Past Talks 歷史演講

We give a brief introduction of the development of the particle algorithms and the dispersion-relation-preserving algorithms for a class of Helmholtz regularization equations. In particular, we focus on the Camassa-Holm equation and the regularized Burgers equation. We will discuss the extensions of the one-dimensional algorithms to solve the Euler-Poincare equations and the regularization-based subgrid-scale turbulence models, which are the higher dimensional extensions of the Camassa-Holm equation.

In this talk I will explain the convexity properties of moment maps for contact manifolds. I will also discuss some examples. This is a joint work with Y. Karshon.

In this talk, we consider a model of a flowing water habitat with a hydraulic storage zone in which no flow occurs. In this habitat, one or two microbial populations grow while consuming a single nutrient resource. Our model is a system of partial differential equations coupled with ordinary differential equations . Conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems, the theory of bifurcation and the theory of monotone dynamical systems. A single population persists if it can invade the trivial steady state of an empty habitat. Under some conditions, persistence occurs in the presence of a hydraulic storage zone when it would not in an otherwise equivalent flowing habitat without such a zone. Coexistence of two competing species occurs if each can invade the semi-trivial steady state established by the other species. We also answer an interesting question: Does a system with storage zones facilitate persistence of planktonic algae in flowing habitats? Numerical work shows that both coexistence and enhanced persistence due to a storage zone occur for biologically reasonable parameters.

For mathematician, probability theory is often considered as a chapter of measure theory. This is in part true, since Kolmogorov in 1933 first laid the foundation for the probability theory based on the measure theory. The probability space is a mathematical framework to describe an experiment of random events. Kolmogorov Extension Theorem gives the existence of probability space for a random experiment which is based on the Caratheodary Extension Theorem in measure theory. Many ideas from measure theory are also used in the analysis in probability theory. However, there is an important concept in probability theory, the concept of independence, that can not be found in general measure theory. This is also the place where the development of probability theory diverges greately from that of measure theory. The filtration, a family of σ-algebras, is also frequently introduced when we want to discuss random events which are not independent. The use of these new concepts can be best seen in the advanced study of stochastic processes. The modern probability theory finds a very wide range of applications, in practical problems and also in the studies of mathematics. In this talk, I hope to mention several examples that some ideas in probability theory give answers to some problems from other branch of mathematics. I will also mention several interesting mathematical problems motivated by the studies in probability theory.