Schedule for Spring 2025
Title: Partial Differential Equations of Mixed Type – Analysis and Applications
Abstract: Three of the fundamental types of partial differential equations (PDEs) are elliptic, hyperbolic, and parabolic, following the standard classification for linear PDEs. Linear theories of PDEs of these types have been considerably better developed. On the other hand, many nonlinear PDEs arising in Mathematics and Science are naturally of mixed type. The solution to several longstanding fundamental problems greatly requires a deep understanding of such nonlinear PDEs of mixed type, particularly those of mixed elliptic-hyperbolic type. Notable examples include the multidimensional Riemann problem (formulated by Riemann in 1860 for the one-dimensional case) and related shock reflection/diffraction problems in fluid dynamics (the compressible Euler equations), and the isometric embedding problem in differential geometry (the Gauss-Codazzi-Ricci equations), among others. In this talk, we will present some old and new underlying connections of nonlinear PDEs of mixed type with these longstanding fundamental problems, from the Riemann problem to the isometric embedding problem. We will then discuss some recent developments in the analysis of these nonlinear PDEs through examples with an emphasis on developing unified approaches, ideas, and techniques for addressing mixed-type problems. Some further developments, perspectives, and open problems in this direction will also be addressed.
Title: Convexity techniques in probability theory
Abstract: In this talk we will discuss a few techniques involving the notion of convexity, which lead to a vast array of results in probability theory. The techniques can be used to develop concentration and deviation inequalities, and moment and entropy bounds, for a large class of random variables; to establish quantitative comparison between statistical distances; as well as to tackle certain problems arising in combinatorics that can be phrased in the language of probability.
Title: Open Systems, Chaos, and Limit Theorems
Abstract: Chaos is a fundamental topic in the theory of dynamical systems. In 1958, Kolmogorov discovered that chaotic dynamical systems exhibit certain statistical properties. In this talk, I will discuss the statistical properties of open chaotic systems. Specifically:
1. The Poisson limit theorem is a useful tool for distinguishing between chaotic and non-chaotic behaviors in billiard systems.
2. The convergence rates of Poisson limit theorems have connections to Riemann-Zeta functions.
3. The polynomial escape rate and its refined property, which describes where orbits are statistically more likely to visit in the phase space. If time permits, I will outline the proof of these results using operator renewal theory. These are the joint work with Prof. Leonid Bunimovich.
Title: Extreme Value Theory for evolving populations with mean-field interaction
Abstract: Classical Extreme Value Theory (EVT) studies the behavior of the tails of probability distributions, which is also reflected in the asymptotic behavior of the largest values in samples drawn from those distributions as the sample size grows large. This behavior turns out to be captured by finitely many real parameters, which can be estimated. The latter allows us to estimate the probability of being above an arbitrarily large threshold by using a single large sample, which helps in the prediction of extreme phenomena that have never occurred in the past. Provided that the asymptotic results of classical EVT extend to stochastic processes which are not necessarily independent, it is possible to estimate the probability of observing various extreme phenomena in the future, like abnormally large returns of stocks in large financial portfolios, or abnormally large electrical potentials in human neurons. This motivates the development of an EVT for Stochastic Differential Equations with Mean-Field interaction, complementing past works which have established Central Limit Theorems and Large Deviation Principles in this framework. A connection with a problem from Random Matrix Theory and a result for Mean-Field Games are also discussed.
Title: Phase Transitions and Algorithmic Aspects of the Binary Perceptron
Abstract: The binary perceptron model, a simple single-layer neural network, has a rich history in theoretical physics and machine learning. This model considers the problem of finding a sign vector that satisfies a set of random halfspace constraints. The two central questions are: for what constraint densities do solutions exist with high probability, and can we efficiently find a solution when one exists?
In this talk, I will discuss my work addressing both questions, guided by long-standing conjectures from physics. These conjectures predict a sharp satisfiability threshold for the existence of solutions, and a strong freezing property (where almost all solutions are isolated, suggesting that finding solutions using polynomial-time algorithms is typically hard). For the symmetric binary perceptron, we rigorously establish both predictions. Furthermore, the strong freezing property is particularly intriguing, because empirical evidence shows that polynomial time algorithms often succeed in finding a solution, challenging the typically hard prediction. This suggests that such algorithms find atypical solutions. We establish formally this phenomenon, showing that at low constraint density, there exists a rare but well-connected cluster of solutions, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability. Additionally, we modify the canonical discrepancy minimization algorithms to solve the binary perceptron problem. We analyze the performance of our algorithm, yielding new algorithmic results.
