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Invited speakers include:
Clifford Lectures 2017
Title
Thursday, Oct 5
Location: Building Room 7:00am-4:30pm
9:00-10:00 Name "Title"
10:00-noon Name "Title"
Lecture I:
2:00-3:00 Name "Title"
3:30-4:30 Name "Title"
5:00-7:00 Reception Building
Friday, October 6
Lecture II
Location: Building 9:00am-6:00pm
9:00-10:00 Name "Title"
10:30-11:30 Name "Title"
2:30-3:30 Name "Title"
4:00-5:00 Name "Title"
Saturday, October 7
Lecture III
Location: Building Room 9:00am-12:00pm
9:00-10:00 Name "Title"
10:30-11:30 Name "Title"
Location: Building 12:00pm-6:00pm
1:30-2:30 Name "Title"
2:30-3:30 Name "Title"
4:00-5:00 Name "Title"
I will review the mathematical achievements of Alex Wentzell: Boundary conditions for diffusion processes and PDEs, Large deviations and long-time influence of random perturbations of dynamical systems, Eigenvalues of stochastic matrices with exponentially small transition probabilities and related elliptic operators, Asymptotic expansions in limit theorems for stochastic processes, Modified averaging principle for stochastic perturbations, and Various problems for PDEs with a small parameter will be considered.
I will present a general overview of several engineered and natural systems with uncertain mathematical models, the multidisciplinary methods required for their analysis, and relevant results. The collection of new mathematical techniques that I will describe lies at the confluence of three important areas: dynamical systems; control and estimation (data assimilation); and information theory. The first part of the talk focuses on the challenges in data assimilation that arise from the interactions between uncertainties, nonlinearities, and observations. I will present rigorous reduced-order data assimilation techniques for high dimensional multi-scale problems. In particular, I will outline how scaling interacts with filtering via stochastic averaging. Optimal sensor placement based on information theoretic concepts will also be discussed. The second part of the talk brings together three interesting themes in dynamical systems — resonances, domains of attraction, and large deviations. The subtleties of their interactions are explored in a canonical way by combining the ideas from dynamical systems, homogenization methods, and large deviations.
Intrinsic noise from molecular fluctuations of voltage-gated ion channels cause spontaneous activity that propagates into and affects local neural network function. A spontaneous action potential is a physical example of a new type of first-exit-time problem: the random time to initiate an excitable event in an excitable system with a single fixed point. I will show how noise induced excitable events in the stochastic Morris-Leccar model are initiated through a predictable sequence of events. In other words, a single mechanism explains how spontaneous activity is generated. Moreover, the generating mechanism contradicts the current understanding of this phenomena. It is widely believed that spontaneous activity in most neurons is driven primarily by fast sodium channels, because these channels govern the fast initiation stage of an action potential. Potassium channels respond much more slowly and are responsible for reseting the membrane voltage at the final stage of the action potential. Contrary to the standard paradigm, metastable dynamics predicts that the primary driving force behind spontaneous initiation of an action potential is the random opening and closing potassium channels.
I will discuss the long-time dynamics of infinite energy solutions to a wave equation with nonlinear forcing. Of particular interest is when these solutions display metastability in the sense that they spend long periods of time in disjoint regions of phase-space and only rarely transition between them. This phenomenon is quantified by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest a regime for which the dynamics are not fundamentally different from that observed in the stochastic counterpart in which random noise and damping terms are added to the equation, as well as a regime for which successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.
Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu