25Fa UArizona Grad Colloquium

In fall 2025, the UA Grad Colloquium is organized by Reese Tanner. This website is maintained by Napoleon Wang.

Previous Talks

Dec 3 Tyler Lendman

Title: Bounded Velocities, Unbounded Scales: The APMC Framework for Transport Equations

Abstract: Monte Carlo methods provide a flexible probabilistic framework for solving PDEs. Yet, their application to multiscale transport equations poses significant challenges—especially in the diffusive regime, where classical particle methods suffer from severe stiffness and restrictive time-step constraints. In this talk, I will begin with a brief review of Monte Carlo techniques for PDEs with emphasis on their interpretation as particle systems and their application to the heat equation via random walks. I will then introduce kinetic transport equations and explain how standard Monte Carlo schemes break down when the characteristic speeds blow up in the diffusive scaling. The main focus of the talk is a review of the asymptotic-preserving Monte Carlo (APMC) method of Dimarco, Pareschi, and Samaey that overcomes these limitations through an implicit reformulation of the underlying kinetic system. This reformulation yields a modified equation with uniformly bounded particle velocities, allowing for time steps independent of the scaling parameter. I will outline the construction of the scheme for the Goldstein–Taylor model and radiative transport, highlighting how the method seamlessly transitions to Brownian dynamics in the diffusion limit. Numerical examples illustrate the method’s efficiency and accuracy across regimes, demonstrating its effectiveness as a robust multiscale Monte Carlo solver.

Nov 19 Shambhavi Srivastava

Title: Testing Helicity Conservation in GRMHD simulations and multi-wavelength observed variability in AGNs

Abstract: Magnetic fields are quintessential to the idea of accretion and jet activity around compact objects. The long-term stability of these relativistic jets, however, is an important and interesting question. There are many explanations and mechanisms on how the jet is formed, but few for how the jet remains collimated on large scales for millions of years. "Magnetic Helicity" measures how organized the magnetic field is, loosely, the twist in magnetic field lines. In ideal MHD, this is a conserved quantity and even in turbulent plasmas it undergoes an inverse cascade. Understanding how helicity behaves in GRMHD is therefore essential for understanding outflows. Complementary to the simulation analysis, time-domain variability in AGNs is examined using the z-transformed discrete correlation function applied to unevenly sampled multi-wavelength light curves. Autocorrelation functions and associated uncertainties reveal characteristic variability timescales and coherence structure in the emission.

Nov 12 Sean Zhu

Title: A brief introduction to analytic number theory

Abstract: Euclid gave the earliest proof for the infinitude of primes about 300 B.C. Then, around the 18th century, Euler gave an analytic expression of the Fundamental Theorem of Arithmetic, which provided further insights into the behavior of primes. This inspired Dirichlet to come up with his L-function analogue of Euler’s zeta function to provide a result about arithmetic progressions. In the 19th century, Riemann took the zeta function and “ran" with it to new places. This brief introduction follows the historical and conceptual development of early analytic number theory from Euclid to Riemann.

Nov 5 Gabriel Black

Title: Pointwise Ergodicity in Combinatorics

Abstract: Many important questions in combinatorics involve looking at what configurations must occur in large subsets of the natural numbers. One such question was posed in the mid 1900's Erdős asked if any (finite) number of large sets must always have a common distance. This question was answered in 1979 by Stewart and Tijdeman in the affirmative, since then, Rusza improved Stewart and Tijdeman's result. In this talk, we will work towards understanding an ergodic viewpoint which leads to a proof of Rusza's theorem and, more importantly, to many generalizations which show the richness of configurations in large sets.

Oct 29 Tyler Kline

Title: Completing Q_p

Abstract: Real analysis is very useful, but there’s also a beautiful analytic theory over the algebraic closure, the complex numbers. For the field of p-adic numbers, Qp, there is a similar story. If you’re doing p-adic analysis (like me!), you might want to work over a suitably closed object above Qp to try and transfer results from complex analysis into the p-adic world. It turns out this is more complicated than taking the algebraic closure. Join me as we travel down the completion rabbit hole in our quest to find the ideal space sitting above Qp for analysis (kind of)! 

