News Archive

Speaker: Zhifu Xie

Title: On the Uniqueness of Convex Central Configurations in the Planar 4-Body Problem

Abstract: A central configuration is a specific arrangement of masses, and a planar central configuration can lead to a homographic periodic solution. It is crucial for understanding the dynamic behavior of the N-body problem, and the question of its finiteness has been a challenge for mathematicians in the 21st century. For the planar four-body problem, its finiteness has been proven by computer-assisted proof in 2006 by Hampton and Moeckel, but there is still much to understand. One conjecture is that there exists a unique convex central configuration for any four positive masses in a given order. Many research paper has attempted this question by assuming either having some equal masses or having restrictions of the geometric shape such as a trapezoid or co-circular shape. In this talk, we provide a rigorous computer-assisted proof (CAP) of the conjecture for four masses belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem. Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its surrounding neighborhood.

Bio: Dr. Zhifu Xie holds the Wright W. and Annie Rea Cross Endowed Chair in Mathematics and Undergraduate Research at the University of Southern Mississippi, where he is a Professor of Mathematics. Zhifu earned his B.S. in Mathematics Education from Chongqing Normal University in 1998, his M.S. in Mathematics from Chongqing University in 2001, and his Ph.D. in Mathematics from Brigham Young University in 2006. He began his academic career as an assistant professor at Virginia State University in 2007 and was promoted to full professor in 2015. Zhifu’s outstanding research, teaching, service, and integration of knowledge earned him recognition as a 2012 SCHEV Outstanding Faculty Award Finalist (SCHEV refers to the State Council of Higher Education for Virginia). He joined the University of Southern Mississippi in 2016 as the Cross Endowed Chair. His research interests range from classical celestial mechanics to differential equations, including their applications in fields such as reaction-diffusion equations and infectious disease models. Zhifu is renowned for his innovative approaches to classroom teaching, which incorporate new techniques and materials. He actively engages undergraduate students in his cutting-edge mathematical research, overseeing their internships and individual projects, and finds great satisfaction in watching their progress.

Title: High-Order Accuracy Computation of Coupling Functions for Strongly Coupled Oscillators

Abstract: We develop a general framework for identifying phase-reduced equations for finite populations of coupled oscillators that is valid far beyond the weak coupling approximation. This strategy represents a general extension of the theory from [Wilson and Ermentrout, Phys. Rev. Lett., 123 (2019), 164101] and yields coupling functions that are valid to higher-order accuracy in the coupling strength for arbitrary types of coupling (e.g., diffusive, gap-junction, and chemical synaptic). These coupling functions can be used to understand the behavior of potentially high-dimensional, nonlinear oscillators in terms of their phase differences. The proposed formulation accurately replicates nonlinear bifurcations that emerge as the coupling strength increases and is valid in regimes well beyond those that can be considered using classic weak coupling assumptions. We demonstrate the performance of our approach through two examples. First, we use the diffusively coupled complex Ginzburg–Landau (CGL) model and demonstrate that our theory accurately predicts bifurcations far beyond the range of existing coupling theory. Second, we use a realistic conductance-based model of a thalamic neuron and show that our theory correctly predicts asymptotic phase differences for nonweak synaptic coupling. In both examples, our theory accurately captures model behaviors that weak coupling theories cannot.

Title: Passing drops and descents

Abstract:

Title: Global Injectivity, Manifold Estimation, and Universality of Neural Networks

Abstract: Abstract: In recent years machine learning, and in particular deep learning has emerged as a powerful and robust tool for solving problems in fields ranging from robotics, to medicine, materials science, cosmology, and beyond. As work on applications has advanced, so too has theory advanced to guide, explain, and interpret deep learning. In this talk, I will provide an overview on some of that theory in three parts. In the first, I will present a connection between injectivity of ReLU layers and vector geometry, which yields a simple criterion for a ReLU network to be end-to-end injective. Second, I will introduce the concept of universality in the context of neural networks and reveal a surprising connection to knot theory sheds light on what kinds of manifold-supported functions can be learned by a neural network. Finally, I will discuss a work that exploits the connections between topological covering spaces and locally bilipschitz maps to develop a recipe for constructing neural networks that can learn to approximate `topologically interesting’ maps between manifolds.

Title: Graphs that allow two distinct eigenvalues

Abstract: Let G be a connected graph on n vertices and let S(G) denote the set of all real symmetric n × n matrices A = [a_ij ] such that aij = 0 if and only if {i, j} is not an edge of G. The diagonal entries of A can take any value. The inverse eigenvalue problem of a graph asks to determine all possible spectra of matrices in S(G). A fundamental subproblem is to determine the minimum number of distinct eigenvalues over all matrices in S(G). This parameter is denoted by q(G). For example q(G) = n if and only if G = P_n, the path on n vertices. The graphs with q(G) = n − 1 have also been characterized. Determining those graphs with q(G) = 2 has been much more difficult. A recent advance has been to determine the minimum number of edges in a graph G with q(G) = 2. The graph G must have at least 2n − 3 edges if n is odd and at least 2n − 4 edges if n is even. The graphs for which equality is attained are characterized.

Title: Active cloaking for the heat equation

Abstract: We present a method for cloaking for the heat equation that relies on using active sources to control the temperature in a region. The same method can be used for mimicking, i.e. giving the illusion that an object or source looks like another one, from the perspective of temperature measurements.

The talk will be accessible to everyone with some multi-dimensional calculus.