Title: Parabolic System of Aggregation Formation in Bacterial Colonies
Abstract: The goal of this talk is to study a fourth-order nonlinear parabolic system with dispersion for describing bacterial aggregation. Analytical solution of traveling wave is found by taking into account the dispersion coefficient. Numerically, we demonstrate that the initial concentration of bacteria in the form of a random distribution over time transforms into a periodic wave, followed by a transition to a stationary solitary wave without dispersion.The goal of this talk is to study a fourth-order nonlinear parabolic system with dispersion for describing bacterial aggregation. Analytical solution of traveling wave is found by taking into account the dispersion coefficient. Numerically, we demonstrate that the initial concentration of bacteria in the form of a random distribution over time transforms into a periodic wave, followed by a transition to a stationary solitary wave without dispersion.
Title: Counting buckyballs
Abstract: You may have heard of Buckminsterfullerene, a spherical molecule made out of 60 carbon atoms. More generally, we can ask about the following counting problem: How many spherical carbon molecules, also called fullerenes, can you make out of 2n carbon atoms? A related question: How many ways are there to build a sphere out of 2n triangular pieces, so that every corner has at most six triangles next to it? There is a beautiful exact formula which is quite surprising!
Title: Mathematical modeling of CD8 T cell search for malaria infection in the liver
Abstract: Malaria, a disease caused by parasites of the Plasmodium genus, begins when Plasmodium-infected mosquitoes inject malaria sporozoites while searching for blood. Sporozoites migrate from the skin via blood to the liver, infect hepatocytes, and form liver stages. In mice, sufficient numbers of vaccine-induced activated or memory CD8 T cells are capable of locating and eliminating all liver stages in 48 hours, thus preventing the blood-stage disease. However, rules of how CD8 T cells are able to locate all liver stages in a limited timeframe remains poorly understood.
I will present results from our work in the past years attempting to understand how CD8 T cells locate the liver stages. I will show how we combined the use of experimental data with math models to narrow down mechanisms of T cell search and how additional experimental measurements challenge earlier conclusions. If time permits, I will also present some philosophical points on how one could best use mathematical modeling to gain insights into biological phenomena.
Title: Stability of small BV solutions to compressible Euler in a class of vanishing physical viscosity limits
Abstract: The stability of a Riemann shock, in the absence of any technical conditions for perturbations, is a major challenging problem even within a mono-dimensional framework. A physically natural approach to justify the stability of such a singularity involves considering a class of vanishing physical dissipation limits (or viscosity limits) of physical viscous flows with evanescent viscosities. I will present the recent results for stability of Riemann shocks in a class of inviscid limits from Navier-Stokes (for the case of isentropic Euler); from Brenner-Navier-Stokes-Fourier (for full Euler). The proofs for those results are based on the a-contraction method. In particular, we use the stability result for the isentropic case, to develop the well-posedness theory of entropy solutions evolving from small BV initial data in the class of inviscid limits. More precisely, small BV entropy solutions to the isentropic Euler can be constructed by inviscid limits from Navier-Stokes, and those are unique and stable among inviscid limits from Navier-Stokes. The proof is based on the three main methodologies: the modified front tracking algorithm; the a-contraction; the method of compensated compactness.
Title: The algorithmic aspects of continuous mathematics
Abstract: We discuss and survey some recent results in effective Polish spaces and topological spaces. We describe how algorithms play a part in understanding and calibrating the effective content of some continuous spaces and processes, allowing us to view these objects in a different light. Many fundamental questions have not been fully explored until very recently and we will describe some directions in clarifying the basic notions of presentation, duality and isomorphism.
Title: Vaught’s Conjecture and Structural Complexity
Abstract: Hilbert’s first of his famous 23 problems concerned the size of the set of real numbers, or what he called the continuum. Cantor had shown years earlier that the continuum is larger than the set of natural numbers; it is uncountable. Hilbert’s first problem, the so-called continuum hypothesis, asked if there is any set with a size between countable and continuum. In the 20th century, Godel and Cohen shocked the mathematical world by showing that the continuum hypothesis is independent of the standard axioms of set theory. In other words, it is neither provable nor disprovable and depends on set-theoretic assumptions.
Vaught’s conjecture is one of the oldest and most well-known open problems in mathematical logic. It is a restricted version of the continuum hypothesis of particular interest because it only concerns sets that are very natural to construct. To be specific, it states that the number of countable models of a given infinitary theory up to isomorphism (e.g. the number of countable groups, countable linear orderings, countable Q-vector spaces, etc.) is either countable or continuum and never in between. Despite all of the work that has gone into this conjecture in the last 60 years, it remains open. That said, there has been significant and recent progress in proving special cases of the conjecture and in building promising mathematical infrastructure to help with a full proof. A particularly fruitful approach has been to use notions of structural complexity, like Scott rank, to break down the problem. This talk will give the context for Vaught’s conjecture, explain the approach to the problem using structural complexity, and describe recent results proven using this approach.
