26Sp UArizona Grad Colloquium
26Sp UArizona Grad Colloquium
In spring 2026, the UA Grad Colloquium is organized by Tanner Reese. This website is maintained by Napoleon Wang.
We meet Mondays, 4–5 PM Arizona time, in Math 402, unless otherwise noted.
In spring 2026, the UA Grad Colloquium is organized by Tanner Reese. This website is maintained by Napoleon Wang.
We meet Mondays, 4–5 PM Arizona time, in Math 402, unless otherwise noted.
May 4 Tony Masso-Rivetti
Title: The Chern-Gauss-Bonnet Theorem
Abstract: The Chern-Gauss-Bonnet theorem is a generalization of the well-known Gauss-Bonnet theorem for 2-dimensional manifolds. The Chern-Gauss-Bonnet theorem is a local to global theorem that relates the Euler characteristic of a closed manifold to a function of its curvature. The Chern-Gauss-Bonnet theorem admits several very interesting proofs that stand at the intersection of differential geometry, algebraic topology, and functional analysis. In this talk, I will cover the basic constructions relevant to the statement of the theorem and give outlines for three of its well-known proofs.
Apr 27 Jackson Zariski
Title: Supervised-Learning for Telescope Control Systems and Performance Assessment via Record-Based Metrics
Abstract: Despite the substantial telemetry produced by modern observatories, telescope acquisition typically continues to rely on static pointing models and manually-tuned heuristics. We present a data-driven alternative for acquisition offset correction that unifies supervised machine-learning (ML), global sensitivity analysis, and an operations-centric software framework. Using archival telemetry logs from a variety of telescopes and observatories, we train regression models to predict on-sky horizon acquisition offsets from a diverse set of features, while also providing a brief examination into guiding forecasts with the aid of recurrent neural networks. To interpret these predictors and guide model tuning, we compute variance sensitivity via traditional Monte Carlo methods, quantifying the contributions of each input to the variance of the predicted offsets. With our pipeline taking historical telemetry and re-purposing it for ML training, we are able to see vast improvement in theoretical offset corrections on a temporally split testing set. For ease of use and greater generalization, we wrap the above trainers and diagnostic tools in a lightweight, Dockerized microservice with a browser-based GUI included, enabling operators to upload new logs, visualize feature sensitivities, and obtain acquisition offset corrections, all provided open-source. In addition, we introduce a new problem in combinatorics, based on probabilistic record-theory, that we call the Disappear Sort procedure. We present novel closed-form expressions for two different versions of this procedure and explore its application as a metric in evaluating an acquisition model's effectiveness in reducing large-localized errors prevalent at certain areas of feature space.
Apr 20 Samuel Herring
Title: Adventures in Statistical Mechanics
Abstract: This talk will explore some of the math and history around statistical mechanics. I intend to introduce some math tools for studying this area of physics, a comparison of classical and quantum statistical mechanics, and a hopefully more down-to-earth explanation of what I accomplished over the last six years.
Apr 13 Chris Mount
Title: Sampling Sticky Brownian Motion via a Ray-Knight Theorem
Abstract: A sticky Brownian motion is a Brownian motion which spends positive Lebesgue time at the origin. We propose a new algorithm for sampling a one-dimensional sticky Brownian Motion, which we will define as a time-changed Brownian Motion. Current sampling methods use Euler-Maruyama methods or approximations using short-ranged potentials that are computationally expensive due to the need for very small timesteps. In the 1960s, Ray and Knight independently discovered that the local time process of a Brownian motion is a Markov process with the same transition kernels as a Bessel process. We use this observation as the basis for our algorithm, and we validate the algorithm with numerical experiments.
Apr 6 Maximilian Rezek
Title: Traveling Waves Revisited
Abstract: We study traveling wave solutions for a Keller–Segel–FKPP system modeling diffusion, logistic growth, and chemotaxis. We prove existence of traveling waves for all parameter values, including the strong chemotaxis regime, by establishing a priori bounds in a uniformly local Lᵖ framework and passing to the limit from finite-domain approximations. Numerical experiments reveal a qualitative transition beyond this regime: while the wave speed remains close to c = 2, strong chemotaxis produces oscillatory structure behind the front. This suggests the emergence of pulsating fronts, that is, solutions that are periodic in a moving frame. We discuss this transition and its connection to stability, along with open questions on the existence and selection of such patterns.
Mar 30 Nick Pilotti
Title: Computing weight 2 paramodular cusp forms via Hecke stability
Abstract: At the Arizona Winter School, we have implemented an algorithm which assists us in constructing examples of weight 2 non-lift paramodular cusp forms. Number theorists are interested in finding examples of such forms in order to better understand the paramodularity conjecture, which generalizes the modularity theorem. Comparatively few examples of weight 2 non-lift paramodular cusp forms are known to exist. Our program can independently construct the previously known examples in just minutes, and it is based off a simple idea called Hecke stability. I will give an accessible introduction to modular forms and discuss the details of the algorithm.
