Summer School 2026

Abstract

Given a smooth manifold, and a distribution (a lower dimensional subspace of the tangent space on the manifold at each point), we are interested in the question whether there is an embedded submanifold of the manifold whose tangent space is precisely this distribution. In the 1-dimensional case, the answer to this is positive. More precisely, if we are given a non-vanishing vector field on the manifold, then the integral curves of this vector field are immersed submanifolds and locally, we can straighten the vector field to obtain that these integral curves are straight lines. We will quickly review this and standard results on flows.  Next, we consider the question of finding higher dimensional analogues of integral curves (locally embedded submanifolds), given a k-dimensional distribution. There is a necessary condition called involutivity that should be satisfied by the distribution. Frobenius theorem states that the involutivity condition is also sufficient. We will prove local and global versions of Frobenius theorem and give some applications to the study of partial differential equations.

Abstract

Lecture 1&2 — Foundations of Optimal Transport
Monge and Kantorovich formulations; Kantorovich duality; cyclical monotonicity; convex potentials and gradients of convex functions; Brenier–McCann theorem; displacement interpolation; Wasserstein geodesics.

Lecture 3— Regularity of Transport Map
Monge–Ampère equation as a fully nonlinear elliptic PDE; a brief overview of the regularity theory for optimal transport maps; McCann’s pointwise formulation of the Monge–Ampère equation; Caffarelli’s contraction theorem.

Lecture 4&5 — Functional and Geometric Inequalities via Transport
I plan to present transport-based proofs of the following inequalities: Sharp Sobolev inequality in the Euclidean isoperimetric inequality; Gaussian functional inequalities; Talagrand, logarithmic Sobolev, and Poincaré inequalities; transfer of functional inequalities to log-concave perturbations of the Gaussian distribution; The HWI inequality; Brascamp-Lieb inequality, Concentration inequalities.

Abstract

This series of lectures provides an introduction to the Liouville equation

−∆u = e^2u in Ω ⊆ R^2

a central model in geometric analysis. We discuss its conformal invariance, explicit radial (bubble) solutions and the classification of finite-mass solutions. The variational structure and the role of the Moser–Trudinger inequality are outlined along with the lack of compactness leading to blow-up and energy quantization in multiples of 4π.

Abstract
My talks will focus on two central themes in harmonic analysis: the convergence of Fourier series and the Lp boundedness of maximal averaging operators.

In the first talk, I will discuss the fundamentals of Fourier series and several notions of convergence. I will then explain the Lebesgue differentiation theorem and its deep connection with maximal averaging operators, including the Hardy-Littlewood maximal function, the spherical maximal function, and maximal averages over compact smooth hypersurfaces.

In the second talk, I will discuss my research problems related to bilinear maximal functions associated with spheres and degenerate hypersurfaces. In particular, I will highlight how the geometric property of curvature plays a fundamental role in proving boundedness results for the corresponding linear and bilinear maximal averaging operators. I will also discuss the Lp and weighted Lp boundedness of bilinear Fourier multipliers, such as the bilinear Bochner-Riesz

Abstract

Solutions of certain partial differential equations (like the Laplace equation) enjoy certain properties called maximum/minimum principle. This property is used to prove many results in PDEs and geometry. We will give a brief introduction to some of these aspects.

Abstract

We begin by motivating the need for options as hedging instruments. We will then see that, no-arbitrage pricing of options in the binomial world is equivalent to risk-neutral pricing, followed by a brief discussion of real world vs risk neutral probabilities. We will then discuss how the continuous limit gives us the real world geometric brownian motion.

Abstract

Integral geometry lies at the intersection of geometry, probability, and analysis, and studies geometric quantities through averaging procedures over groups of transformations. Originating from classical questions of Buffon, Crofton, and Blaschke, the subject provides ele-gant probabilistic interpretations of geometric phenomena and powerful tools for understanding length, area, volume, and curvature.

We will discuss the role of valuations and kinematic formulas, which form the modern foundation of integral geometry, and briefly indicate connections to convex geometry, stochastic geometry, and applications in contemporary data analysis and imaging. The aim of the talk is to provide an accessible overview of the subject, highlighting both its classical origins and its continuing relevance in modern mathematics.

Abstract

Treatment of homogeneous media is classical and well-understood with respect to their approximation. In contrast, in-homogeneous media threw a challenge, especially in approximating them. The way out is homogenization. . Theory of Homogenization in PDE has been in development for the last 60 years or so. I plan to go through some highlights in the initial development of the theory without entering into too many details. I plan to illustrate these developments on a basic PDE model which, I hope, would be accessible even to beginners.

Abstract

The energy industry presents a unique frontier for applied mathematics, where abstract frameworks are required to bridge the gap between massive industrial data streams and complex physical structures. This presentation is structured in two parts: an overview of the mathematical landscape in the energy sector, followed by a detailed exploration of semi-supervised learning for material characterization.

The first part of the talk provides a high-level survey of how diverse mathematical tools address industrial challenges. We will briefly highlight applications ranging from Stochastic Programming in global logistics to the use of advanced polynomial approximations for diagnos-ing the degradation of catalysts during processes. These examples illustrate how mathematics serves as a foundational “industrial bridge” for enhancing operational reliability and accelerating materials discovery.

The second part of the talk focuses on the ‘Continuous Iterative Guided Spectral Class Rejection (CIGSCR)’ algorithm. Designed for the multiphase segmentation of 3D Micro-CT digital material images, CIGSCR addresses critical challenges such as grayscale overlap and radial imaging artifacts. We will discuss the mathematical foundation of this iterative splitting approach, demonstrate its ability to achieve high-fidelity segmentation in volumes exceeding two billion voxels with minimal human guidance, and the integration of spatial semantics. The presentation concludes with a discussion on future directions.