Department of Mathematics

The Landau equation models the motion of gas particles in the specific case where there are Coulomb (repulsive) intermolecular forces. This equation was derived heuristically from the Boltzmann equation, heuristically by Landau himself in 1937 and the rigorous mathematical justification came over fifty years later with Degond, Lucquin - Desreux, Desvillettes, Villani and others. For some time, solutions of the Landau equation were studied through the Boltzmann equation before taking the limit to transition into the Landau equation. However, the Boltzmann equation turns out to be very complicated especially in the Coulomb case, and the only global existence theory we have without any extra regularity assumptions is that of very weak solutions (H-solutions) of Villani from 1998. In more recent years, therefore, the Landau equation has been studied directly via techniques for general parabolic equations.

In this talk, we will discuss an abstract but simple family of homogeneous Boltzmann equations for which we can develop an existence theory in $L^p$, and we will see what happens in the limit as the Boltzmann equation transitions into the Landau equation.

In 1977, K. Steffens proved a theorem which gives a necessary and sufficient condition for a countable graph to have a perfect matching (we have named this "Steffens' Theorem"). We investigate the proof-theoretic strength of a number of variations of Steffens' Theorem in the context of reverse mathematics. In this talk, we will encounter each of the "Big Five" subsystems, and more. This work is joint with Stephen Flood, Oscar Levin, and Tyler Markkanen.

Abstract: The Ricci iteration is a discrete analogue of the Ricci flow. Introduced
in 2007, it has been studied extensively on Kahler manifolds, providing a new
approach to uniformisation. In this talk, we will discuss the Ricci iteration on
spheres that are equipped with transitive Lie group actions. Joint work with
Timothy Buttsworth (Queensland), Yanir Rubinstein (Maryland) and Wolfgang
Ziller (Pennsylvania).

It was once believed that our world is governed by rules and that all observed outcomes are predictable. This Newtonian dream of having an entirely predictable world is no longer true. We now know that there are incredibly simple mathematical models with dynamics so sensitive to initial conditions that long-term prediction is impossible. This realization gave birth to what is known as "Chaos Theory''.

In this presentation we will explore what chaos is in a mathematical sense and will look at several specific examples that illustrate the mystery of chaos. We will also look at chaos as it is observed in the world around us and discuss what lies ahead in this field.

Complex chemical processes such as the decomposition of energetic materials are typically modeled at the atomistic level using large-scale molecular dynamics simulations (MD), generating a wealth of data. It is natural to wonder whether these rich and expensive datasets can be used to do more than simply describe the particular system being studied. In our work, we have developed a statistical framework for extracting information about the fundamental underlying reaction pathways observed from MD data, using it to build kinetic Monte Carlo models (KMC) of the corresponding chemical reaction network. Our KMC models can not only extrapolate the behavior of the chemical system by as much as an order of magnitude in time, but can also be used to predict the dynamics of entirely different chemical trajectories. Additionally, we have built a new and efficient data-driven algorithm for interpretable model reduction of nonlinear dynamical systems. This allows us to reduce the complex chemical reaction networks from our learned KMC models, consisting of thousands of reactions, in a matter of minutes.

In recent work, we seek to extend the extrapolation capabilities of our framework to arbitrarily long timescales by using the fast KMC models as a numerical integrator with a prediction-correction scheme. Our "leapfrog" approach has the potential to truly enable computational simulation to reach experimental timescales.
One can easily imagine a future in which MD simulations used for research are routinely archived and analyzed in order to add to and modify an existing repository of learned chemical reactions and reaction rates. This repository would form a "chemical genome" that can then be used to quickly simulate all kinds of new chemical systems. This data-driven approach has the potential to break through long-standing barriers in the accuracy versus system size and time scale tradeoff that is at the core of computational materials science.


Short Bio:
Qian Yang joined the Computer Science and Engineering Department at the University of Connecticut in August 2018 as an Assistant Professor. She received her PhD in Computational and Mathematical Engineering from Stanford University in January 2018, where she was a member of the Materials Computation and Theory Group advised by Dr. Evan Reed. She holds a B.A. in applied mathematics/computer science from Harvard College. Her research focuses on the development of machine learning and computational methods for the physical sciences, and her work has been published in journals such as Chemical Science, Energy & Environmental Science, among others, as well as in a book chapter.