Mar 23 Xinran Qian
Title: Modularity of theta series and Vertex algebras
Abstract: The notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel. They are, by now, ubiquitous in the representation theory of infinite-dimensional Lie algebras. They have also found applications in such fields as algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. The theory of vertex algebras also serves as the rigorous mathematical foundation for two-dimensional conformal field theory and string theory, extensively studied by physicists. In my talk, I will give some examples of vertex algebras and discuss the celebrated work of Yongchang Zhu on the modular invariance of characters of vertex algebras.
Mar 16 Roberto Castillo
Title: Cohomology of cluster varieties
Abstract: Given a set of polynomials in several variables, how can we begin to study the topology of the algebraic variety they define? In particular, what is the cohomology of this variety? In general, these questions can be extremely difficult to answer. By considering a particular class of varieties, it might happen that this question becomes more approachable. A cluster algebra is a special type of algebra with additional combinatorial structure that provides more tools for its study. In this talk, we will explain the algorithm given by Lam and Speyer to compute the cohomology of varieties defined by this type of algebra, more specifically the cluster varieties associated to acyclic quivers and of completely full rank.
Mar 2 Christian Cooper
Title: An Introduction to Moser's Iterative Procedure
Abstract: Hilbert's 19th problem asked whether the solutions to Euler-Lagrange equations associated with certain functionals are analytic. Along the road to the affirmative result came the goal of proving the Hölder continuity of solutions to such equations — and along that road the task of showing that solutions were locally bounded. With a roadmap to a proof, how can one actually show that solutions to these equations have such properties using only the assumption of a solution and the initial problem parameters? In this talk, I will introduce the simplifying iterative procedure used by Jürgen Moser to show that solutions are bounded on the interior by their L2 norm — a powerful technique in the analysis of elliptic partial differential equations.
Feb 23 Illia Hayes
Title: From Relativity without Coordinates to Conformal Geometry without Coordinates
Abstract: In the latter half of the 20th century, problems in general Relativity were growing incredibly complex. In an attempt to develop new tools to approach these problems, T. Regge proposed a discrete counterpart to David Hilbert's formulation of general relativity. The idea was to replace a smooth manifold with a complex of simplicial blocks glued together face to face and to replace the metric with a choice of edge lengths in each simplex. In Hilbert's formulation, the field equations arise from the calculus of variations applied to the total scalar curvature functional. Regge proposed an analog to Hilbert's functional and derived the corresponding vacuum equations. In this talk, we will introduce a notion of conformal deformation to Regge's simplicial spaces and discuss the conformal variations of Regge's functional.
Feb 16 Tanner Reese
Title: Crystal Growth on Locally Finite Partially Ordered Sets
Abstract: One can model a crystal growing in a corner using a set-valued Markov process. Using this model, one can understand the asymptotic behavior of the growth and hence its macroscopic properties. Additionally, this model has close connections to other statistical models such as Totally Asymmetric Simple Exclusion Processes (TASEP), Last-Passage Percolation (LPP), and random matrices. We will investigate how one can generalize this model and prove bounds for its mean and fluctuations.
Feb 9 Gabriel Black
Title: An Introduction to Knot Theory
Abstract: At the end of the 18th century the mathematical study of knots arose to answer questions arising in physics and since has been a flourishing field of mathematics. The purpose of this talk is to rigorously introduce knots as a mathematical object and show off some of the techniques used to distinguish knots. Specifically, we will investigate tricolorability, the Jones polynomial, and a new polynomial invariant generalizing the Jones polynomial introduced by Dr. Boninger in 2023.
Feb 2 Joseph Ruiz
Title: Representation type of small monoids over finite fields
Abstract: Monoids are objects similar to groups. A monoid is a set with an associative binary operation and an identity element. An algebra over a field is a vector space with an associative bilinear product. In the field of representation theory, one way we study these objects is by studying their module categories. An algebra is of finite representation type when its module category has a finite number of indecomposable objects. Surprisingly, every algebra of finite representation type is a summand of a monoid algebra. To understand the way monoids classify algebras of finite representation type, we look at the case of algebras over finite fields.
Jan 26 Abhirup Basu
Title: Some cool things about particles in a magnetic field
Abstract: I'll talk about some interesting phenomena that happen when you put charged particles in magnetic fields. For instance, we have the Aharonov-Bohm effect where a particle seemingly knows about a magnetic field even though it does not interact with it. I'll also talk about Dirac monopoles and how the existence of a single monopole can explain the quantization of charge we already see in nature. So why haven't we detected monopoles...or have we?