Fridays in Skye 284
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Social Time: 3.30-4p
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Talk: 4-4.50p
Scheduled Talks
Spring 2026
5 June 2026
The magic of positivity
In Lie Theory we have 2 perspectives: the complex vs the real, and the algebra vs the geometry. We'll look at two concepts: magical sl_2 triplets, and theta-positivity. The first one lives in the complex algebra, and the latter in the real group. These two classify the same families of semisimple objects. We will be showing examples of both, and attempting to show the relation between these two.
22 May 2026
Triangularity and the Schur Algebra
A "triangular basis" gives a basis for an algebra in which elements factor through "special idempotents." We'll take a look at "codeterminants" which are a well-known example of a triangular basis for the Schur algebra.
15 May 2026
Multiscale Modeling for Virtual Hair Follicle (Oral Practice)
Hair follicle is a mini-organ in the skin which undergoes cyclic growth. In recent years it has emerged as a leading system for studying general mechanisms of stem cell biology, tissue growth and regeneration, and cell fate regulation. In this L’Oreal sponsored research project, we develop a Virtual Hair Follicle, dedicated for the study of androgenic alopecia, a common type of hair loss associated with aging. We create a multiscale computational model which integrates numerical PDEs for signaling dynamics and an agent-based submodel for cellular movement and kinetics. We model multiple types of cells and diffusive signaling pathways and are currently working to extend the model into successive hair cycles. This work is in collaboration with my advisor Dr. Qixuan Wang and Dr. Plikus at UC Irvine, and funded by L’Oreal Research Group.
8 May 2026
A Brief Introduction to Physics-Informed Neural Networks
Physics-informed neural networks (PINNs) have been a hot topic of study recently. We will go over the basics of PINNs and some newer versions of them. We will also talk about other models used in scientific machine learning as time permits.
1 May 2026
Uniformly positive Lyapunov exponents with monotone potentials along local unstable leaves
Anderson localization is a physical phenomenon that was observed by Phillip Anderson. One definition of localization is when the spectrum of the Schrodinger operator has pure point spectrum with exponentially decaying eigenfunctions. The Lyapunov exponent plays a central role in studying the phenomenon, as uniformly positive Lyapunov exponents paired with a large deviation estimate has been a large indication of localization. Our focus is when one can prove uniform positivity, as Kotani theory would imply that the family of Schrodinger operators has an empty absolutely continuous spectrum. We will discuss the setting and the methods used to show uniform positive Lyapunov exponents for lower Holder potentials along local unstable leaves when generated by hyperbolic dynamics with at least one expanding direction.
24 April 2026
One Thousand And One Arabian Carpets
We investigate modified Sierpiński Carpet fractals, constructed by dividing a square into a uniform nxn grid, removing a subset of the squares at each step, and then repeating that process for each square remaining in that grid. We study these fractals using the fractal zeta functions, first introduced by Michel Lapidus, Goran Radunović, and Darko Žubrinić in their book "Fractal Zeta Functions and Fractal Drums". We give a full classification of all the possible Sierpiński Carpet modifications, and we use that classification to demonstrate algorithms for computing the complex dimensions of every Sierpiński Carpet modification on the nxn grid. This is a practice talk for my Oral Exam, so any feedback would be hugely appreciated.
17 April 2026
Solving PDEs by doing Algebraic Geometry
A certain grad student, who I love, talks a lot about how every major theorem in geometry is actually a theorem in PDEs. Wouldn't it be nice to have a major theorem in PDEs that's actually a theorem from geometry? Even better, what about a major theorem in PDEs that's actually a theorem in algebraic geometry! In this talk we'll sketch a proof of the Malgrange–Ehrenpreis Theorem, a major theorem saying that every differential operator with constant coefficients admits a Green's function. The proof will go through the Bernstein-Sato polynomial, whose existence follows from the theory of D-modules in algebraic geometry. In fact, this theory is constructive and has been implemented in algebraic geometry software such as Macaulay2 and Singular. Along the way we'll discuss a pretty relationship between distributional inverses and "normal crossing" singularities in algebraic geometry which puzzled the speaker for at least a year.
3 April 2026
Connections between Quiver Flow Polytopes and Combinatorial Hodge Theory.
Using the clique complex, we are able to construct a cochain complex from a graph. The Hodge decomposition on the first cochain space allows us to induce a total order on the vertices of the graph. Additionally, in the case of quivers, we study which components contain flows parametrized by an associated quiver flow polytope.
Winter 2026
6 March 2026
A proof of the Hairy Ball theorem via the Stokes theorem
Exterior calculus does not only allow us to do integration on a manifold, but also yields the magical formula of the Stokes theorem. As an extremely global result, the Stokes Theorem has many applications. For examples: the Cauchy's integral theorem, the Brower's fixed point theorem, and the de Rham theorem. In this talk, we will see a trivial proof of the Hairy Ball Theorem by applying the Stokes Theorem on the unit 2n-spheres, which is not present in a usual textbook (as far as I am aware of). Familiarity of the exterior calculus will not be assumed as I will tell you what differential forms are and list all calculation rules used in the proof.
27 February 2026
A trivial proof of a known inequality relating mlds and multiplicities using toric method.
In this talk I will briefly introduce the minimal log discrepancies(mlds) and multiplicities (in the sense of Hilbert-Samuel). I will then present a toric way to explain and prove the known inequality $$mlds \le 2/ mult$$, and more importantly, share the experience of solving this problem.
20 February 2026
Faculty advisors at UCR
We'll discuss what different faculty members' research fields are and how they're related. We'll then talk about what differentiates faculty in similar research areas and how one goes through choosing an advisor in our department.
13 February 2026
Continuity of the Lyapunov exponent for cocycles over subshifts of finite type.
