杰出学者讲座

In this talk, we introduce our recent result on the finite time blow-up for the supercritical defocusing nonlinear wave equation (NLW) in  

The proof of this result is based on a surprising connection between complex-valued NLW and relativistic Euler equations, and the construction of self-similar imploding solutions of the relativistic Euler equations.

在众多数学分支中，数学对象的分类问题往往具有核心的地位。直观上，有些分类问题比较简单，有些则非常复杂。在这个报告中我们通过一些实例来介绍一个可以严格探讨数学中分类问题的相对复杂性的数学理论，即等价关系的描述集合论。我们将展示如何将这一理论应用到数学分类问题的研究中去。

Herwig and Lascar, in their celebrated paper of 1992, proved the EPPA for some general classes of structures by showing that finitely generated free groups have certain combinatorial property. We call this property the Herwig-Lascar property (HL-property for short) and explore further results. For instance, we show that the HL-property is closed under free products. We also establish explicit equivalences between the HL-property of groups and extensions of structures on which the group acts. As an application, we prove that the automorphism group of any Henson graph contains a dense locally finite subgroup that is isomorphic to Hall's group. This is joint work with Mahmood Etedadialiabadi, François Le Maître, and Julien Melleray.

猜想：每个可数无穷图都有一个不友好顶点二剖分，即此二剖分使每个顶点在自己所属的类里的邻居数不超过在另一个类里的邻居数。

这是无穷图论中有35年历史的最著名猜想之一，见R. Diestel [2016. Graph theory (5th edition). Springer.]第8.1节。它易于陈述，能被稍有数学基础的人理解，但又极具挑战性。从Open Problem Garden (http://www.openproblemgarden.org/)可以得知，它的重要性为3星；概率论中的 KPZ Universality Conjecture、数论中的The 3n+1 Conjecture (The 3x+1 Problem)的重要性也是3星；此处重要性的最高等级为4星，黎曼猜想是4星。 

“20世纪数学的一个了不起的发展是认识到有时候可以用概率方法来证明一些看起来并没有概率本性的数学命题。” 见N. Alon（Wolf数学奖得主）和M. Krivelevich（2008. Extremal and probabilistic combinatorics, in The Princeton Companion to Mathematics）。 

此未决猜想能否被概率方法攻克？我们的答案是肯定的。此报告基于多年的探索[Xiang Kainan, (2019-2024), Every countable infinite graph admits an unfriendly partition, Preprint.]， 其中位置渗流、概率测度的弱收敛理论及一个新颖的拓扑与度量构造起着关键的作用。

Distributions of Gaussian process extremes (supremum, infimum and absolute supremum) is a subject long studied in probability theory, yet its relevance to statistical inference is inadequately understood. I will illustrate with examples of simultaneous confidence regions (SCRs), and present some recent results on exact quantiles of Gaussian process extremes which provide the indispensable theoretical underpinning of the SCR methodology.

In a financial system, firms hold debts both within and outside the system. The reimbursement abilities are intertwined, thereby potentially generating coordination failures and a cascade of defaults. To avoid such failures and account for the links in the system, an orderly resolution liquidates all debts simultaneously and assigns the amount each firm receives and reimburses both inside and outside the system.  I present properties that determine which firms are left bankrupt and how much the non-bankrupt ones reimburse and receive.  The main properties are the priority of external creditors at the system level and the proportionality principle.

We develop a general model for independent random matching of a large population in a continuous time dynamical system. We work with a general (and possibly infinite) type space for the agents and construct a continuum of independent continuous-time Markov processes that is derived from random mutation, random matching with enduring partnership, random type changing and random break up. lt follows from the exact law of large numbers that the deterministic evolution of the agents' realized type distribution for such a continuous-time dynamical system can be determined by a system of measure-valued differential equations. The results provide the first mathematical foundation for a large literature on continuous-time search-based models of labor markets, money, and over-the-counter markets for financial products.

