Mathematics Colloquium
Bounds on the Entanglement entropy of droplet states in the XXZ model
When
March 22, 2023 | 2:30 p.m. - 3:30 p.m.
Refreshments at 2:00 in UH 4004
Where
University Hall 4004
Speaker
Christoph Fishbacher, Baylor University
Abstract
In this talk, I will give an overview over recent results on the entanglement entropy of the one-dimensional Heisenberg XXZ model. For the spin-1/2 case, Beaud and Warzel showed that generic low-energy states satisfy a logarithmically corrected area law. I will talk about the extension of this result to higher-energy states (joint work with H. Abdul-Rahman and G. Stolz) and about a logarithmic lower bound (joint work with R. Schulte).
Frustration in the packing of soft materials
When
March 20, 2023 | 3:30 p.m. - 4:30 p.m.
Refreshments at 3:00 in Lobby
Where
University Hall 1005
Speaker
Keith Promislow, Michigain State
Abstract
Many processes in material science involve entropic contributions from packing-the constraints imposed by volume occupied by other material. Diblock polymers offer a rich environment to study the packing of soft materials as gradient flows of a system energy. Ideas from $\Gamma$ convergence provide powerful tools to extract simplified models in certain singular limits. We present examples of packing dichotomies in both continuous and discrete formulations and identify cases in which limiting problems may be more complex. We present a derivation of a random phase reduction of self-consistent mean field models, identify regimes in which they converge to functionalized Cahn-Hilliard energy, and provide a discrete system for the packing of soft balls that exhibits large-system frustration: the inability of gradient flows to obtain the global energy minimum, that significantly complicates the extraction of limiting processes.
Computing Nonlinear Waves in Bose-Einstein Condensates and Beyond: Adventures in Applied Mathematics
When
February 17, 2023 | 3:30 p.m. - 4:30 p.m.
Refreshments at 3:00 in Lobby
Where
University Hall 1005
Speaker
Stathis Charalampidis
Abstract
A Bose-Einstein Condensate (BEC) is a state of matter formed by a collection of bosons that are cooled to temperatures close to absolute zero. On the theoretical side, BECs can be described by a nonlinear PDE known as the NLS equation. In this talk, a mathematical and computational analysis of the NLS equation will be presented using Newton’s method for nonlinear systems together with penalty methods. We have developed the Deflated Continuation Method (DCM) to trace disconnected branches of solutions by factoring out previously computed ones. Finally, the study of nonlinear waves will be extended to various physical systems, including water waves and meta-materials which have a plethora of engineering and industrial applications.
Combining networks and mathematical models to improve epidemiological predictions
When
February 15, 2023 | 3:30 p.m. - 4:30 p.m.
Refreshments at 3:00 in Lobby
Where
University Hall 1005
Speaker
Nao Yamamoto
Abstract
Mathematical models have played a prominent role in predicting, assessing, and controlling potential outbreaks in the current coronavirus disease 2019 (COVID-19) pandemic. Many modeling studies using statistics or differential equations have been proposed to analyze the COVID-19 dynamics. Network analysis and cluster analysis have also been adapted along with these methods to understand the dynamics of disease transmission in more detail and from multiple perspectives. The decision of which method to choose should be based on the principle of simplicity and the ability of the model to answer the question of interest. In some cases, new observations, new types of data, or new technologies may require modifications to the model. My research aims to improve epidemiological predictions by combining network science and various mathematical/statistical methods.
Membrane Mechanics Meet Minimal Manifolds
When
February 13, 2023 | 3:30 p.m. - 4:30 p.m.
Refreshments at 3:00 in Lobby
Where
University Hall 1005
Speaker
Leroy Jia
Abstract
A time-honored problem in mathematical physics is to show that an area-minimizing fluid interface such as a soap film forms a special surface whose mean curvature vanishes. In the axisymmetric setting, this surface is called a catenoid. We recap Euler's original solution to the catenoid problem and then proceed to discuss a generalization of this problem where the fluid interface is replaced by a membrane with a bending stiffness. The shapes that such a membrane can make, which are critical surfaces of an energy called the Willmore functional, are intimately related to the catenoid and also to the shapes of a compressed rod (i.e., the elastica) in a precise mathematical way that informs us about their mechanics.
Deep Learning Meets Optimal Control and Dynamical Systems: A Mathematical Approach
When
February 6, 2023 | 3:30 p.m. - 4:30 p.m.
Refreshments at 3:00 in Lobby
Where
University Hall 1005
Speaker
Deepanshu Verma
Abstract
Deep neural networks, or DNNs, have become a popular tool in various fields due to their universal approximation properties. This talk delves into the intricacies of designing efficient DNN algorithms and architectures, as well as exploring their use in solving optimal control problems. DNN architectures are known to be difficult to design and train, often resulting in challenges such as the exploding or vanishing gradients issue. These networks also exhibit a large number of equivalent optimal solutions in the parameter space. A neural network architecture designed using fractional derivatives will be introduced to tackle the former challenge. Additionally, the notion of bias ordering, with theoretical guarantees, will be introduced to narrow down the parameter search space. The second half of the talk will focus on the applications of DNNs in high-dimensional optimal control problems. Traditional methods for solving these problems often suffer from the Curse-of-Dimensionality, where computational complexity increases exponentially with the dimension of the problem. Utilizing DNNs to approximate the value function of control problems can effectively tackle this issue. One of the key challenges in training is discovering the relevant parts of the state space. To address this, techniques from control theory will be employed to devise a self-supervised training algorithm. Several numerical experiments, including applications to PDE constrained optimization, will be presented.
