Title: On the distance function in the presence of an obstacle
Speaker: Paolo Albano（University of Bologna）
Datetime: 2022-02-23 16:00 — 17:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：153 623 8696
Password：246118
Abstract
Some results on the distance function from a point-wise target in the presence of an obstacle are presented. We are mainly interested in the regularity and in the singularities of such a distance.
More precisely, assuming that the obstacle is an open bounded set (with suitably smooth boundary), we have that the distance is a semiconcave function in a generalized sense (with a "fractional" modulus of semiconcavity). We show, with an example, that this regularity result is the best one can hope for. Furthermore, in the Euclidean setting, we give some results on the "propagation" of singularities of the distance function.
Title: On Sarnak's Moebius Disjointness Conjecture
Speaker: 魏菲（清华大学丘成桐数学中心）
Datetime: 2022-03-04 14:00 — 15:00 Beijing/Shanghai
Venue: Tencent meeting APP
Meeting ID：159 944 662
Password：1859
Abstract
In this talk, I will first give a brief introduction on the background of Sarnak's Moebius Disjointness Conjecture. Secondly, I will talk some progress on this conjecture, specially on the Moebius Disjointness for rigid dynamical systems, which is closely related to the distribution of the average value of the Moebius function in short intervals. Thirdly, I will talk about some unsolved problems related to Sarnak's conjecture.
Title: Modelling moving contact lines on elastic membrane
Speaker: 任维清（National University of Singapore）
Datetime: 2022-05-06 16:00 — 17:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：132 839 0882
Password：293981
Abstract
We consider the system of two immiscible fluids on an elastic membrane, where the fluid interface intersects with the membrane at a contact line. We first study the static profiles of the interfaces by minimising the total energy of the system, which consists of the interfacial energies and the membrane bending energy. Asymptotic solutions are obtained in the limits as the bending modulus tends to infinity (stiff limit) and zero (soft limit), respectively. Then we consider the dynamical problem and derive the hydrodynamic model, particularly the boundary conditions, from generalised thermodynamics. Numerical solutions are presented for the relaxation of droplets on a membrane and transport of droplets by bendotaxis.
Title: Reducible Operators in von Neumann Algebras
Speaker: 沈隽皓（University of New Hampshire）
Datetime: 2022-05-10 10:00 — 11:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：137 406 2542
Password：438315
Abstract
Let \(H\) be a complex separable Hilbert space and \(B(H)\) the set of all bounded linear operators on H. An operator A in \(B(H)\) is reducible if there exists a nontrivial projection \(P\) in \(B(H)\) such that \(TP=PT\). In 1968, P. Halmos asked whether every operator in \(B(H)\) is a norm-limit of reducible operators. This question was answered affirmatively by D. Voiculescu in 1976 by his noncommutative Weyl-von Neumann theorem. A self-adjoint subalgebra of \(B(H)\) that is closed in the weak operator topology is called a von Neumann algebra. Von Neumann algebras are further classified by Murray and von Neumann into type I, type II, and type III. By definition, \(B(H)\) is a von Neumann algebra with type I. In the talk, we will introduce a concept of reducible operators in a von Neumann algebra and discuss the question whether each operator in a von Neumann algebra of type II is a norm-limit of reducible operators.
Title: An Eikonal equation with vanishing Lagrangian arising in Global Optimization
Speaker: Martino Bardi（University of Padova）
Datetime: 2022-05-16 16:00 — 17:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：154 880 5188
Password：159292
Abstract
We show a connection between global unconstrained optimization of a continuous function \(f\) and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution v of the critical Hamilton-Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then \(v\) is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of \(v\). Such trajectories are proved to converge to the set of minima of \(f\), using tools in control theory and occupational measures. This is joint work with Hicham Kouhkouh.
Title: On transport problems related to the Hamilton-Jacobi equation
Speaker: Konstantin Khanin（University of Toronto）
Datetime: 2022-05-18 16:00 — 17:00 Beijing/Shanghai
Venue: Umeet APP
Meeting ID：189 726 1319
Password：193992
Abstract
In this talk I'll discuss several problems related to transport maps for the Hamilton-Jacobi equation. In the first part of the talk I'll discuss the case of non-quadratic Hamiltonians. In the second part I'll consider the transport problems arising in the setting of the random forced Burgers equation in the 1D case. This case is strongly related to the KPZ problem.