时间：2025年7月6日-16日
地点：西湖大学云谷校区E10-215
课程信息
讲师：Mimi Dai, University of Illinois Chicago
题目：Constructions of solutions to fluid equations
摘要：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.
讲师：Dennis Eriksson, Chalmers University of Technology
题目1：From Rational Solutions to Topology: Why is Fermat's Equation Hard to Solve?
摘要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.
题目2：From Polynomial Roots to Modular Forms: The Global Geometry of Cubic Curves
摘要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.
讲师：Daniel Faraco, Universidad Autónoma de Madrid
题目：Introduction to Differential Inclusions and Convex Integration by Geometry
摘要：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.
讲师：Hervé Pajot, Institut Fourier, Université Grenoble Alpes
题目：Around the Kakeya problem: an introduction to geometric measure theory and harmonic
摘要：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, ...).
讲师：Yu Yuan, University of Washington
题目：Rigidity for minimal/maximal surface and Monge-Ampere equations in two dimensions
摘要：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.
讲师：Jun Zhang, University of Michigan
题目：Geometrization of Science of Information
摘要：Information Geometry is the differential geometric study of the manifold of probability models, and promises to be a unifying geometric framework for investigating statistical inference, information theory, machine learning, etc. Central to such manifolds are “divergence functions” (in place of distance) for measuring proximity of two points, for instance Kullback-Leibler (KL) divergence, Bregman divergence, etc. Such divergence functions are known to induce a beautiful geometric structure of the set of parametric probability models. The Serial Lectures will introduce the basic ingradients of information geometry. Lecture 1 will use two motivating examples (the probability simplex over discrete sample space and the family of univariate normal distributions over continuous sample space) to introduce exponential representation, mixture representation, and their duality linked through Shannon entropy. The emphsis will be placed on parameter-indepenedent representation of probability mass (density) functions as points on a manifold, and pameter-independent definition of entropy and KL divergence as functions on the manifold. Lecture 2 will review convex analysis and the role of Legendre duality in constructing divergence, especially the KL divergence and Bregman divergence. Then local approximation of divergence functions will be shown to relate to the second- and third-order invarnaces of the manifold of parametric probability models. Lecture 3 will investigate the exponential family and the mixture family of probability models, and explore the full e/m-duality with underlying biorthogonal affine coordinates and Hessian metric (“dually flat” structures) on a manifold. Duality in e/m-projection reveals the essence of non-symmtric KL divergence. Lecture 4 will introduce recent progress in deformation to the dually-flat structure, especially the lambda-deformed model (variously named Box-Cox deformation in statistics, Tsallis deformation in theoretical physics, and Renyi deformation in information theory). Lambda-deformed exponential and lambda-deformed mixture families will be constructed, and their resulting geometry explained. The Serial Lectures are intended for an audience with little background in differentiable manifold, only assuming their solid background in multi-variable calculus.
讲师：Emmanuel Lecouturier, Westlake University
题目：An introduction to modular forms and some applications
摘要：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".
讲师：Hao Shen, University of Wisconsin-Madison
题目：An introduction to statistical physics
摘要：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.
讲师：Weijun Xu, Peking University
题目：Finding Lyapunov functions
摘要：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.
讲师：Stan Palasek, IAS/Princeton
题目：Well-posedness theory for the Navier-Stokes equations
摘要：The incompressible Navier-Stokes equations are a system of PDEs modeling the flow of a viscous fluid. The purpose of this course is to introduce the question of well-posedness of the Navier-Stokes equations. Given an initial vector field, does there exist a unique solution forward in time? We will see that the answer can be "yes" for certain classes of initial data. Then we will turn to a useful toy system known as a dyadic model. These models allow us to isolate much of the interesting behavior of the PDE and discover new phenomena, while avoiding many technical complications of the full system. This will lead us toward a counterexample showing that well-posedness can in fact fail for initial conditions with a "critical" amount of smoothness.
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