Past Colloquia

Title: Making LaTeX PDFs ADA Compliant: Setup, Workflows, and Verification Tools

Abstract: Upcoming legal requirements mandate that all instructional materials provided to students be fully accessible and ADA compliant. Historically, LaTeX—despite being the standard tool for producing high-quality mathematical and technical documents—has been unable to generate PDFs that meet modern accessibility standards in a reliable and verifiable way.

This talk focuses on the Phase 3 release of the LaTeX tagging project, officially released on November 1, 2025. Developed by the core LaTeX development team, this release represents a fundamental rewrite of key components of the LaTeX engine to support the semantic document structure and metadata required for accessible, standards-compliant PDFs. As a result, LaTeX can now natively produce properly tagged PDFs suitable for screen readers and accessibility validation tools.

The presentation will demonstrate where and how to obtain a sufficiently recent LaTeX engine—since most commonly used distribution channels do not yet provide a version that supports Phase 3 tagging—and will outline the specific changes required to update existing LaTeX documents for accessibility compliance. Particular attention will be given to commonly used document features and packages, including which are currently supported and which require modification or replacement.

There will also be reference materials, including a demonstration LaTeX source file and curated resources that will assist with updating previous documents. The session will conclude with a discussion of tools and workflows for verifying that compiled PDFs are genuinely ADA compliant, allowing faculty to confidently assess their materials before distribution to students.

Here is the talk recording and the speaker’s site with more details.

Title: Toward a “GPT” Moment for Scientific Computing

Abstract: Foundation models such as ChatGPT have reshaped AI by learning reusable representations that transfer across tasks. This talk asks whether a similar shift is possible in scientific computing: moving beyond solvers for a single partial differential equation (PDE) toward foundation models for families of PDE-governed systems. A central obstacle is that high-fidelity PDE data are expensive—often requiring hours to millions of CPU-hours per simulation—making purely data-driven scaling impractical. I present a physics-first roadmap that replaces data scale with physical structure, using governing equations as supervision.

I will first focus on the single-PDE setting and show how physics-informed neural networks (PINNs) can be made reliable by diagnosing and addressing key training pathologies, leading to substantial accuracy improvements and successful simulations of challenging problems including 3D turbulence. I will then extend physics supervision from learning individual PDE solutions to learning solution operators for parametric PDE families. I will introduce the framework of physics-informed DeepONet and improve its scalability with continuous vision transformers. Finally, I will discuss how these advances motivate a longer-term direction toward unified models that can generalize across heterogeneous PDEs. Together, these results provide practical and theoretical steps toward PDE foundation models, with implications for accelerated simulation,design and control in computational science and engineering.

FSU/UF Topology & Geometry Meeting

Title: Intersec(K)tions

Abstract: What do we associate to a cycle? A classical fact is that if $X$ is a nice complex manifold or variety, and $Y\subset X$ can be (locally) given by one equation, then we can attach to $Y$ a line bundle on $X$. The situation is a lot murkier if $Y$ has higher codimension, that is, roughly speaking, you need more than one equation to describe it—what to attach to it is a lot less clear. I would like to argue that one can indeed attach to $Y$ a certain object (a sheaf of spaces or homotopy types) computed in terms of the $K$-Theory of $X$.

$K$-Theory usually gets a bad rap because of its (deserved?) reputation for being abstruse, so, on my way to show what to attach to $Y$, I will illustrate some simple ideas on how to get to the $K$-Theory space’s lower homotopy types.

One conjectures, and this is in fact a theorem in lower codimensions, that the proposed assignment can account for the intersection of cycles. This is work, partly in progress, in collaboration with Niranjan Ramachandran (UMD) and Maxime Ramzi (U Münster). No previous knowledge of $K$-Theory, Sheaf Theory, or Algebraic Geometry is assumed.

Title: Exploiting Low-Rank and Related Structures in Data Science Compute

Abstract: Most machine learning and AI models combine large data, intense computations and special hardware. At the intersections mathematical problems emerge, which enable systematic solutions. In this talk, I discuss three novel methods, with open-source software, that exploit certain problem properties, such as kronecker form or low-rankness. In particular, when tensor data is a combination of continuous and discrete information (hybrid), standard decompositions either don’t exploit this structure fully or become computationally very expensive. Through careful preprocessing, I decouple large factor subproblems into a sequence of smaller ones or apply effective iterative solvers. Second, for high throughput streaming data, it is typically important to process this information rapidly. For large systems, recomputing the best low-rank approximation is prohibitive. Therefore, I develop an updating method that is closely related to updating the singular value decomposition, however is more effective at tracking a low-rank subspace. Third, nonlinear optimization is a mathematical foundation for training AI models. When only gradients are available, it is often effective to approximate 2nd order curvature information for the iterates. Thus, I propose a parametric family of 2nd derivative Hessian estimates, which applies to deterministic or stochastic optimization. I conclude with a set of future directions.

