2013 Stanisław Łojasiewicz Lecture

Preparatory Workshop

Kraków, 15 May 2013 (Wednesday)

Schedule

9.30-10.15

 Wojciech Słomczyński (UJ) From Monge-Kantorovich transportation problem to "classical" distance between quantum states

Abstract: The original Monge problem (1781) that emerged from studying the most efficient way of transporting soil from one place to another was formulated as follows: Split two equally large volumes into infinitely small particles and then associate them with each other so that the sum of products of these paths of the particles over the volume is least. Along which paths must the particles be transported and what is the smallest transportation cost? In the forties of the twentieth century Kantorovich proposed a relaxation of this problem allowing a piece of mass to be split between two or more target points. Both closely related problems have attracted a lot of attention in recent years finding a wide variety of applications in many areas of mathematics. We use the Monge-Kantorovich metric in the space of measures on the sphere induced by the value of the optimal transportation cost to introduce and compute the "classical" distances between quantum states.
 

10.30-11.15

 Paweł Strzelecki (University of Warsaw) A priori estimates for surfaces of prescribed mean curvature

Abstract: One of old papers of Neil Trudinger adresses gradient estimates for solutions of the equation of surfaces of prescribed mean curvature. We will recall this work and discuss some of its modern sequels, including the solution of Hildebrandt's conjecture (regularity of solutions under very weak assumptions on the mean curvature functions) and a few open problems.
 

11.15-11.45

 Coffee break

11.45-12.30

 Grzegorz Karch (University of Wrocław) Trudinger-Moser Inequality and chemotaxis equations

Abstract: The Trudinger-Moser inequality corresponds to the limit case of the Sobolev inequalities. In the talk, I will explain an important role of this inequality in the study of properties of solutions of a certain system of partial differential equations modelling chemotactic aggregation of cellular slime molds which move towards relatively high concentrations of a chemical secreted by the amoebae themselves.
 

12.30-13.30

 Lunch break

13.30-14.15

 Zbigniew Błocki (UJ) Hessian measures

Abstract: We will discuss the problem of defining real and complex Monge-Ampere operators, as well as more general Hessian operators, for non-smooth admissible functions. In case of real Hessian operators a highly non-trivial theorem of Trudinger and Wang (Ann. of Math. 150 (1999), 579-604) asserts that it is possible for an arbitrary admissible function. The methods are somewhat similar (although harder) to Bedford-Taylor's approach to the complex Monge-Ampere operator. One of surprising tools is the p-Laplacian.
 

14.30-15.15

 Sławomir Kołodziej (UJ) Isoperimetric inequalities for quermassintegrals and a priori estimates for Hessian equations

Abstract: N. Trudinger extended isoperimetric inequalities for quermassintegrals from the convex sets setting to certain families of non-convex sets. I shall describe their application (also due to Trudinger) to a priori estimates for Hessian equations.
 


All lectures will take place in Room 1016 of the Department of Mathematics and Computer Science of Jagiellonian University (New Campus, Kraków).

Please contact Zbigniew Błocki for further details.