In this talk we extend the fundamental gap comparison theorem of Andrews and Clutterbuck to the infinite dimensional setting. More precisely, we proved that the fundamental gap of the Schrödinger operator   (  is Ornstein-Uhlenbeck operator) on the abstract Wiener space is greater than that of the one dimensional operator   , provided that   is a modulus of convexity for  . Similar result is established for the diffusion operator Furthermore, we give the probabilistic proofs of fundamental gap conjecture and spectral gap comparison theorem of Andrews and Clutterbuck in finite dimensional case via the coupling by reflection of the diffusion processes.

I will first review the recent derivation of the backbone exponent for 2D percolation via Liouville quantum gravity, based on the joint work with Nolin, Qian, and Zhuang. Then I will present a few further results obtained in this fashion, based on joint works with Ang, Liu, Xu, Yu, Zhang. It remains a challenge to understand these results using a more direct method without quantum gravity.

The classical Hilbert transform has a natural analogue on the nonabelian free groups by decomposing the free group into disjoint subsets according to the first letter of the reduced words. Mei and Ricard prove that such a transform (decomposition) is unconditional with respect to the noncommutative Lp norm associated with the free group von Neumann algebras for all 1<p<\infty. I plan to talk about a possible extension of the Lp unconditionality of such “transforms” to the general hyperbolic groups.

In this talk we consider  nonlinear Klein-Gordon equations with potential. We prove that spatially localized and time-periodic bound states of the linear problem may be destroyed by generic nonlinear Hamiltonian perturbations, via energy transfers from the discrete to continuum modes and slow radiation of energy to infinity. We explore the underlying mechanism  (generalized Fermi's Golden Rule) of such phenomenon and give  descriptions on the transfer rate in the full generality: small or large and single or multiple eigenvalues, high dimensional eigenspaces. This settles a long-standing problem raised in the paper of Soffer-Weinstein 1999, in which single and large eigenvalue case was first treated. This is a jiont work with my students Jie Liu and Zhaojie Yang.

Sharp convergence rates in Wasserstein distance are derived for empirical measures of diffusion processes on Riemannian manifolds, and the renormalization limits are explicitly formulated by using eigenvalues and eigenfunctions of the associated elliptic operator. For explosive diffusion processes, the convergence is described by conditional expectations.  

We consider the action of finitely truncated singular integral operators on functions taking values in a Banach space. Such operators are bounded for any Banach space, but we show a quantitative improvement over the trivial bound in any space renormalizable with uniformly convex norm, which is equivalent to probabilistic estimates known as martingale type and cotype. The proof, which is based on the representation of a singular integral as an average of dyadic model operators over a random choice of the dyadic decomposition of the domain, follows the broad outline of recent works on similar results for genuinely singular (non-truncated) operators in the narrower class of UMD (unconditional martingale differences) spaces, but our setting, the main theorem, and some aspects of its proof, are new. This is based on arXiv:2310.08926.

The hydrodynamics stability has been a main theme in the fluid mechanics since Reynolds's famous experiment in 1883. This field is mainly concerned with the transition of fluid motion from laminar to turbulent flow. On one hand, the eigenvalue analysis showed that the plane Couette flow and pipe Poiseuille flow are linearly stable for any Reynolds number. On the other hand, the experiments showed that these flows could be unstable and transit to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradoxes. The resolution of these paradoxes is a long-standing problem in the fluid mechanics. In this talk, we introduce some recent progress towards understanding the transition mechanism by two means: pseudospectra and transition threshold problem.