Title: Algebraic foundations of machine learning

Abstract: This talk explores how algebraic geometry offers powerful tools for understanding models in statistics and machine learning. I will explain how viewing statistical models as algebraic varieties can address practical problems such as maximum likelihood estimation. Building on this perspective, I will focus on modern machine learning models and how their algebraic and semi-algebraic structure gives rise to polynomial invariants and complexity measures, such as Euclidean Distance degree. These tools provide theoretical guarantees for neural network verification, optimization, and robustness, supporting safety-critical applications.

Title: A complex scaling method for junctions of semi-infinite interfaces

Abstract: Scattering problems involving unbounded interfaces occur frequently in physics and engineering settings. Due to this prevalence, there exist many numerical methods for solving such problems. Unfortunately, the complicated behavior of solutions in the vicinity of infinite interfaces can make it challenging to deriving explicit error bounds for these methods. Many of these methods also require a large computational domain and so require a large number of discretization points to accurately solve the problem.

In this talk, I present a class of decomposable scattering problems. For this class of problems, the PDE domain can be decomposed into a collection of simple subdomains. The fundamental solutions for these simple regions can then be used to reduce the scattering problem into an integral equation on the interfaces between these subdomains. These integrals equations can be analytically continued into the complex plane, where they can be safely truncated. I demonstrate this procedure for a junction of two dielectric waveguides. For this problem, I show that the fundamental solutions and integral equation densities decay exponentially in the complex plane, and so the analytically continued integral equation can be truncated with provable exponential accuracy. I will also demonstrate how this method can be applied to a number of other scattering problems, including junctions of periodic gratings and corrugated waveguides.

Title: A Universal Description of Stochastic Oscillators

Abstract: Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. In joint work with Alberto Perez, Benjamin Lindner, and Boris Gutkin, we have introduced a nonlinear transformation of stochastic oscillators via a complex-valued function, Q, that greatly simplifies and unifies the mathematical description of the oscillator’s spontaneous activity, its response to external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The Q function is the eigenfunction of Kolmogorov’s backward operator (also called the stochastic Koopman operator) with the least negative (but non-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around $\omega_1$; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of both weakly and strongly coupled stochastic oscillators.

Joint work with:
Alberto Perez-Cervera (Universitat Politècnica de Catalunya, Barcelona)
Boris Gutkin (Ecole Normale Supérieure, Paris)
Benjamin Lindner (Humboldt University, Berlin)
Max Kreider (Penn State University)

Title: Random commuting matrices

Abstract: The study of the eigenvalue distribution of random matrices is a well-established field, dating back to the 1920’s. It became popular with the work of Wigner and Dyson in the 1950’s and 60’s, and today is a major field in both probability and theoretical physics. Typically a random matrix is generated by choosing the entries identically and independently distributed.

What can one say about random d-tuples of commuting matrices? What does it even mean, since one can no longer choose entries independently?

We will describe one approach to defining a random d-tuple of commuting matrices. We shall show that in the Hermitian case, the description of their eigenvalue distribution parallels to some extent the single matrix theory, though there is a qualitative change when d ≥ 5. In the non-self adjoint case the eigenvalue distribution is quite unlike the single matrix case.

Title: Progressive Decoupling and the Proximal Method of Multipliers in Optimization

Abstract: The proximal method of multipliers in convex optimization applies the proximal point algorithm to find a saddle point of the convex-concave Lagrangian function associated with the problem. The progressive decoupling algorithm in optimization can be derived as a specialization in various forms. A recently developed variable-metric version of the proximal point algorithm with generic linear convergence has a key role in supporting error analysis and understanding how the convergence rate depends on proximal parameters.

South Eastern Logic Symposium (SEALS) 2026

Title: Isometry groups of ultrametric Polish spaces

Abstract: A long-standing open problem, formulated by Krasner in the 1950’s, asks for a characterization of the isometry groups of ultrametric spaces. We characterize the isometry groups of Polish ultrametric spaces, using suitable forms of generalized wreath products of full permutation groups. Our solution is actually developed in the finer context of topological (Polish) groups, and thus solves a problem of Gao and Kechris from 2003. Furthermore, we provide correspondences between the isometry groups of Polish ultrametric spaces belonging to natural subclasses (such as homogeneous and exact spaces) and different versions of generalized wreath products proposed in the literature by Hall, Holland, and Malicki. This is joint work with Riccardo Camerlo and Luca Motto Ros.

