## Spring 2022

#### 3 June 2022

The p-Adic Fields are of interest to researchers as widely varied as number theorists, fractal analysts, and physicists. We will introduce these spaces and explore some oddities that arise from their topological structure. We will also discuss some of the analytic constructions on these spaces. Time permitting, we will also motivate studying these settings through current work.

### Moving anchor acceleration methods in minimax problems

Minimax problems, a classical problem setting within optimization, have recently gained more attention in the machine learning community due to their use in Generative Adversarial Networks (GANs) and adversarial training. A variety of algorithms exist to approach the optimal point of such problems, as do acceleration methods that improve their convergence rates. In this talk, we explore a novel acceleration technique, the so-called moving anchor, applied to a variant of the extra-gradient method. On the theoretical side, we demonstrate a comparable rate of convergence to state-of-the-art algorithms that achieve an O(1/k^{2}) rate of convergence on the squared gradient norm, with minimal assumptions. This is known to be the best rate possible in the family of first order algorithms. With optimal parameter choices, we expect our moving anchor algorithm to be faster than state-of-the-art algorithms by a constant factor. Numerical experiments illustrate the effectiveness of our moving anchor algorithm. We briefly explore a proximal version of the algorithm with some promise, and some other potential avenues of exploration.

### Macdonald polynomials and Graded characters of generalized Demazure modules of $$\mathfrak{so}_{2n}[t]$$

In recent work published by Biswal, Chari, Shereen, and Wand the authors defined a family of symmetric polynomials indexed by pairs of dominant integral weights, $$G_{\nu, \lambda}(z,q)$$ where $$z=(z_1, \cdots. z_{n+1})\in\mathbb{C}^{n+1}$$, and determined that $$G_{0, \lambda}(z,q)$$ is the graded character of a level two Demazure module for $$\mathfrak{sl}_{n+1}[t]$$. The aim is to construct analogues of these polynomials for the generalized Demazure modules for $$\mathfrak{so}_{2n}[t]$$ as they are presented by Chari, Davis, and Moruzzi. We do this by constructing modules which interpolate from this presentation and local Weyl modules. We then create short exact sequences between them to relate their graded characters. This allows us to identify coefficients in the corresponding graded characters with the coefficients in $$G_{\nu,\lambda}(z,q)$$ and create closed formulas to describe them.

### Visualizing and Understanding the 3-Sphere via the Hopf Fibration and Symmetries

The Hopf fibration: it's that example you *know* Fred is going to put on your 205c final. In this talk, I'll show you some techniques I've thought about for visualizing and understanding the 3-sphere (the domain of the Hopf fibration map), present a nifty and relevant mathematical toy, and try to give you a feeling for why Fred has spent so much time thinking about this object and where it sits in the study of the elusive class of positively curved manifolds.

We will use this example as a window into what has been one of the most powerful tools in the study of positive curvature: isometric group actions, and the decompositions (foliations, stratifications, and fiber bundle structures) they induce on manifolds as well as on their orbit spaces. So-called "symmetry techniques" have revitalized the otherwise difficult positive curvature field, and have resulted in a number of theorems that give us some clues about how the larger class of positively curved manifolds might behave. Join us for what I hope will be a talk accessible to non-geometers, and motivate interest in the mystery of positive curvature.

### On the Virtual Betti Numbers of Some Fibered 4-Manifolds

A manifold has a positive virtual first Betti number if it admits a finite cover whose first Betti number is positive. Surprisingly, a large class of 3-manifolds have this seemingly simple, but powerful property--even if their own first Betti numbers vanish. In this talk, we explore an analogous property coined virtual excessive homology for a class of 4-manifolds known as surface bundles over surfaces.

