Intro to Research at UCR seminar 2026

Thursday 3:30-4:30 in 284 Skye Hall

Schedule:

• April 2: Zhenghe Zhang

Title: Introduction to Ergodic Schrödinger operators.

Abstract: Ergodic Schrödinger operators arise naturally in the study of Schrödinger equations, which model the motion of a quantum particle in a disordered medium. On the other hand, the spectral analysis of such operators can often be reformulated as problems in dynamical systems. In this talk, I will briefly introduce how these three areas--mathematical physics, spectral analysis, and dynamical systems--interact in my research. 

• April 9: Sarah Yeakel

Title: Homotopy theory: a framework for topology

Abstract: In this talk, we will introduce fundamental concepts in homotopy theory, the study of spaces under continuous deformations, and the long-term project of computing algebraic invariants of spaces, including homology and homotopy groups. We will focus on the history of a class of examples called lens spaces, nontrivial 3-manifolds which are completely classified up to homeomorphism. 

• April 16: Siting Liu

Title: From ICON to GenICON: In-Context Operator Learning with Uncertainty Quantification

Abstract: I will introduce In-Context Operator Networks (ICON), a framework in which a single neural network learns solution operators for differential equations directly from a few prompted input-output examples at inference time, without any weight updates. ICON acts as a few-shot learner across forward and inverse problems for ODEs, PDEs, and mean-field control. I will then present a probabilistic interpretation: under a random differential equation data model, ICON implicitly computes the posterior predictive mean given the context, linking operator learning to Bayesian inference. This motivates GenICON, a generative variant that samples from the posterior predictive for principled uncertainty quantification, yielding a unified Bayesian view of in-context operator learning.

• April 23: David Weisbart

Title: Scaling Limits of Discrete-Time Random Walks

Abstract: This talk introduces scaling limits in the study of stochastic processes and outlines a broader program in which they serve as a tool for analyzing discrete approximation in a variety of settings. After a brief overview of my current research program, I will discuss several recent papers on Brownian motion in local fields. The most recent work points toward a new direction, suggesting that diffusion in non-Archimedean spaces may capture critical behavior through an underlying geometric mechanism.

• April 30: Wee Liang Gan

Title: Persistence modules

Abstract: I will talk about two results on persistence modules which I find interesting: the decomposition theorem and the algebraic stability theorem. 

• May 7: Qixuan Wang Special time and place: Skye 268 at 2pm.

Title: Virtual organ: hair follicles and skin

Abstract: Our lab specializes in developing multi-scale models for biological systems. In this talk, I will present two virtual organ models that we developed in recent years. The first is a L’Oreal sponsored research project, where we develop Virtual Hair Follicle, dedicated for the study of androgenetic alopecia, a commonly seen hair loss associated with aging. The second is Digital Skin, and we started by applying it to keloid scar.

• May 14: Agnieszka Zelerowicz

Title: Lorentz gases on quasicrystals

Abstract: The Lorentz gas was originally introduced as a model for the movement of electrons in metals. It consists of a massless point particle (electron) moving through Euclidean space bouncing off a given set of scatterers $\mathcal{S}$ (atoms of the metal) with elastic collisions at the boundaries $\partial \mathcal{S}$. If the set of scatterers is periodic in space, then the quotient system, which is compact, is known as the Sinai billiard. There is a great body of work devoted to Sinai billiards and in many ways their dynamics is well understood. In contrast, very little is known about the behavior of the Lorentz gases with aperiodic configurations of scatterers which model quasicrystals and other low-complexity aperiodic sets. This case is the focus of our joint work with Rodrigo Treviño.

We establish some dynamical properties which are common for the periodic and quasiperiodic billiard. We also point out some significant differences between the two. The novelty of our approach is the use of tiling spaces to obtain a compact model of the aperiodic Lorentz gas on the plane. 

• May 21: Fred Wilhelm

Title: 13 ways to say lower curvature bound

Abstract: We will deÖne what it means for an intrinsic metric space to have a lower curvature bound, by comparing the spaceís geometry to that of a space of constant curvature. We will start with a description of the three model, two dimensional, spaces with constant curvature, the hyperbolic plane, the euclidean plane, and the two sphere. We will then discuss the wide open problem:

Problem: Which closed smooth nñmanifolds admit Riemannian metrics with nonnegative sectional curvature.

• May 28: Yat Tin Chow

Title: An introduction to iterative optimization algorithms and an example

Abstract:  In this talk I will give a brief exposition of what research on optimization algorithm may focus on, and why and how these conclusions matter. After a brief introduction to provide  some high-level ideas, I will provide one specific example of research that one of my graduated PhD students was working on with myself (namely extra gradient and Popov’s scheme with a moving anchor) to illustrate how these high-level ideas can help obtain specific results when they are particularized.

The research presented in this talk is a joint work with J. K. Alcala (USC) and M. Sunkula (Purdue).

• June 4: Jose Gonzalez