Fridays in Skye 284
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Social Time: 2.30-3p
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Talk: 3-3.50p
Upcoming Talk
TBD
TBD
TBD
Scheduled Talks
Spring 2025
17 April 2024
Introduction to Riemannian Hodge Theory
Riemannian Hodge Theory is a crowning achievement of 20th century mathematics, and sees use in a variety of fields of math. In this talk, we'll give a gentle introduction to the main objects and concepts, as well as go through some basic examples from undergraduate calculus. If time permits, we'll discuss how this finds footing in Kähler geometry, and the results one achieves there.
10 April 2024
Kinematic Interaction Graphs of Classical Mechanical Systems
If anybody else is bored of me talking about this I am because I unintentionally gave this talk three times already this week so I can derail and tell you how I’d build an automated trading system inside-out from scratch and an interesting financial math research question I’m curious about but haven’t had time to explore, or I can talk about my other research project on a categorical approach to the compositional generalization problem in deep reinforcement learning (although I don’t have many results yet). A classical mechanical (CM) system is specified by a collection of point-mass particles, called \textbf{actors} mediated by a collection of interactions between pairs of actors and form an \textbf{interaction graph}. Joint configuration spaces of pairwise interacting actors may be restrictions from the product of configuration spaces of actors, kinematically encoded by \textbf{constraints}. When a CM system has feedback loops, the configuration space of the total system may fail to be a manifold without specific conditions between actors and constraints. The framework classifies lower kinematic pairs as joints in linkages and establishes a universal joint must constructed from at least three actors as a lower kinematic system. Future work will reinterpret prior results composing dynamics in open CM systems under this generalized kinematic framework.
Winter 2025
07 March 2024
Infinitesimal Rigidity of $G$-Cyclic Surfaces
Something something Higgs bundle, something something cyclic, something something Nonabelian Hodge, something something Lie algebra, something something doesn’t move. At the end, I showed that derivative equals to 0 means constant.
28 February 2025
Keloid and Skin Aging Agent-Based Models
Digital skin models can be a useful resource to assist biologists in uncovering regulating mechanisms of various skin conditions. This presentation focuses on 2 different agent-based skin models – one on keloid scars and another on aging skin. Keloids are debilitating skin scarring disorders, triggered by an aberrant wound healing program and exhibit continuously spreading growth. Recent single cell RNA-sequencing experiments reveal heterogeneity in human keloid fibroblasts, and interactions among various sub-clusters of fibroblasts together with interactions with immune cells might be key factors regulating the keloid propagation. We developed an agent-based model to investigate the propagative dynamics of keloids, where single cell data inferred cell communications are implemented into the model. Using the model, we propose the potential regulating mechanisms behind keloid propagation. There are many environmental factors that result in aged skin such as the sun’s UV rays or air pollution. Human skin takes decades to age so using one person’s young and aged skin in an experiment is not feasible. Although mice skin ages in just 18 months, there are too many differences in their structure to obtain human skin aging mechanisms. Current digital skin models only look at the skin without paying attention to individual cells or they model individual cells, but only for one layer of skin. We are developing a digital human skin agent-based model, which models cell interactions in both layers of skin, the epidermis and the dermis. In this presentation, I will show our preliminary results from the skin model.
21 February 2025
Introduction to Ramsey Theory
Ramsey Theory is a combinatorial field which discusses colorings of certain countable structures and monochromatic subcolorings. The field was started by Ramsey's theorem, which applied to graphs, but has since expanded to a general categorical notion. We will discuss the basics of graphical Ramsey Theory and talk about generalizations to other categories.
14 February 2025
How to rank? From Choice to Statistics
I want to cover the problem of how to rank/order a set, in this particular context I want to go over some of the ranking methods applicable to the game of Go: from traditional ones, different servers and variations of ELO. I also want to talk about how this problem of ordering is related to the axiom of choice and what could be the issue for using this with go players.
7 February 2025
The Magic Square of Squares (and Other Ways to Avoid Homework*)
Let’s ignore midterms and take a stroll through the mathematical landscape with an infamous puzzle. The Magic Square of Squares asks whether a 3×3 grid can be filled with nine distinct perfect squares so that every row, column, and diagonal sums to the same total. Having already stolen thousands of hours from distracted mathematicians, the puzzle will claim another 45 minutes this Friday. We'll meander from Euler's letters, to Lunar arithmetic, visual proofs, hexagonal tessellations, Pythagorean triples, inversive tri-sums, and birational raycasting -- glimpsing the foothills of Algebraic Geometry along the way. (*Those grad students who have transcended the mortal duty of homework may be assured: this puzzle is also an effective deterrent of research authorship, journal readings, and thesis planning).