### The Complex Dimensions of Space-Filling Curves

Space-filling curves have been colloquially referred to as “fractals” since the term was coined and defined by Benoit Mandelbrot in the late 1970s. However, space-filling curves themselves do not satisfy Mandelbrot's definition, and other definitions of fractality also neglect to properly classify space-filling curves as fractals. This is due to the fact that, as sets, they are topologically simple because they fill N-dimensional space and, thus, coincide with N-dimensional space. In the 1990s through the present, Distinguished Professor Michel L. Lapidus and various collaborators have developed a more accurate definition of fractality based on a larger theory of “complex dimensions” of fractal sets. The theory has been incredibly fruitful over the years, even allowing for the Riemann hypothesis to be recast in terms of the complex dimensions of the geometric zeta function of an appropriate fractal string. Today, it has allowed the speaker to prove that a class of space-filling curves are, indeed, fractals, and to provide insight into the oscillatory properties of these curves via their complex dimensions.

This talk will motivate the study and utility of complex dimensions of fractal sets, briefly introduce the theory of space-filling curves, and then detail the construction of a collection of relative fractal drums capable of detecting the complex dimensions of a class of space-filling curves. Next, the oscillatory properties of curves in this class will be highlighted through the existence of nonreal complex dimensions, and a conjecture for a generalization of the main result to a larger class of curves will be presented.

### Bounded domains on Kobayashi hyperbolic manifolds covering compact complex manifolds

Since there is no Uniformization Theorem in several complex variables, there is a desire to classify all of the simply connected domains. We use a result of Zimmer and a localization technique of Lin and Wong to extend a result of Cheung et al. In particular, we show that if a domain with $C^{1,1}$ boundary on a Kobayashi hyperbolic complex manifold contains a totally real boundary point and covers a compact manifold, then its universal cover must be the Euclidean ball.

### Stated Skein Theory

Quantum groups (ooooo buzzwords) are a big topic in representation theory these days. However, anyone who's ever had the displeasure to do any computations with them will surely vent to you at some point about how disgusting the computations can get. (Stated) skein theory is a more topological way of thinking about these objects that somehow magically make their calculations more fun to work with. In this talk I'll discuss the basics of stated skein theory and how representation theorists use them to study quantum groups.

### Level Set Method for Modeling Cell Polarity in Pavement Cells

Mechanisms that generate the polarity of individual cells and mechanisms that coordinate tissue polarity over neighboring cells are often regarded as separate processes, and their interdependencies are poorly understood. One model system used to study this relationship is the jigsaw-puzzle-like pavement cells in Arabidopsis leaves. In this talk I will be discussing the preliminary modeling approach we are using to unravel how an initially spatially uniform signal can induce the correct polarization to produce the signature puzzle-piece-shaped cells.

### The Number e and What It Means to Me

We will discuss the number e and some of its wonderful properties. Viewed in the right way, these properties serve as links between many concepts in differential equations, linear algebra, and complex analysis. These links turn the study of differential equations from a nearly impossible task into a reasonably tractable one. This will be explored through several examples.

## Winter 2022

### Agent Based Modeling of Neural Stem Cell Migration

Neural Stem Cells (NSC’s) hold the remarkable ability to respond to chemotactic signals, pass through the blood brain barrier, and localize around tumors as well as other injury sites. This provides an exciting treatment option due to their ability to repair damage in the brain as well as deliver therapeutics directly to the site. However, the efficacy of these therapies are contingent upon enough viable cells reaching the desired locations. Since experiments for such treatments are often expensive; computational models of NCS migration are needed to optimize dosing/injection strategies. We extend the model established in Gomez et al. (2022) and fit this to experimental data in the settings of Glioblastoma and Traumatic Brain Injury. In the case of TBI the proposed model proves to be inadequate to explain the experimental data. Thus we look to further extend the model to include a repair mechanism as well as the effects of cytokine distributions on the proliferation and survival of NSCs.