所谓域上的平凡绝对值，是指将域中任何非零元素映为1，将零元映为0 的映射。这样的度量体现的是域的离散拓扑，在数论中往往被排除在研究的兴趣之外。本报告中我将以代数曲线的情形为例介绍与森脇淳的一个工作，展现平凡度量下域上代数簇丰富的算术性质。

What happens to an $L_p$ function when one truncates its Fourier transform to a domain? This is in the root of foundational problems in harmonic analysis. Fefferman’s ball multiplier theorem (1971) shows that $L_p$-preserving Fourier truncations are associated to domains with flat boundary. What if we truncate on a curved space like a Lie group? And if we truncate the entries of a given matrix? How does it affect its singular numbers? These apparently unrelated problems are interestingly connected. Our talk will be based on a joint work with M. de la Salle and E. Tablate. 

 In this talk I will present my recent joint work with Professor Xing-Tang Dong of Tianjin University. We introduce and study a two-parameter family of integral operators on the Fock space.We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group SL(2,R) on the Fock space. 

Permutation patterns is a popular area of research introduced in 1968, but with roots going to the work of Leonhard Euler in 1749. 

In this talk, I will present a brand-new notion of a singleton mesh pattern (SMP), which is a multidimensional mesh pattern of length 1. It turns out that avoidance of this pattern in arbitrary large multi-dimensional permutations can be characterised using an invariant of a pattern called its rank. This allows to determine avoidability for an SMP P efficiently, even though determining rank of P is an NP-complete problem. Moreover, using the notion of a minus-antipodal pattern, one can characterise SMPs which occur at most once in any d-dimensional permutation.

I will also discuss several enumerative results regarding the distributions of certain general projective, plus-antipodal, minus-antipodal and hyperplane SMPs. 

This is joint work with Sergey Avgustinovich, Jeffrey Liese, Vladimir Potapov and Anna Taranenko.

统计学“始于度量、兴于相关”。度量相关性和检验独立性是统计学领域基本问题，预测和推断是统计学和机器学习领域核心问题。本报告将会回顾统计学领域线性相关和非线性相关各类度量准则，以及独立性检验的最新进展。

In this talk, I will first review some classical results of scattering theory for Schrodinger operator ,including limiting absorption principle, generalized eigenfunction expansions and the existence of wave operators in the context of . These concepts/operators also have similar  counterparts or  estimates related with harmonic analysis, including uniform Sobolev estimates of resolvent,  bounds of wave operators and so on. Finally, we also address some related -works associated with higher order Schrodinger operators . These are adjoint-works with Haruya Mizutani and Zijun Wan.

Local unitary equivalence of quantum states is a basic theoretical problem in quantum information. Lots of efforts have been made on finding complete invariants for pure and fixed states. So far only bipartite mixed qubits are known by the Macklin's 18 invariants. We report on some recent progress using hypermatrix algebras and hyperdeterminants on 3-qubits. Joint work with Dobes.

Given a closed hyperbolic surface, a systole is a closed geodesic of shortest length. A generic surface of genus  has a unique systole which cuts it into one or two noncontractible pieces. However, there are closed hyperbolic surfaces of genus , whose set of systoles fills the surface, i.e. it cuts the surface into polygons. In general, it is easy to find surfaces with a large number of systoles which fill. A classical example is the Bolsa surface of genus 2 with 12 systoles. Here we are interested to find surfaces for which the set of systoles is filling while being small. In fact, our approach provides surfaces with significatively less systoles than the previously known examples. Then we will briefly outline a consequence in Teichmüller theory. Our construction is based on the theory of Coxeter groups, especially the arithmetic properties of the Tits representation. From a geometric viewpoint, it involves hyperbolic tessellations. Although this work mixes hyperbolic geometry and representation theory, our presentation will be quite elementary.

  We try to give geometric interpretations on Navier-Stokes equations

I shall discuss some new recent striking results that appear in the analysis of problems with unbalanced growth. In the isotropic case, I shall point out an interesting discontinuity property of the spectrum. In the second part my talk, I will discuss the anisotropic case and I shall discuss a new case corresponding to equations with mixed regime. Several new research directions will be highlighted in the final part of this talk.

Let  be a bounded and regular domain. In this talk we will discuss some uniqueness results of positive solutions of the following nonlinear elliptic equation

via a kind of convexity of the associated functional

where . Some related results for the -Laplacian equation and other equations will also be discussed. 