South Eastern Logic Symposium (SEALS) 2026

Title: Computability on R and Gal(Q)

Abstract: We begin by recalling the notion of a computable function on the real numbers R, developed independently by Gregorczyk and Lacombe over sixty years ago.  Using this notion, we note that the real numbers that are themselves computable form a countable subfield of R with exactly the same first-order properties as R itself. (Logicians would therefore call it an elementary subfield.)  So, in a first-order sense, everything that happens in R is already exemplified in this much nicer subfield.  However, even when one knows that an existential statement holds for all parameters, it may be impossible (both in R and in the subfield) to give a computable procedure for producing witnesses.  Similar results hold in C.

We will then turn to a different continuum-sized structure:  the absolute Galois group Gal(Q) of the rational numbers.  Once again the computable elements of this group form a subgroup, but now it is an open problem whether the group and the subgroup have the same first-order theory, let alone whether this is an elementary subgroup.  (If they do have the same theory, this would put nice upper bounds on the complexity of the theory of Gal(Q).)  However, using joint work with Kundu, we can show that once again there is no computable procedure for producing witnesses to the truth of (true) existential statements, either in the full group or in the subgroup.

Number Theory Conference

Title: AI generated proofs of mathematical statements

Abstract: There is an upswing in natural language artificial intelligence (AI) models for high-level mathematics. These models certify their findings using Automated Theorem Provers (ATP). We will give an overview of the nowadays interactions between mathematics, ATPs and AI. Later Over an explicit example, we discuss and criticize how it might be possible to verify wrong theorems and their possible ramifications.

Title: New unlikely intersections on families of elliptic curves

Abstract: We consider families of elliptic curves, their sections, and tangencies between them, mainly over the complex numbers.  Such a tangency is an “unlikely intersection” and the main result we discuss is a bound on them.  Although all the objects involved and the statement live in the world of algebraic geometry, our proof surprisingly make a a brief detour into the real analytic category. The talk will assume only basic differential geometry and a passing acquaintance with elliptic curves.  If there is time, I’ll also mention a characteristic p analogue (non-obvious given the non-algebraic nature of the previous discussion), and an interesting inaccuracy in a famous old paper of Manin.

Title: Regularity for nonlinear elliptic PDEs and surface design problems

Abstract: The design of surfaces that are “free from bulges, ripples and wrinkles” and are “aesthetically pleasing to the eye” is a central problem in industrial design, data science, computer-aided design, computer-aided manufacturing, architecture and engineering. In the 1990’s, UC-Berkeley computer scientists Henry Moreton and Carlo Séquin addressed this problem by introducing the so-called surfaces of minimum mean curvature variation. Since then, many numerical algorithms have shown that such surfaces were indeed smooth, as desired in applications. We will first review the regularity theory for second order elliptic PDEs and then present the first mathematical results on existence and regularity of surfaces of minimum mean curvature variation, proving that the numerical conjecture is indeed true. This is based on a joint work with Luis A. Caffarelli and Hernán Vivas.

Ulam Colloquium

Title: Data driven reduced order modeling for waveform inversion

Abstract: Waveform inversion seeks to estimate an inaccessible heterogeneous medium by using sensors to probe the medium with signals and measure the generated waves. It is an inverse problem for a hyperbolic system of equations, with the sensor excitation modeled as a forcing term and the heterogeneous medium described by unknown, variable coefficients. The traditional formulation of the inverse problem, called full waveform inversion (FWI), estimates the unknown coefficients via nonlinear least squares data fitting. For typical band limited and high frequency data, the data fitting objective function has spurious local minima near and far from the true coefficients. This is why FWI implemented with gradient based optimization can fail, even for good initial guesses. We propose a different approach to waveform inversion: First, use the data to “learn” a good algebraic model, called a reduced order model (ROM), of how the waves propagate in the unknown medium. Second, use the ROM to obtain a good approximation of the wave field inside the medium. Third, use this approximation to solve the inverse problem. I will give a derivation of such a ROM for a general first order hyperbolic system satisfied by all linear waves in lossless media (sound, electromagnetic or elastic). I will describe the properties of the ROM and will use it to solve the inverse problem for sound waves.

Title: From centralized to federated learning of neural operators: Accuracy, scalability, and reliability

Abstract: As an emerging paradigm in scientific machine learning, deep neural operators pioneered by us can learn nonlinear operators of complex dynamic systems via neural networks. In this talk, I will present the vanilla deep operator network (DeepONet) and several extensions of DeepONet, such as DeepONet with Fourier decoder layers and manifold operator learning. I will demonstrate their effectiveness on diverse multiphysics and multiscale 3D problems, such as geological carbon sequestration, full waveform inversion, and topology optimization. Deep learning models are usually limited to interpolation scenarios, and I will quantify the extrapolation complexity and develop a complete workflow to address the challenge of extrapolation for deep neural operators. Moreover, I will present the first operator learning method that requires only one PDE solution, i.e., one-shot learning, by introducing a new concept of local solution operator based on the principle of locality of PDEs. I will also present the first systematic study of federated scientific machine learning (FedSciML) for approximating functions and solving PDEs with data heterogeneity. Lastly, I will present FunDiff, a novel framework of diffusion models over function spaces for physics-informed generative modeling.

