Time：July 6-16, 2025
Venue：E10-215
Speaker：Mimi Dai, University of Illinois Chicago
Title：Constructions of solutions to fluid equations
Abstract：For nonlinear partial differential equations, one does not expect to find solutions explicitly in general. We will discuss several constructions to obtain weak solutions for fluid equations, including Euler equation, surface quasi-geostrophic equation, magnetohydrodynamics systems, etc. Such solutions are usually referred as wild solutions since they violate physics laws.
Speaker：Dennis Eriksson, Chalmers University of Technology
Title1：From Rational Solutions to Topology: Why is Fermat's Equation Hard to Solve?
Abstract 1：Fermat's Last Theorem, as proven by Andrew Wiles, claims there are no non-trivial integer solutions to x^n +y^n = z^n when n > 2. But why is this equation so hard? Starting with Pythagorean triples (for n=2 case), we explore rational parametrizations and why they fail for higher-degree Fermat curves. Using the implicit function theorem, we show that local holomorphic parametrizations always exist. These local structures hint at the curves' global topology. Through topological arguments, we identify the genus, an integer invariant, as a key obstruction: a positive genus implies no rational parametrization. We then show how this reveals deeper reasons behind Fermat's complexity.
Title2：From Polynomial Roots to Modular Forms: The Global Geometry of Cubic Curves
Abstract 2：Classical discriminants detect multiple roots in polynomials. Starting from quadratics, we trace this idea through various examples. We focus in particular on cubic curves—elliptic curves—whose discriminants control their geometry. Although such curves lack rational parametrizations, they admit global ones via the complex plane, expressed through Weierstrass equations depending on a point in the upper half-plane. This reframes the discriminant as a function on that space—and remarkably, it turns out to be a very special such function - namely a modular form. This reveals a striking link between polynomial algebra, complex analysis, and the arithmetic of elliptic curves.
Speaker：Daniel Faraco, Universidad Autónoma de Madrid
Title：Introduction to Differential Inclusions and Convex Integration by Geometry
Abstract：This course offers an accessible introduction to the theory of differential inclusions and convex integration, emphasizing their geometric origins and analytical applications. These methods, stemming from Nash's isometric immersion theorem in the 1950s, have evolved into powerful tools for constructing highly irregular and non‑unique solutions to PDEs, challenging classical expectations of regularity and uniqueness.
We will explore how many PDEs can be reformulated as differential relations, and how the geometry of matrix spaces—particularly rank constraints and laminates—governs the emergence of singular solutions. Applications include the vectorial calculus of variations, elliptic systems, non-uniqueness in fluid dynamics, and energy relaxation in unstable flows and magnetohydrodynamics.
The course emphasizes intuition and key constructions—Young measures, staircase laminates, and convex‐hull methods—over full technical proofs. Target audience: advanced undergraduate and graduate students with foundational knowledge in analysis and PDEs.
Speaker：Hervé Pajot, Institut Fourier, Université Grenoble Alpes
Title：Around the Kakeya problem: an introduction to geometric measure theory and harmonic
Abstract：In 1920, Besicovitch proved that there exist compact sets in the n-Euclidean space (n>1) which contains a line in every direction and with zero measure in the sense of Lebesque (Besicovitch or Kakeya sets). This gives an answer to the initial question of Kakeya (1917): What is the minimum area required to turn a needle of length 1 through 180 degrees?
A natural question is "How small can be a Besicovich set ?" This leads to the famous Kakeya conjecture: Every set in the n-Euclidean space containing a line in every direction has Hausdorff dimension n.
The case n=2 was solved by Davis in 1971. A solution in the case n=3 was announced very recently by Wang and Zahl (2025).
In these lectures, we will consider the following questions:
- How to solve the initial Kakeya problem?
- How to construct a Besicovich set?
- How to solve the Kakeya conjecture in dimension 2?
- What are the connections between the Kakeya conjecture and some well known (and open!) problems in harmonic analysis (Restriction problem, Bochner-Riesz problem, ...)?
For this, we will introduce basic tools in geometric measure theory (Hausdorff measures and dimension, rectifiability theory, ...) and harmonic analysis (Schwartz spaces, Fourier transform, ...).
Speaker：Yu Yuan, University of Washington
Title：Rigidity for minimal/maximal surface and Monge-Ampere equations in two dimensions
Abstract：We present Bernstein/Calabi, Jorgens theorems, which states that all entire solutions
to the minimal/maximal surface and Monge-Ampere equation in two dimensions are linear/linear, quadratic polynomials respectively. Recall the Liouville theorem: every entire harmonic function with polynomial growth is a polynomial. In the course of proving these results, we go over the necessary background and indicate further developments.
Speaker：Jun Zhang, University of Michigan
Title：Geometrization of Science of Information
Abstract：TBA
Speaker：Emmanuel Lecouturier, Westlake University
Title：An introduction to modular forms and some applications
Abstract：In this series of lectures, we will give a brief introduction to the theory of modular forms in the simple setting of level 1 and weight k. We shall see some classical examples of modular forms such as Eisenstein series, the Ramanujan Delta function and theta functions. As an application, we shall give results on the representation of integers by integral quadratic forms. Our reference is Serre's book "A course in arithmetic".
Speaker：Hao Shen, University of Wisconsin-Madison
Title：An introduction to statistical physics
Abstract：This is a lecture series on the basics of statistical physics. We will mostly focus on the Ising model, including topics such as thermodynamic limit, correlation inequalities, phase transitions, etc. If time permitted, I will also discuss some results on models with continuous symmetry.
Speaker：Weijun Xu, Peking University
Title：Finding Lyapunov functions
Abstract：Existence of invariant measures for Markov processes are often established by finding suitable Lyapunov functions. In practice, constructing such a function is often more of an art than science. We introduce some techniques in finding Lyapunov functions, and demonstrate their usage in one particularly interesting example.
Speaker：Stan Palasek, IAS/Princeton
Title：TBA
Abstract：TBA
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