The Lyapunov exponent is a central concept in dynamical systems. In particular, we are often interested in when the Lyapunov exponent is stable under perturbation. This talk will provide background and discuss some techniques we can use to analyze the continuity of the Lyapunov exponent. This is in preparation for my oral exam.
6 February 2026
The Energetic Variational Inference Method and Some Related Topics
I will talk about the energetic variational inference method and some extensions of it for my current research. We will also talk about some other fun numerical algorithms that come up as time permits.
30 January 2026
Determining numbers of Praeger-Xu graphs
Let G = (V, E) be a finite simple graph. One might say that G is "very symmetric," if the order of Aut(G) is very large relative to the cardinality of V. Alternatively, we can quantify the symmetry of G with more interesting parameters. A subset S of V is a determining set if the action on V of any automorphism of G is determined by its action on just S. The determining number of G is the minimum cardinality of a determining set of G. The Praeger-Xu graphs PX(n, k) are a family of cool-looking graphs, and in this talk, we compute their determining numbers. If there's time, we'll discuss other interesting symmetry parameters. This talk is based on the paper Determining and distinguishing numbers of Praeger-Xu graphs by Sally Cockburn and myself.
23 January 2026
Higgs bundles, opers and conformal limits.
In 2014 Davide Gaiotto conjectured a connection between two different structures on a Riemann surface: Higgs bundles and opers. We will discuss these objects along with the outline of the proof of this conjecture in a slightly different setting. This is in preparation for my oral exam.
16 January 2026
Reading Course Chronicles
A panel of second-year graduate students recount their reading course experiences. The speakers will share both math they learned and what it was like to work with each professor.
9 January 2026
Intro to Biological Circuits
I aim to introduce the fundamentals of Biological circuits towards the direction of my specific research problem.This talk should be approachable to anyone with a background in calculus, some differential equations (particularly ODEs/bifurcations) would be good, but the latter materials are not absolutely necessary. I’ll start by covering basic biological circuit design principles or motifs, beginning with autoregulation (negative and positive) and delving into their effects on expression (response times, bistability, hysteresis). Furthermore, we’ll discuss more general feedforwards (coherent, incoherent) and negative feedbacks (oscillatory behavior). Lastly we’ll cover time scaling and delays. If time permits, I’ll show how I use each of these in my own research.
Fall 2025
5 December 2025
Introduction to Ergodic Schrodinger Operators
We will introduce Ergodic Schrodinger Operators and discuss how math physics, spectral theory, dynamical systems, and functional analysis play a role. In particular, we discuss how uniformly positive Lyapunov exponents can play a central role in a physical phenomenon Anderson Localization.
21 November 2025
[It's a surprise!]
You'll just have to come and find out!
14 November 2025
Escher: Geometry, art, and tiling
I want to talk about Escher's interest in hyperbolic geometry and how he came up with inspiration for tiling the space with art. Also I want to talk about fun and interesting tilings like the Einstein tile.
7 November 2025
What happens in the boundary layer?
What happens in the boundary layer for solutions to the incompressible Navier-Stokes equations, asymptotically for small viscosity (high Reynolds number), is not generally known. Heuristic and numerical methods have been proposed and, in special circumstances, verified. I will discuss aspects of the history of this problem, and demonstrate a connection between the work of Chorin (with Marsden and others) in the 1970s, Kato in the 1980s, and Maekawa and others over the past decade or so. This talk is based on joint work with Prof. Jim Kelliher.
31 October 2025
The Big C Seminar: A crash course in ergodic theory and hyperbolic dynamics.
As a part of The Big C Seminar, this two-part talk will serve as an introduction to the 2025 Victor Shapiro Distinguished Lecture, delivered by Professor Amie Wilkinson. With a gentle introduction to ergodic theory and (partially) hyperbolic dynamics, the talk will provide relevant definitions, examples, and theorems so we can better engage with the Shapiro Lecture.
24 October 2025
A Very Brief Introduction to Diffusion Models
In recent years, diffusion models have become an extremely popular type of generative model. In this talk, we will discuss the basic mathematical principles behind diffusion models. We will also discuss some recent scientific and engineering applications.
17 October 2025
On the hyperkahler nature of the fundamental group
The fundamental group is one of the most basic topological invariants of a space. However, it also encodes a list of flat structures. Hitchin, in his seminal work, by studying a special class of solutions of the self-dual Yang-Mills equations, showed the correspondence between the space of solutions and the moduli space of certain holomorphic/harmonic bundles, called Higgs bundles. On the other side of story, The correspondence between representations and Higgs bundles can be viewed as a Hodge theorem for non-abelian cohomology. This powerful machinery was later extended by Simpson to give a uniformization theorem for kahler manifolds.
10 October 2025
Lightning Talks
A series of quick pitches from graduate students across the department sharing their research and experiences in the Ph.D. program.
3 October 2025
Brownian Motion in Financial Markets
I derive Brownian motion as a scaling limit of a random walk, the Black-Scholes model from Ito's Lemma, discuss Donsker's Invariance Principle applied to financial markets, and speculate about theoretical approaches to studying market microstructure. That said, I couldn't decide what to talk about so the topic is tentative; every talk I give is improvised so I am always happy to change topics per audience request. Please ask questions and heckle me. Topics I can talk about: My dissertation research on a categorical framework for classical mechanical physics; a theory for a co-inclusion exclusion principle and how that shows up in my dissertation work and in reinforcement learning problems; how to get a job in quant finance (except I don't have one yet so what do I know).
The UC Riverside Math Graduate Student Seminar (GSS) is brought to you by the UCR student chapter of the American Mathematics Society. GSS is organized by the officers of the chapter.
Officers
Chris Grossack, President
David Weed, Vice President
Cynthia Nguyen, Secretary
Thoon Nguyen, Treasurer