Title: Semi-retractions and Ramsey degrees

Abstract: Ramsey degrees and their transfer principles form a branch of combinatorics that has received much recent interest. As a brief illustration of definitions I will present in the talk, the ordered pair $a<b$ has small Ramsey degree 1 in the class of all finite linear orders by Ramsey’s classical theorem for finite sets. In contrast, the ordered pair $a<b$ has big Ramsey degree 2 in the rational order $(\mathbb{Q},<)$: if we color ordered pairs from $\mathbb{Q}$ some finite number of colors, there is a suborder isomorphic to the rational order whose pairs take on at most two colors; however, there is a 2-coloring of pairs from $\mathbb{Q}$ such that any suborder isomorphic to the rational order has pairs of each color.

In a 2021 paper, I introduced the concept of a “semi-retraction”, which is a pair of maps $(g,f)$ between infinite mathematical objects that has the necessary architecture to preserve facts about Ramsey degrees. This notion was further developed in a 2024 paper joint with Dana Bartošová at University of Florida. In this talk, I will survey some results from this paper and extensions of these results.

Title: Policy iteration for inverse mean field games

Abstract: We propose a policy iteration method to solve an inverse problem for a mean f ield game (MFG) model, specifically to reconstruct the obstacle function in the game from the partial observation data of value functions, which represent the optimal costs for agents. The proposed approach decouples this complex inverse problem, which is an optimization problem constrained by a coupled nonlinear forward and backward PDE system in the MFG, into several iterations of solving linear PDEs and linear inverse problems. This method can also be viewed as a fixed-point iteration that simultaneously solves the MFG system and inversion. We prove its linear rate of convergence and present some numerical examples to demonstrate the effectiveness of the method. This is based on a joint work with Nathan Soedjak and Shanyin Tong.

Title: From Spheres to Bands: New Rigidity Phenomena in Scalar Curvature

Abstract: When does a lower bound on scalar curvature force a space to be “as round as a sphere”? A landmark theorem of Llarull says: if a closed spin manifold maps onto the unit sphere without increasing areas and with nonzero degree, then having the same scalar-curvature lower bound as the sphere already forces the map to be an isometry-so the manifold itself is a round sphere. I will describe a new band version of this phenomenon for compact manifolds with boundary that map into a spherical band (the sphere with two caps removed-the region between two latitudes). We obtain sharp inequalities controlling how “long” such bands can be and we characterize the equality case. Two payoffs: (1) a proof of Llarull’s theorem in dimension four without the spin assumption, and (2) a rigidity theorem for manifolds with conical ends mapping into punctured spheres. I’ll emphasize the geometric mechanisms behind these results and keep technicalities to a minimum.

Title: How to make sense of sonic boom and other discontinuities in fluid mechanics

Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities.

Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question.

In this presentation, we will provide the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent method of a-contraction with shifts. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation, and to solve the Bianchini and Bressan conjecture in this special case.

Title: Physics-Informed Gaussian Process for ODE/PDE Simulation and Calibration

Abstract: Parameter estimation for nonlinear dynamic system models, represented by ordinary differential equations (ODEs) or partial differential equations (PDEs), is a fundamental challenge across science and engineering. Traditional methods rely heavily on numerical integration, which is computationally expensive and often ill-suited for noisy and sparse data. We propose a new method, the Physics-Informed Gaussian Process (PIGP), that combines statistical rigor with physics-based constraints. Specifically, we model system components as Gaussian processes, explicitly conditioned on the governing ODE/PDE system. This approach bypasses traditional numerical integration entirely, leading to substantial computational savings while providing uncertainty quantification and enabling inference even for unobserved system components. Our framework offers a principled Bayesian alternative to physics-informed neural networks, with unique strengths in transparency, interpretability, and robust uncertainty awareness.

At a high level, many real-world scientific and engineering systems, ranging from disease spread to weather patterns, from fluid flows to acoustic waves, are described by differential equations. Our method introduces a way to apply uncertainty-aware machine learning directly to these equations using limited data. In plain terms, we can use small amounts of experimental observations, combined with known physics, to make accurate predictions while quantifying confidence in the results. This ability is especially valuable in high-stakes applications where uncertainty must be explicitly managed.

Ulam Colloquium

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.

South Eastern Logic Symposium

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.

South Eastern Logic Symposium

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.

Ramanujan Colloquium

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.

Erdős Colloquium

Title: Two and a half millennia of irrationality in mathematics

Abstract: The discovery of irrational numbers – especially the fact that the square root of 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.

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.