Title: On the geometric aspects behind classical mechanics.Abstract: In this talk about physics, we will not talk about physics, but the applications of mathematical tools used in that field. From a functional to symplectic geometry, passing by abstract algebra and using differential forms, we will explore some of the foundations of classical mechanics and apply them to the famous example of the2D harmonic oscillator. No knowledge of any physics is required.
Title: Mathematical modeling of mosquito-borne disease transmission in wildlife
Mosquito-borne diseases (MBDs) cause enormous losses of human lives and health throughout the world. Recent environmental changes, including global climatic warming and increased human encroachment on natural areas, are predicted to increase the spread of MBDs. The emergence of novel zoonoses (diseases originating in wildlife and capable of spreading in human populations) is also expected to accelerate, stretching thin our ability to respond effectively to future epidemics. These emerging crises call for substantial advancement in our understanding of how ecological processes drive MBD transmission within and between wildlife populations. But understanding the underlying factors driving MBD outbreak risk is a substantial challenge. For these diseases, transmission occurs during blood feeding, an essential step in the mosquito reproductive cycle. Blood feeding itself is an interactive ecological process: the animals upon which mosquitoes feed may defend themselves, leading to a back-and-forth struggle that can significantly alter transmission rates.
Mathematical models have long been used to study MBD transmission, dating back to the late 19th century when it was first discovered that mosquitoes vectored malaria. However, recent advances in our understanding of mosquito ecology call into question some long-held assumptions underlying these classical models. I will discuss my work which focuses on incorporating ecological interactions into mathematical models of MBD transmission, including resource competition between host animals and defensive behaviors against mosquito biting. These interactions will be further connected to recent work incorporating the effect of ambient temperature on mosquito biology into MBD transmission models.
Title: The Connections Between Discrete Geometric Mechanics, Information Geometry, Accelerated Optimization and Machine Learning
Abstract: Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.
Biography: Melvin Leok is professor of mathematics at the University of California, San Diego. His research interests are in computational geometric mechanics, computational geometric control theory, discrete differential geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry. He received his Ph.D. in 2004 from the California Institute of Technology in Control and Dynamical Systems under the direction of Jerrold Marsden. He is a Simons Fellow in Mathematics, three-time NAS Kavli Frontiers of Science Fellow, and has received the DoD Newton Award for Transformative Ideas, the NSF Faculty Early Career Development (CAREER) award, the SciCADE New Talent Prize, the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis. He has given plenary talks at Foundations of Computational Mathematics, NUMDIFF, and the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, and is the coauthor of a research monograph entitled, “Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds.”