note that a transparent argument for this miracle is not yet available, and remains a challenging unsolved problem.

The approach uses very elementary mathematics involving only linear algebra of matrices, and should be accessible to all undergraduate students.

Reference : Preserving quantum states: a super-zeno effect, D.Dhar, L.K. Grover and S. M. Roy, Phys. Rev. Lett., 96, 100405 (2006).

Title: Inequivalence of the Ball and Polydisc in ℂ²

Speaker: Sivaguru R, TIFR CAM , Bangalore

Abstract: Riemann mapping theorem tells us that all simply connected domains in ℂ (except ℂ itself) are biholomorphic to the unit disc 𝔻. This is no longer true in ℂ² (and in higher dimensions) – for instance, the unit ball in ℂ² is not biholomorphic to 𝔻 × 𝔻. We will prove this result by studying the Bergman kernel of these domains. The Bergman kernel of a domain is the reproducing kernel for the Hilbert space of square integrable holomorphic functions on that domain.

Title:  Egorychev method for computation of combinatorial sums

Speaker : Venky Krishnan, TIFR CAM, Bangalore

Abstract : In this talk, we will explore the method of Egorychev based on contour integration in the complex plane to analyze and simplify combinatorial terms. Several examples will be presented. The talk will be accessible to a wide audience. Basic knowledge of complex analysis is all that is required.

Title: Sorting a poset using the Brunn-Minkowski inequality

Speaker : Jaikumar Radhakrishnan

Abstract :  We consider the problem of sorting under partial information. We are given a partially ordered set X= {x_1,x_2, ... x_n}. We are told that {x_1, x_2, ..., x_n} have an underlying total order Q that is consistent with X. We wish to determine Q using the fewest comparisons between elements  of X. The number of comparisons needed has a lower bound log_2 e(X), where e(X) is the number of linear extensions of X. We will present an argument based on the Brunn-Minkowski inequality due to Jeff Kahn and Nati Linial that shows that O(log_2 e(X)) comparisons suffice.

Schedule of Lectures

Selection List

List of candidates shortlisted for Summer Programme 2025 at TIFR CAM Bangalore. Shortlisted candidates will receive the communication with regard to this program by email sent from summerprog@tifrbng.res.in