Seminars and Colloquia by Series

Heat semigroup approach to isoperimetric inequalities in metric measure spaces

Series
Stochastics Seminar
Time
Thursday, January 30, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Patricia Alonso-RuizTexas A&M University

The classical isoperimetric problem consists in finding among all sets with the same volume (measure) the one that minimizes the surface area (perimeter measure). In the Euclidean case, balls are known to solve this problem. To formulate the isoperimetric problem, or an isoperimetric inequality, in more general settings, requires in particular a good notion of perimeter measure.

The starting point of this talk will be a characterization of sets of finite perimeter original to Ledoux that involves the heat semigroup associated to a given stochastic process in the space. This approach put in connection isoperimetric problems and functions of bounded variation (BV) via heat semigroups, and we will extend these ideas to develop a natural definition of BV functions and sets of finite perimeter on metric measure spaces. In particular, we will obtain corresponding isoperimetric inequalies in this setting.

The main assumption on the underlying space will be a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in many examples of interest, also in fractals such as (infinite) Sierpinski gaskets and carpets. The results are part of joint work with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev.

Mixed-mode and relaxation oscillations in slow-fast predator-prey problems

Series
CDSNS Colloquium
Time
Monday, January 27, 2020 - 11:15 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Susmita SadhuGeorgia College & State University Milledgeville

The entry-exit function for the phenomenon of delay of stability loss (Pontryagin’s delay) arising in certain classes of slow-fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations. In the first part of my talk, I will present an elementary approach to study the entry-exit function for a general class of slow-fast systems, and apply this function to a broad class of slow-fast planar systems to obtain existence, global uniqueness and asymptotic orbital stability of periodic solutions that exhibit relaxation oscillations. The obtained results will then be applied to some predator-prey models. This research was conducted in collaboration with Dr. Shangbing Ai. In the second part of my talk, I will present a slow-fast system comprising of three species where further complex oscillatory patterns such as mixed mode oscillations (MMOs) are observed. MMOs are concatenations of small amplitude oscillations and large amplitude oscillations which are of relaxation types. In a neighborhood of singular Hopf bifurcation, these types of oscillations occur as long lasting chaotic transients as the system approaches a periodic attractor. The transients could persist for thousands of generations, reflecting that dynamics on an ecological timescale can be completely different than asymptotic dynamics. The goal of the talk is to find conditions that will determine whether a trajectory exhibits another cycle of MMO dynamics before reaching its asymptotic state.

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