Plenary Talks

  Speaker: Rachel A Ward, University of Texas at Austin

Title: All hands on deck: applied mathematics in the era of ready-made proofs

Time: 8:30-9:30 am, Saturday, October 10, 2026

Abstract: We should plan now as though AI systems with enough compute will soon be able to answer any mathematical question that can be settled within existing theory. But which questions are worth answering? I argue that this presents an exciting opportunity for applied mathematicians to lead the charge. We can now pursue bigger and bolder research questions, collaborate across scientific disciplines without worrying about speaking different languages, and interact more directly with the applications we are trying to understand mathematically.

To frame this perspective, I will first give an overview of what large language models are, why recent improvements in post-training and inference-time scaling have enabled systems of LLMs to answer research-level mathematics questions, yet not improved in abilities regarding attribution and exposition.  I will discuss my experience over the past year as an organizer of the First Proof Project, an ongoing independent assessment of AI capabilities in research mathematics. Finally, I will discuss three applications where mathematicians are crucial for deciding which questions need answering.

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  Speaker: Xiaobing Feng, University of Tennessee, Knoxville.

Title: Full Moment Uncertainty Quantification for Numerical Solutions to Stochastic Navier-Stokes Equations

Time: 1:30-2:30 pm, Saturday, October 10, 2026

Abstract: The stochastic Navier-Stokes (and Stokes) equations provide a fundamental mathematical framework for modeling fluid flows subject to random influences arising from uncertainty, unresolved scales, external forcing, or environmental fluctuations. Compared with their deterministic counterparts, these stochastic PDEs pose significant analytical and computational challenges due to the presence of noise, the low regularity of their solutions, and the need to compute and quantify various important quantities of statistic interest.

While many successful numerical methods for deterministic fluid equations can be adapted to stochastic models, their performance often deteriorates, and their analysis requires fundamentally different techniques. In particular, classical deterministic techniques/arguments are generally insufficient for obtaining strong norm error estimates in the stochastic setting. To overcome these difficulties, new numerical analysis tools must be developed to address the intricate interplay among stochasticity, incompressibility, nonlinearity, and the propagation of higher-order moments.

In this talk, I will discuss some recent advances in the numerical analysis of the stochastic Navier-Stokes (and Stokes) equations driven by both additive and multiplicative noise. The focus will be on establishing strong convergence results, especially full-moment error estimates for numerical approximations. Particular emphasis will be given on the development of numerical and analytical tools and techniques that are successfully used to design and analyze optimal or near-optimal numerical methods for the stochastic Stokes and Navier-Stokes equations. These results apply not only to velocity approximations but also to pressure approximations, thereby providing a more complete numerical theory for incompressible stochastic flows.

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  Speaker: Lisa Fauci, Tulane University

Title: Explorations of active matter: self-avoidant droplets and swimming cups

Time: 9:00-10:00 am, Sunday, October 11, 2026

Abstract: At the microscale, individual cells comprising a protist colony wave their flagella, causing the colony to move through the fluid.  Droplets in an aqueous solution can spontaneously "swim” as they secrete oil, by causing local surfactant gradients.   In each case, their collective behavior is mediated by their environment, such as the surrounding incompressible fluid or chemical field. A mathematical description of these systems should capture the interaction of the Lagrangian objects (e.g. microorganisms, droplets) with an underlying Eulerian description of the environment.  Here we discuss the  immersed boundary framework used in the context of these two models.  First, we examine the dynamics of  droplets that move in response to a self-produced chemical gradient using a reaction-diffusion system. The particles have an unlimited supply of a chemical, secrete it at a given rate, but are anti-chemotactic and move in the direction of its maximal decrease.  In both one-and two-dimensional periodic domains, we find intriguing long-time behavior of collections of particles.  Our second example is a model of a multicellular microbial choanoflagellate colony, Choanoeca flexa.  Individual cells form cup-like colonies that can turn inside-out so flagella line the cup’s interior or cover its outside surface. We present a reduced model that examines the hydrodynamic implications on swimming and feeding for these different states.