Speaker: Nick Gismondi
Title: Integrable Weak Solutions of Stationary Active Scalar Equations
Abstract: In this talk, I will discuss the construction of nontrivial integrable weak solutions for certain classes of stationary active scalar equations using a convex integration framework. We focus in particular on stationary active scalar equations with non-odd drift, as well as the stationary surface quasi-geostrophic equation. A central feature of the scheme is measuring the error in a negative-regularity homogeneous Sobolev norm, which enables the use of generic intermittent building blocks and allows us to consider arbitrary dissipation exponents. This talk is based on joint work with Alexandru Radu.
Speaker: Howen Chuah
Title: On a Nonlinear Model for Long Range Segregation
Abstract: We consider a system of fully nonlinear elliptic equations, depending on a small parameter, that models long-range segregation in population dynamics. The diffusion is governed by the negative nonlinear Pucci operator. We establish the existence of solutions and prove convergence, as the parameter goes to zero, to a free boundary problem. In the limit, high competition forces the species to segregate at a positive distance. Geometric properties of the free boundaries will be discussed, including directions for future research. This talk is based on joint work with Professors Monica Torres and Stefania Patrizi.
Speaker: Ben Doyle
Title: Weyl's Equidistribution Criterion and Discrepancy
Abstract: A classical result of Weyl relates the equidistribution of a sequence of real numbers modulo 1 to cancellation in the related exponential sums. Weyl's criterion is inherently qualitative. In this talk, we discuss various theorems which quantify this relationship. Namely, we introduce Koksma's inequality and the Erdos-Turan inequality, which provide essentially optimal bounds. Time permitting, we will compare the Erdos-Turan inequality to a competing bound of LeVeque, and show that the Erdos-Turan bound is always at least as strong as that of LeVeque.
No Talk
Speaker: Atal Bhargava
Title: The Circle Method and Additive Problems in Number Theory
Abstract: Some of the oldest unsolved problems in mathematics come from number theory. We will discuss some famous examples of problems in additive number theory, such as Waring's and Goldbach's problems. So far, the set of techniques that has been most successful in making progress on these has come to be known as the Hardy-Littlewood Circle Method. In this expository talk aimed at general audiences, we will describe the circle method and how it works, and show how analysis (particularly Fourier analysis in this case) turns out to be instrumental in tackling longstanding problems in number theory.
Speaker: General Ozochiawaeze
Title: PDEs and Inverse Problems
Abstract: Inverse problems for PDEs arise when one seeks to determine unknown properties of a medium from indirect measurements—like recovering a potential from scattering data or a conductivity from boundary voltages. In this talk, I will introduce the general framework of inverse problems for PDEs, with examples from scattering and EIT. The focus will be the three-dimensional Calderón problem: determining an unknown conductivity from boundary measurements. I will outline Calderón’s uniqueness result, the reduction to a Schrödinger-type equation, and sketch the construction of complex geometrical optics solutions that underpin the proof. Time permitting, I will also discuss stability aspects of the problem.
Speaker: General Ozochiawaeze
Title: PDEs and Inverse Problems, Part 2
Abstract: Inverse problems for PDEs arise when one seeks to determine unknown properties of a medium from indirect measurements—like recovering a potential from scattering data or a conductivity from boundary voltages. In this talk, I will introduce the general framework of inverse problems for PDEs, with examples from scattering and EIT. The focus will be the three-dimensional Calderón problem: determining an unknown conductivity from boundary measurements. I will outline Calderón’s uniqueness result, the reduction to a Schrödinger-type equation, and sketch the construction of complex geometrical optics solutions that underpin the proof. Time permitting, I will also discuss stability aspects of the problem.
Spring Break
Speaker: Paula Luna-Velasco, University of Sevilla - University of Grenada
Title: Global Dynamics of Two-Phase 2D Navier–Stokes Flows with Nonnegative Density
Abstract: The inhomogeneous Navier-Stokes equations governing two immiscible fluids present severe analytical challenges, particularly when there are regions with zero density. In this talk, I will present new global-in-time well-posedness results for this 2D free boundary problem in the nonnegative density setting. We establish the preservation of the C^{1+\gamma} interface regularity under sharp low-regularity assumptions on the initial velocity. Furthermore, we extend our analysis to density-dependent viscosities, proving global regularity for small viscosity contrasts.
Speaker: Martin Hsu
Title: Smoothing Inequalities for Triangular-Type Bilinear Averages
Abstract: We study the averaging operators associated with curved models of the triangular Hilbert transform. Our main result provides purely geometric criteria on curves that guarantee smoothing inequalities for the associated averaging operators. Connections to o-minimal theory are also explored. In particular, uniformity of the estimates is derived via tools from o-minimal theory. Such uniformity enables various applications in bilinear singular integral theory, maximal operator theory, and pattern finding. As one specific example, we prove the boundedness of the curved models of the triangular Hilbert transform for a wide class of curves. The talk is based on joint work in preparation with Fred Yu-Hsiang Lin.
No Talk
Speaker: Claudia Peña Vázquez de la Torre, BCAM Bilbao
Title: On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3
Abstract: The purpose of this talk is to present the article [1], in which we establish two results: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in R^3 with initial data (u0, b0) of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference u0 − b0. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field b under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.
References:
[1] Lucà, R. and Peña, C. (2025) On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in R^3 . Preprint: arXiv:2505.09340.
No Talk
Speaker: Mark Ma, Columbia University
Title: An Introduction to Obstacle and Alt-Caffarelli Free Boundary Problems.
Abstract: This is an introductory talk on Free Boundary Problems. In this talk, we'll introduce the basic setup and strategy to study the Obstacle and Alt-Caffarelli (One Phase Bernoulli) Free Boundary Problems. We'll start with a discussion on `solutions': basic properties of minimizers, the optimal regular, non-degeneracy. Then we look at the Free Boundary, which is the boundary of the positive set, with a focus on the Regularity of the Free Boundary and its Singular set. Along the way we introduce the Weiss Monotonicity formula and characterize the homogeneity of blowup limits. We shall also discuss potential classification of blowup profiles, and emphasize the difference between the two problems.