### Modeling of glioma tumors in CAR T-cell treatment

Chimeric antigen receptor (CAR) T-cell based immunotherapy has shown its potential in treating blood cancers, and its application to solid tumors is currently being extensively investigated. For glioma brain tumors, various CAR T-cell targets include IL13Rα2, EGFRvIII, HER2, EphA2, GD2, B7-H3, and chlorotoxin. In our work, we are interested in developing a mathematical model of IL13Rα2 targeting CAR T-cells for treating glioma. We focus on extending the work of Kuznetsov et al. (1994) by considering the binding of multiple CAR T-cells to a single glioma cell, and the dynamics of these multi-cellular conjugates. Our model more accurately describes experimentally observed CAR T-cell killing assay data than a model which does not consider cell binding. Moreover, we derive conditions in the CAR T-cell expansion rate that determines treatment success or failure. Finally, we show that our model captures distinct CAR T-cell killing dynamics at low, medium, and high antigen receptor densities in patient-derived brain tumor cells.

### Heegaard Splittings and Dehn Surgery

Our eventual goal is to sketch a proof showing how to construct any closed orientable 3-manifold out of performing Dehn surgery on S^3. Dehn surgery is a way of decomposing, modifying, and gluing back together some 3-manifold in order to construct a different 3-manifold. In order to motivate this and understand the proper decomposition, we will first discuss Heegaard splittings. These topological constructions are closely related to knot theory, homology, and mapping class groups. There's some fun to be had decomposing these manifolds into multiple pieces and gluing them back together in interesting ways.

### In Search for Invariant Polynomials

In this talk we will try to find and classify all invariant polynomials under a specific algebraic map. This is a classic problem seen at the start of algebraic geometry, which can be translated combinatorially into a much easier to approach problem. We will introduce some notions from algebra, graph theory, and geometry to help us on our endeavor. Our discussion will be constrained to mostly 2-dimensional graphs, as the majority of this problem is still open research.

### An energy conserving Local discontinuous Galerkin Method for solving power-law beam equations

The Euler-Bernoulli Beam equation for power-law materials describes the mechanics of a material of which its stress-strain curve exhibits a power-law relationship. While this equation is commonly introduced to undergraduate engineers, its analytic general solution is an open problem. Restricting studies numerically still ends up being difficult. In this talk I will introduce the local discontinuous galerkin (LDG) method, and work done to approach this equation as well as possible future work to improve behavior.

### Solving An Easy Analysis Problem: Together

*asmr voice* Let's all take a break. Unwind. And solve a nice friendly problem in elementary analysis.

Now that I've lulled you into a false sense of security, you should know that we'll be solving this (very concrete!) problem by using heavy duty machinery from category theory and logic. In particular, we'll be using the language of topos theory. In the process, we'll see why people care about constructive mathematics, how category theory can solve real problems, and whether topos theory really is as scary as its reputation makes it out to be.

We assume no background in topos theory, or indeed anything but basic category theory.

### The infinite boundary

As far as topological spaces go, metric spaces are very nice. However, the notion of equivalence used in geometric group theory, quasi-isometry, is usually not a continuous function! In general, it is difficult to determine if two metric spaces are quasi-isometric. In this talk, I'll introduce a very powerful invariant for hyperbolic metric spaces called the visual boundary; not only does this boundary compactify the space, it is also a homeomorphism invariant for quasi-isometric spaces.

### A Glimpse into the Understanding of Cell Polarity

One of the major tasks that a cell faces during its lifecycle is how to spatially localize its components. Cell polarity is fundamental for correct cell shape, as well as carrying out essential cellular functions, such as spatial coordination of cell division. Fission yeast are a model organism to study in the attempt to better understand cell polarity. We will discuss a couple models developed to explain how fission yeast properly regulate their shape and division cycle. If time allows, potential applications of these models will be discussed.

### Divergence is only the Beginning

When Èmile Borel brought his pioneering work on the resummation of divergent series to Mittag-Leffler, the established mathematican put his hand on the complete works of his teacher, Karl Weierstrauss, and told Borel sternly in Latin, The Master forbids it." In this talk, we will delve into the very same methods that made mathematicians of yore tremble.