本报告介绍近期我们用拓扑斯理论(Topos theory)研究人工智能的工作。依据图灵测试对智能给出的科学定义, 我们将人工智能系统定义为由拓扑斯理论所描述的物理系统(Isham等人在2008年建立的物理理论), 它们具有自身的高阶形式语言及逻辑推理系统。特别是,我们用Hilbert空间上算子理论构造相应的拓扑斯以给出量子人工智能系统的数学描述。我们还将介绍量子计算的拓扑斯理论模型。

In the talk, I wish to stress the link between branched transportation theory, and some issues in the study of Sobolev maps between manifold. In particular, I will present a counterexample to the sequential weak density of smooth maps between two manifolds 𝑀 and 𝑁 in the Sobolev space 𝑊1,𝑝 (𝑀, 𝑁), in the case 𝑝 is an integer. It has been shown quite a while ago that, if 𝑝 < 𝑚 = 𝑑𝑖𝑚(𝑀) is not an integer and the [𝑝]-th homotopy group 𝜋[𝑝] (𝑁) of 𝑁 is not trivial, [𝑝] denoting the largest integer less than 𝑝, then smooth maps are not sequentially weakly dense in 𝑊1,𝑝 (𝑀, 𝑁). On the other hand, in the case 𝑝 < 𝑚 is an integer, examples of specific manifolds 𝑀 and 𝑁 have been provided where smooth maps are sequentially weakly dense in 𝑊1,𝑝 (𝑀, 𝑁)with 𝜋[𝑝] (𝑁) ≠ 0, although they are not dense for the strong convergence. This is the case for instance for 𝑀 = 𝐵 𝑚. Such a property does not hold for arbitrary manifolds 𝑁 and integers 𝑝.

The counterexample deals with the case 𝑝=3, 𝑚 ≥ 4 and 𝑁 = 𝑆 2 , for which 𝜋3 (𝑆 2 ) = 𝑍 is related to the Hopf fibration. We provide an explicit map which is not weakly approximable in 𝑊1,3 (𝑀, 𝑆 2 ), by smooth. One of the central ingredients in our argument is related to issues in branched transportation and irrigation theory in the critical exponent case.

“... But the most general and direct method for resolving questions of probability consists of making them depend on difference equations ...” 

The aim of this talk is to illustrate this quote from Laplace (Philosophical essay on probabilities) by developing certain aspects of the theory of discrete potential theory attached to a random walk in 𝑍 𝑑 ; the difference equations playing in this framework the same role played by the PDEs in the theory of the classical potential theory. Although the presentation will be mainly limited to simple walks in quadrants, some extensions to walks in discrete Lipschitzian domains and to inhomogeneous walks will be discussed. The emphasis will be placed on the role that tools from discrete potential theory can play (harmonic functions and discrete caloric functions, maximum principle, Harnack inequalities, boundary Harnack inequalities at the boundary) in establishing optimal estimates for the number of paths confined to a region as well as the number of excursions.

Definitions form a fundamental part of the mathematical study. I will discuss the requirements that mathematicians put on `good' definitions and explain how such definitions develop, based on the notion of a locally compact quantum group, originating in 1970s and 1980s, and reaching a (possibly?) final form in the work of Kustermans and Vaes in 2000. Later developments will also be mentioned, but in general the talk will be accessible to general mathematical audience.

About the speaker: Adam Skalski is a full professor and the scientific director of the Institute of Mathematics of of the Polish Academy of Sciences, and the chairman of ERCOM (European Research Centers on Mathematics). His areas of interest include topological quantum groups, operator algebras and quantum stochastic processes. He has been awarded the Kuratowski Prize of the Polish Mathematical Society in 2008 and the Sierpinski Prize of the Polish Academy of Sciences in 2014. He is one of the executive editors of Studia Mathematica and Banach Center Publications.