Title: Euler’s constant: Euler’s work and modern developments
Abstract: The first part of the talk surveys Euler’s work on his constant gamma = 0.57721… and related constants, starting from 1731. It has a cousin, the Euler-Gompertz constant, delta = 0.59634… which will put in a guest appearance. The second part reviews a selection of subsequent developments from the following 300 years, which exhibit its appearance in many fields of mathematics. This mysterious constant is conjectured to be transcendental, but it is not even known to be irrational. It appears in many striking analytic formulae; some were contributed by Ramanujan. The problem of computing it was raised in Turing’s 1937 paper defining Turing machines. This constant has a particularly strong and elusive relation to prime number theory and the Riemann hypothesis. In a certain sense it knows (something) about every individual prime.
Title: Data Driven Modeling for Scientific Discovery and Digital Twins
Abstract: We present a data-driven modeling framework for scientific discovery, termed Flow Map Learning (FML). This framework enables the construction of accurate predictive models for complex systems that are not amenable to traditional modeling approaches. By leveraging measurement data and the expressiveness of deep neural networks (DNNs), FML facilitates long-term system modeling and prediction even when governing equations are unavailable.
FML is particularly powerful in the context of Digital Twins, an emerging concept in digital transformation. With sufficient offline learning, FML enables the construction of simulation models for key quantities of interest (QoIs) in complex Digital Twins, even when direct mathematical modeling of the QoI is infeasible. During the online execution of a Digital Twin, the learned FML model can simulate and control the QoI without reverting to the computationally intensive Digital Twin itself. As a result, FML serves as an enabling methodology for real-time control and optimization of the physical twin, significantly enhancing the efficiency and practicality of Digital Twin applications.
Title: The Partition Parity Problem
Abstract: One of the major motivating problems in partition theory is the conjecture that half of the partition numbers are even, and half odd. An enormous array of authors have been working on this question for decades, and we are very far from a solution – but generating a great deal of interesting work on the way! This talk will give an overview of what we know and don’t know, and a selection of some of the current suggested approaches, including some work of the speaker and co-authors.
Title: Some algebraic aspects of quantum computing
Abstract: Compared to classical computing, the basic theory of quantum computing leads fairly quickly into deeper mathematical waters: matrix theory, Lie groups, representation theory, number theory, algebraic geometry, …. Rather than attempt a survey, I will describe a few specific problems in which interesting mathematics appeared unexpectedly. First, an encoding scheme for qubits which mitigates magnetic noise by passing through representations of symmetric groups; second, a new and more efficient decomposition of 3-qubit operations into 1- and 2-qubit operations which leverages the exotic, octonion-related “triality” automorphism of the Lie group PSO(8). No prior knowledge of quantum computing or quantum physics will be assumed.
Title: Two and a half millennia of irrationality in mathematics
Abstract: The discovery of irrational numbers – especially the fact that √2 is not rational – is often attributed to the Pythagorean philosopher Hippasus. According to legend, this revelation so disturbed his fellow Pythagoreans that they drowned him in a lake. In this talk, we will explore the long (often unsuccessful) history of our attempts to understand irrationality, with the goal of explaining how mathematicians now really think about these questions. Our journey will span from ancient Greece to the ancien régime, from 17th-century Basel to the present day. This talk will be accessible to undergraduate math majors.
Schedule for Fall 2024
The previously scheduled colloquia on Sep 27 and Oct 14 were rescheduled (due to hurricanes, etc) for the spring.
Title: My trajectory towards mathematical modeling of pulmonary infections
Abstract: The immune response to respiratory infections is highly complex and multiscale, making it amenable for mathematical modeling. Fungal respiratory infections are becoming increasingly prevalent and pose the threat of antimicrobial resistance. The immune response to respiratory pathogens is highly complex and multiscale, making it difficult to predict how to manipulate the host’ immune system to better fight off pathogens. Mathematics provides an excellent framework for integrating knowledge and de novo data and provides a platform for systematic interrogation of host interventions to improve the outcome of disease. In this talk, I will survey recent advances in mathematical modeling of respiratory infections, including some of our own work in invasive pulmonary aspergillosis. Moreover, I will discuss my career trajectory from number theory to working in a medical laboratory mixing mathematical modeling and experimentation.
Title: Interactive Theorem Proving
Abstract: For many years, mathematicians have used computers to perform computations that inform their research and lead to new conjectures. More recently, computers have been used to formally verify correctness of proofs via software known as proof assistants. This talk will be an introduction to proof assistants – what they are, why they are used, and what the mathematical community has accomplished with them.
