杰出学者讲座

  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.