Oct 22 Tony Masso-Rivetti

Title: Mathematical Scattering Theory and the Spectral Shift Function

Abstract: Mathematical scattering theory is concerned with how the absolutely continuous spectrum of a self-adjoint operator changes under small perturbations.   Under sufficiently nice assumptions, we are able to unitarily relate the subspaces on which the operators’ spectra are absolutely continuous and describe the change in spectrum with a function of a single variable.   In this talk I will cover the basic tools used in scattering theory, with a focus on the wave operators and spectral shift function, as well as providing detailed examples.

Oct 15 Tanner Reese

Title: Particles from Quantum Fields

Abstract: The development of QED and other field theories to describe the mechanics of elementary particles has produced the most precise predictions known to man. Despite this, a mathematically rigorous treatment of interacting quantum fields has remained illusive for the last half century. We will see how the concept of quantized particles is not an ad hoc assumption, but instead flows directly from the non-commutative nature of the underlying fields. This will be derived for the particular case of phonons and considered more broadly for any quantum field. We will also see the connection between particles and representations of fundamental symmetry groups. 

Oct 8 Nick Pilotti

Title: Theta functions are automorphic forms 

Abstract: Theta functions have a mysterious way of appearing in seemingly unrelated results. In number theory, they are used to prove the functional equation of Riemann's zeta function, the quadratic reciprocity law, and Lagrange's four-square theorem. Each of these theorems follows from the fact that theta functions are automorphic forms meaning they satisfy an important functional equation. More specifically, theta functions are automorphic forms on the so-called metaplectic group, which is a group of unitary operators coming from harmonic analysis. In this context, we reinterpret the classical number theoretic uses of theta functions.

Oct 1 Samuel Herring

Title: Quasilocality and Spectral Perturbation Theory in Quantum Mechanics

Abstract: In quantum lattice systems, we say that the dynamics supplied by the Schrödinger equation are quasilocal. While the dynamics generally do not keep information local, this term allows us to precisely describe the spread of information in a quantum lattice system. In this talk, I intend to discuss another quasilocal map called the spectral flow. The spectral flow can be used as a tool to describe the spectrum of a perturbed system. I will discuss the spectral flow and how to use it to analyze the perturbed spectral gap of well-behaved models.

Sep 24 Xinran Qian

Title: What is ALGEBRAIC analysis? The theory of D-modules

Abstract: The 2025 Abel prize was awarded to Masaki Kashiwara ''for his fundamental contributions to algebraic analysis and representation theory, in particular the development of the theory of D-modules and the discovery of crystal bases". D-modules arise naturally from studying systems of linear partial differential equations with algebraic tools. An outline of D-module theory was proposed in the early 60's at the University of Tokyo by Mikio Sato with whom Kashiwara completed his Master's thesis. This thesis established the foundations of D-module theory. In my talk, I will give a brief introduction to D-modules and try to explain its applications to PDEs, algebraic geometry, and representation theory.

Sep 17 Illia Hayes

Title: The space of Riemannian metrics and special geometries

Abstract: Every smooth manifold admits Riemannian metrics, and choosing one amounts to endowing the manifold with geometry. This choice determines quantities like lengths of curves, volumes, and curvature. The family of all metrics on a given closed orientable manifold forms an infinite dimensional space which we can view as a manifold modeled on Fréchet space. In this talk, we will discuss some of the basic properties of the space of metrics. In particular we will discuss a remarkable result of Ebin: the action of the diffeomorphism group on the space of metrics admits a slice, analogous to the slice theorem for smooth actions of compact Lie groups on finite-dimensional manifolds. We will also look at the Einstein Hilbert functional on the space of metrics and its critical points, which correspond to metrics with special geometric significance.

Sep 10 Chapman Howard

Title: Is Your Birthday in π? Statistical Miracles in Number Theory & Connections to Langlands

Abstract: Does every finite sequence of digits appear in the digits of π? This is a famously unanswered question about normal numbers. We address the existence of such numbers and explore a famous analogue: the Sato-Tate Conjecture, which reveals a hidden statistical law in the arithmetic of elliptic curves. Such a connection (between objects of a priori completely different significance) is the key idea in the Langlands Program. We'll tie together the major ideas and display the connections between the accessible question of birthday-in-π, and the leading edge of research in Langlands.