Borel's work essentially amounts to formally applying a term-by-term inverse Laplace transform, which divides each coefficient by a factorial. As such, it turns divergent series whose coefficients grow at most factorially into possibly convergent ones. Under suitable conditions, this new function admits a Laplace tranform to undo the first Borel transform, and this function is asymptotically equivalent to the original divergent series.

Today, we shall explore Borel's work on resumming divergent series and see that this is only the beginning of a much larger field of active research known as resurgence.

## Fall 2021

### Sites & Bytes: Website Workshop!

Bring your laptop and you’ll walk away with your own personal website! This week, we’ll walk you through making a completely free website on a platform of your choice. Options include Google Sites, Wordpress, and Github pages. We’ll help you pick the right choice and then launch it to the internet during this interactive workshop.

### Melting Snowflake Fractals

The von Koch snowflake is one of the first studied fractals and thus a great starting point for investigating fractal and spectral geometry. In this talk, we'll study how much heat flows into a fractal shape from its heated boundary. To that end, we'll discuss von Koch fractals, the heat equation, and how to find the total amount of heat which flows into the fractal. In particular, we build and solve an approximate functional equation in order to estimate the total amount of heat in the fractal after small amounts of time.

### Colors, Strings, and Beads: Categories for Children

There is a graphical language for formal category theory [Myers 2016]. The language empowers users to understand concepts and theorems with striking facility and clarity. There is also a logical interpretation: colors are types, strings are judgements, and beads are inferences. Together, category theory and logic can be taught in a simple, colorful, and engaging way. To demonstrate, we explore how the concepts of adjunction and extensions (in the category of relations) might be taught as early as elementary school.

### Boundaries at Infinity and Stable Subgroups

Given a proper, geodesic, Gromov-hyperbolic metric space X, one can construct a boundary on the space called the visual boundary. The topology of this visual boundary is an invariant under a natural equivalence for hyperbolic spaces, and has been extensively studied with respect to quasi-convex subgroups of Iso(X). There has been much recent work towards constructing an analogue to this boundary in non-hyperbolic spaces.

In this talk, I'll relive the trauma of my oral exam slides as we review some of the known properties of the visual boundary, some of the issues this boundary has with non-hyperbolic spaces, and we'll introduce a new analogue which can be used instead.

### Six Operations on Diamond Topos

We present preliminary work conjecturing six operations on the topos of a nonarchimedean object towards a new reciprocity law.

### Intro to Random Walks: From an electrifying problem to modeling forest fires

This talk is a basic introduction into random walk theory, mainly focused on the Monte Carlo method for approximations and Markov Chains for exact solutions. Throughout the study of mathematics, physics has always been a huge source of problems and starting points of inquiry. However, as we generalize mathematics, we often leave the realm of “real world” application. BUT! There is a well-known connection between random walks and electric networks, and we will get to investigate some different techniques on arriving at a solution. Furthermore, if time permits, we explore a couple different ideas of how to apply the techniques from this talk, including the forest fire model mentioned in the title.

### This is not the title of this talk

You may have heard of the birthday paradox, but what about a birthday full of paradoxes? Join me as I begin my quarter life crisis with some mathematical anomalies! Can we find the person at Getaway such that if they're drinking, everyone else is drinking? Did you know that most of your friends probably have more friends than you? Were you aware that if this claim is true, you'll get a thousand bucks just for showing up?

In this talk, we'll fill half of the whiteboard with paradoxes, and then we'll fill half of the remaining boardspace with paradoxes, and then we'll fill half of the remaining boardspace with paradoxes, and then we'll...

### Top 5 Undecidable Problems: Number 4 will shock you!

Undecidable problems are problems which are, in a certain sense, provably unsolvable. In this talk we’ll survey some undecidable problems in mathematics, both famous and surprising, and discuss how these problems witness the close ties between mathematical logic and other branches of math.

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

Will Hoffer, President

Jacob Garcia, Vice President

Alysha Toomey, Secretary

Raymond Matson, Treasurer