31 January 2025
A large deviation theorem for C^1 monotone potentials generated by the doubling map
A physical phenomenon, Anderson localization, was discovered by Phillip Anderson in 1958, but it was only in the 2000's that Bourgain-Goldstein found that a uniform large deviation theorem combined with positive lyapunov exponents were strong indicators of localization. We will give an introduction to the setting to study this phenomenon, which revolves around the dynamics associated with the eigenvalue equation of the Schrodinger operator. We will then discuss the main goal of the project and ideas of the proof in the setting of iid generated potentials. By the end, we will be able to compare the iid case and the doubling map case to understand the complications one will have to face. We will conclude with future directions if this project were to be completed. This talk is in preparation for my oral exam.
Fall 2024
6 December 2023
Topological Data Analysis: How to apply topology
There is often the question in pure mathematics of when is this ever useful. I hope to provide one example of where very abstract seeming ideas are pivotal to real world scenarios. In particular I will discuss persistent homology in regards to topological data analysis. This talk will be accessible to anyone in the program as all definitions will be provided and no prior knowledge is really needed. (If you have always wondered why topology matters this is a good example to see that.)
22 November 2024
Nonstandard Calculus
In this talk, we'll explore the world of analysis without any of the pesky (ε, δ)-definition of limits and instead in the language of the hyperreals. As a detour, in the talk, we'll also explore the "shadowy" techniques of umbral calculus as yet another formulation of the subject we all know and love.
15 November 2024
The Subdivision of the Orbit Category and the Link Orbit Category (for Z/2Z)
The orbit category and the link orbit category are categories that keep track of the equivariant and isovariant maps, respectively, between the quotient sets of G (when they are viewed as G-sets). In this talk, I will outline the proof of my first tiny result concerning the weak equivalence of the subdivision of the orbit category and the link orbit category for G=Z/2Z. (This is practice for my oral in December, so all critiques and advice are much appreciated!)
08 November 2024
Euler characteristic, a simple but useful number
For this talk, I want to cover the Euler Characteristic from its more simple definitions to some applications of it. I will review the topological perspective, and how it appears in other contexts such as differential equations, algebra, geometry, topological data analysis, and even measure theory! The usefulness brings a light on how invariants are useful throughout their simplicity!
Nov 1 2024
Mostow rigidity theorem
Consider for a moment this fairy tale scenario: any two Riemannian manifolds with isomorphic fundamental groups are isometric. In this imaginary world, topological invariants will be the same as geometric invariants. Mostow rigidity theorem states that any two hyperbolic manifolds ,of dimension>=3, with isomorphic fundamental groups are isometric. In this talk, I will talk about Gromov’s proof of Moscow rigidity theorem and how to measure volume using homology.
Oct 25 2024
Complex Hyperbolic Structures Arising from SU(2, 1)-Higgs Bundles
In this talk I will outline the background needed to follow the main ideas of the talk I will be giving at the AMS on Saturday, then conclude the final 20 minutes with a practice of my AMS talk. The key ideas we will cover are Higgs bundles, geometric structures and then a new result about the existence of complex hyperbolic structures.
Oct 18 2024
Choose Your Own Adventure!
When I was a kid I used to love books with branching storylines, where you could flip to different pages in order to read different possible stories. I hope to give you all the same sense of interactivity by presenting a talk that branches! At the start of the seminar, I'll ask for a vote in order to decide between four possible topics: 1. applications of topos theory, 2. an introduction to condensed mathematics, 3. what are infinity categories, and why should you care?, and 4. topological quantum field theories and you. Depending on the topic, there may be other points in the talk where we will have further votes for further branches, for instance in the topos theory talk we could give applications to either algebra, analysis, or logic. I can't wait to see you there!
Oct 11 2024
Mild Solutions of the Navier-Stokes Equations
One way to construct solutions to the Navier-Stokes equations is to treat the nonlinearity as a perturbation of the linear part iteratively. Solutions constructed like this are called mild solutions. In this talk, we'll discuss how mild solutions are formally build and some fun facts about them.
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
We are sharing this role, Vice President
Jialin, Jennifer Wang, Secretary
Rahul Rajkumar, Treasurer