A short introduction to the most commonly used proof assistant, Lean, will also be given. Together, we will use Lean to formally verify that every natural number is either even or odd, a fact that shouldn’t surprise many but is a good illustration of what working with a proof assistant is like. Those interested in following the proof on their own computers are encouraged to install Lean ahead of the talk by going to: https://leanprover-community.github.io/get_started.html. The process is straightforward and will certainly help get the most out of the talk.
Title: Including human behavior in infectious disease models
Abstract: The COVID-19 pandemic has revealed the good and the bad of infectious disease models. While a well-developed model provides invaluable insights needed to understand and combat the pandemic, many models suffer from imperfect or simplistic assumptions that result in inaccurate or even completely wrong predictions. In this talk, I will present several infectious disease models my group has developed since the beginning of the COVID-19 pandemic, using both classical differential equations as well as individual-based network models. All presented models incorporate certain aspects of human behavior (e.g., rates of mask wearing or vaccine uptake that depend on age, education, etc.) and social processes (e.g., homophily in social interaction patterns). A particular focus will be on heterogeneous mixing patterns. People with similar characteristics are more likely to interact, a phenomenon called assortative mixing or homophily. Empirical age-stratified social contact matrices have been derived by extensive survey work. We lack however similar empirical studies that provide social contact matrices for a population stratified by attributes beyond age, such as gender, sexual orientation, or ethnicity. Accounting for heterogeneities with respect to these attributes can have a profound effect on infectious disease model dynamics. I will present a new method, which uses linear algebra and non-linear optimization, to expand a given contact matrix to populations stratified by binary attributes with a known level of homophily and a known population-wide prevalence. As an example, I will show how accounting for ethnic homophily in the United States can give rise to interesting and non-trivial trade-offs that should be taken into consideration when developing prioritization strategies for future mass vaccine rollouts.
Title: The number and nature of subgroups of the symmetric group
Abstract: The symmetries of any object are described by a group, so it is natural to ask: What does a random group look like? This talk will start with a brief survey of how we might go about counting various algebraic structures. We’ll then go on to see what a random group might be, in various different contexts.
Every group arises as a subgroup of a symmetric group. An elementary argument shows that there are at least 2^{n^2/16} subgroups of the symmetric group on n points, and it was conjectured by Pyber in 1993 that up to lower order error terms this is also an upper bound. The same year, Kantor conjectured that a random subgroup of the symmetric group is nilpotent. This talk will present a proof of one of these conjectures, and a disproof of the other.
The new results in this talk are joint work with Gareth Tracey (Warwick).
Title: Fundamentals and Applications of Diffusiophoresis and Diffusioosmosis: Particle and Fluid Motion Induced by Solute Concentration Gradients
Abstract: Diffusiophoresis and diffusioosmosis are the deterministic motion of particles and fluids induced by a concentration gradient of solute, respectively. Diffusiophoresis and diffusioosmosis receive much attention in recent years given its relevance in natural settings such as metamorphic transformation, and in applications, including particle separation, enhanced oil recovery, and nanoparticle drug delivery. In this talk, I present recent projects in my group concerning the fundamentals and applications of diffusiophoresis and diffusioosmosis. First, recent experiments demonstrated diffusiophoresis in porous media for nanoparticle drug delivery through hydrogels, but existing theories cannot predict the particle motion. We open a new area of research by developing a foundational mathematical model that can predict diffusiophoresis in porous media. A comparison between our model predictions and experiments demonstrates excellent agreements. We show surprising results which arise from diffusiophoresis in different mixtures of electrolytes. Second, existing theories of diffusioosmosis have focused on dilute electrolytes, but theories for diffusioosmosis of concentrated electrolytes are lacking. We develop a predictive mathematical model for diffusioosmosis of concentrated electrolytes, where ion-ion electrostatic correlations are important. We predict a novel reversal in the direction of diffusioosmosis due to electrostatic correlations. We demonstrate how this reversal gives rise to new flow responses when coupled with different interfacial properties. Our models will motivate future theories and experiments, and enable efficient design of current and emerging applications.
Prior Semesters
The schedules from the math department colloquia from past semesters are linked below.
- Fall 2023, Spring 2024
- Fall 2022, Spring 2023
- Fall 2021, Spring 2022
- Fall 2020, Spring 2021
- Fall 2019, Spring 2020
- Fall 2018, Spring 2019
- Fall 2017, Spring 2018
- Fall 2016, Spring 2017
- Fall 2015, Spring 2016
- Fall 2014, Spring 2015
- Fall 2013, Spring 2014
Some recorded colloquium talks are available.