Abstract: We study the elastic behaviour of prestrained plates, phenomenon observed for instance in plastic deformation, natural growth of soft tissues or manufactured polymer gels. When actuated, the prestrained plates reduce their internal stresses by undergoing (possibly large) deformations. Their mathematical modeling consist of a geometric nonlinear fourth order problem with a nonlinear riemanian metric constraint. A discrete gradient flow is proposed to decrease the system energy and is coupled with finite element approximations of the plate deformations based on discontinuous Galerkin finite elements.
In this talk, we give a general description of the model, introduce the numerical scheme and discuss some of its properties, such as the Gamma-convergence of the finite element approximations. We illustrate the performance of the proposed methodology through several numerical experiments involving different prestrain metrics.
Abstract: We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bueltel and Pappas, and further studied by Lau. We use this framework to generalize the purely group-theoretic definition of Rapoport-Zink spaces given by Bueltel and Pappas, and to show that this definition coincides with the classical one in the case of unramified EL-type local Shimura data.
Abstract: Let X be a closed, connected, hyperbolic surface of genus 2. Is it more likely for a simple closed geodesic on X to be separating or non-separating? How much more likely? In her thesis, Mirzakhani gave very precise answers to these questions. One can ask analogous questions for square-tiled surfaces of genus 2 with one horizontal cylinder. Is it more likely for such a square-tiled surface to have separating or non-separating horizontal core curve? How much more likely? Recently, Delecroix, Goujard, Zograf, and Zorich gave very precise answers to these questions. Surprisingly enough, their answers were exactly the same as the ones in Mirzakhaniâs work. In this talk we explore the connections between these counting problems, showing they are related by more than just an accidental coincidence.
Abstract: We derive aposteriori error bounds in time-maximum-space-squared-sums and time-mean-squares-of-spatial-energy norms for a class of fully-discrete methods for linear parabolic partial differential equations (PDEs) on the space-time domain based on hp-version discontinuous Galerkin time-stepping scheme combined with conforming spatial Galerkin finite element method. The proofs are based on a novel space-time reconstructions, which combines the elliptic reconstruction Georgoulis, Lakkis & Virtanen (2011), Lakkis & Makridakis (2006), and Makridakis & Nochetto (2003) of and the time reconstruction for discontinuous time-Galerkin schemes Makridakis & Nochetto (2006), SchÃ¶tzau & Wihler (2010) into a novel tool, allows for the user's preferred choice of aposteriori error estimates in space and careful analysis of mesh-change effects.
Abstract: Classical Iwasawa theory studies a relationship, called the Iwasawa main conjecture, between a $p$-adic $L$-function and a Selmer group. This relationship involves codimension one cycles of an Iwasawa algebra. This talk will discuss results on the topic of higher codimension Iwasawa theory. We will consider the restriction to an imaginary quadratic field of an elliptic curve defined over the rational numbers with good supersingular reduction at an odd prime. We shall also consider the tensor product of Hida families. This is joint work with Antonio Lei.
Abstract: A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).
Abstract: The close connection between global temperature variation and atmospheric carbon dioxide concentration has been central to the issue of climate change. The lag/lead between sets of longitudinal data on the two variables has implications for the causality of that connection. We consider this problem as one of curve registration. Most of the available solutions for this problem have been designed for the growth data application, where the number of observations is small and the number of replicates is large. We argue that a different emphasis is needed for the paleoclimatic application. We provide a new method, which is able to pool local information without smoothing and to match sharp landmarks without manual identification. We prove the consistency of the proposed method under fairly general conditions. Simulation results show superiority of the performance of the proposed method over two existing methods. Use of the proposed method to Antarctic ice core data leads to some interesting conclusions.