Old Friends & New Trends

Mapping Theorems: From Riemann to the Present Uniformization, Rigidity, and the Geometry of Metric Surfaces

The Riemann Mapping Theorem and the Uniformization Theorem are among the foundational results of complex analysis, providing canonical conformal models for planar domains and Riemann surfaces. This mini-course surveys the development of mapping theorems from the classical work of Riemann, Koebe, Hilbert, Grötzsch, and de Possel to recent advances in geometric function theory and the geometry of metric surfaces.

We begin with the Riemann Mapping Theorem and its extensions to multiply connected domains by Koebe, Hilbert, Grötzsch, and others. We discuss Koebe's circle-domain theorem, Hilbert's slit-domain theorem, and the role of extremal methods in classical uniformization theory. Particular attention will be given to the evolution of these ideas into Schramm's theory of transboundary modulus, which has become one of the principal tools in modern uniformization theory. As an application, we present the celebrated He–Schramm theorem and discuss the still-open Koebe Circle Domain Conjecture from 1908.

The second lecture focuses on rigidity and removability phenomena for circle domains and related fractal sets. We discuss transboundary modulus as a unifying tool underlying several recent developments, including new proofs of the He–Schramm theorem, uniformization results for domains bounded by quasitripods, and counterexamples to rigidity conjectures. We also survey recent advances in the geometry of metric surfaces, including quasisymmetric rigidity, metric analogues of Koebe-type uniformization theorems, and conformal mappings with a dense set of conformal blowups.

Throughout the course we highlight the interplay between uniformization, rigidity, and modulus methods, as well as several open problems that continue to shape the field.

Topics include:

• The Riemann Mapping Theorem
• Uniformization of Riemann surfaces
• Koebe's circle-domain program
• Hilbert's slit-domain theorem
• Grötzsch, de Possel, and extremal methods
• Transboundary modulus
• The He–Schramm theorem
• The Koebe Circle Domain Conjecture
• Rigidity and removability
• Domains bounded by quasitripods
• Geometry of metric surfaces
• Quasisymmetric Koebe uniformization
• Conformal mappings with dense conformal blowups
• Open problems

The Cauchy–Poisson problem

The aim of this course is to give the proof of a short time existence of 2D fluid motion in which the displacement and velocity of the surface, called free boundary, are specified at some instant, say t = 0.

The fluid occupies the infinite region in the xy plane below the free boundary, which is assumed to be a graph over the x axis. At initial time the free boundary and its velocity are given and one seeks the subsequent motion. In the classical setting the pressure over the free boundary is constant, say zero. The fluid is subject to gravitational force but the surface tension is neglected.

The problem of the 3D waves produced by local disturbances of the surface was investigated in two classical memoirs by Cauchy (1827) and Poisson (1815). This is a foundational fluid dynamics problem that describes the evolution of surface gravity waves generated by initial (localized) perturbations or a surface pressure impulse.

The plan of the lectures is as follows:

1. Derivation of the main equations
2. An introduction to the pseudo-differential operators
3. Linearization of the Cauchy–Poisson system
4. The proof of short time existence for the linearized hyperbolic system of nonlocal PDEs

The methods discussed in this course have broad applications, and if time permits I will give another application to Prandtl's problem of minimal drag for incompressible subsonic flow.

A Mean-Field Game model for large-scale attrition in attacker–defender systems

In this talk I present a novel Mean-Field Game (MFG) framework for large-scale attacker–defender systems aimed at protecting one or multiple High-Value Units (HVUs). Motivated by classical agent-wise attrition models, we introduce a population-wise attrition mechanism formulated by statistical distance between populations, enabling a macroscopic description of weapon-based interactions between large populations. Leveraging this and Lions derivative on the space of probability measures, we derive the associated MFG system, which characterizes optimal strategies and the evolution of population distributions in attacker–defender interactions. For numerical investigation, we develop a numerical scheme combining physics-informed neural networks with the Sinkhorn algorithm to solve our attacker–defender MFG system.

Delayed collective dynamics in Nomadic Mean-Field Games

In this work, we investigate the emergent collective dynamics of nomadic populations subject to environmental memory lag. We formulate the problem as a Mean Field Game (MFG) in R^d, where agents modulate their velocity to either align with or avoid a delayed population average, representing distinct ecological strategies such as harvesting or depletion avoidance. To address the infinite-dimensional nature of the delayed coupling, we introduce a first-order Taylor approximation in the mean delay that reduces the system to a tractable set of local differential equations.

By applying the Stochastic Maximum Principle, we derive explicit analytical laws for the evolution of the population moments. We establish that the interaction type fundamentally alters the collective kinematics: in the finite-horizon equilibrium (specifically for vanishing terminal costs), harvesting couplings induce an effective inertial drag that causes the population mean to lag seasonal drifts, while repulsive depletion couplings can generate propulsive overshoot. Furthermore, we prove that the population variance satisfies a third-order linear ODE, which predicts a synchronizing regime for harvesting interactions and the emergence of oscillatory “breathing modes” in depletion regimes. Finally, we provide numerical simulations of the stochastic particle system in ℝ². These experiments corroborate our analytical findings, confirming the existence of memory-induced drag and rhythmic spatial dispersal.

A missing Euler–Lagrange condition for the Mumford–Shah functional

Some old results about the asymptotics of the Mumford–Shah minimizers near the crack-tip, as well as the curvature, will be presented, which leads to a new Euler–Lagrange condition.

We also develop a new numerical method to compute the minimizers in 2D, which is very accurate near the crack-tip.

In 3D we construct a continuous family of stationary solutions which are not global minimizers. We make a conjecture that the dependence of the function on the point running over the crack-front is related to the behavior of the co-normal tangent at the crack-front (joint work with John Andersson, Antoine Lemenant and Zhilin Li).

Some questions in non-linear Fourier analysis

Nonlinear Fourier analysis is a field at the intersection of harmonic analysis, integrable PDEs, orthogonal polynomials and quantum computing among others. We will present some open problems and recent advances emphasizing, for the most part, the parallel with the linear Fourier analysis.

Controllability as a key tool for studying long-time behaviour of random dynamical systems

This talk is devoted to the study of the long-time behaviour of random dynamical systems under highly degenerate non-Gaussian forcing. I will review recent results showing how the controllability properties of the underlying deterministic dynamics can be used to handle degeneracy and to establish ergodic and chaotic behaviours.

Singular energies and free boundaries

In this talk I will discuss the Alt–Phillips free boundary problem, focusing on the classical Laplacian case. The problem arises from the minimization of an energy functional with a singular lower-order term, leading to a rich interaction between variational methods, elliptic regularity, and geometric properties of free boundaries.

The talk will focus on the main qualitative features of minimizers: optimal regularity, non-degeneracy, growth estimates, density properties, and the structure of the free boundary. The goal is to give an accessible introduction to the Alt–Phillips model while emphasizing the ideas that make it a central prototype in modern free boundary theory.

On irregularity of free boundaries

In this talk, we present a novel technique to obtain estimates from below for quantities related to the free boundary. In particular cases, these prove the irregularity of the free boundary. The technique presented has room for improvement and hence new future results.