## Spring 2021

### Bialgebra

Algebra is about forming structures: an operation A x A => A combines two parts into one whole.

Coalgebra is about observing behaviors: a co-operation A => A + A splits one thing into two possibilities.

Bialgebra is their synthesis: in practice, we live in complex systems of interacting parts. A distributive law SB => BS intertwines an algebra and a coalgebra by expressing "given the behaviors of the parts, here's how to infer the behavior of the whole".

We discuss a few examples of coalgebra and then move on to bialgebra, giving just enough definitions to support the intuition. We explore two main applications: programming languages (combining how to build programs and how they behave in a network) and open dynamical systems.

### Stability for Current Density Impedance Imaging and Uniqueness for the Inverse Sturm-Liouville Problem

In a joint effort with my advisor, we studied stability of reconstruction in current density impedance imaging (CDII), that is, the inverse problem of recovering the conductivity of a body from the measurement of the magnitude of the current density vector field in the interior of the object. Our results show that CDII is stable with respect to both errors in interior measurements of the current density vector field as well as errors in voltage potential along the boundary and confirm the stability of reconstruction which was previously observed in numerical simulations, and was long believed to be the case.

Additionally, we study the Inverse Sturm-Liouville problem which is the problem of reconstructing the coefficient function $$q$$ from the second order elliptic differential operator $$-\nabla + q$$ using the boundary spectral data. While there are several results in one dimension and higher dimensions using complete spectral data and even finitely many terms omitted, none have explored results for a subsequence of spectral data. We aim to establish such results in one dimension and higher dimensions by using the asymptotic behavior of eigenfunctions on the boundary.

### Blowing Up Planes

In this talk we will learn how to blow...[KNOCK KNOCK KNOCK* FBI open up!]...That's strange. They must have the wrong door. Let me go see what they want...[indistinct accusatory conversation]...I'm sorry officers, there's been a misunderstanding!. Those aren't the types of planes I'm talking about; I'm referring to two-dimensional affine space. You see, a lot of theorems in algebraic geometry fail for singular varieties so we have come up with a way to resolve those singularities in a way that preserves the geometry of the variety as much as possible. Consider an affine curve on a plane with a nodal singularity where it folds over itself. We can resolve that singularity by unraveling the node and stretching the curve into higher dimensional space. We call this the blowing-up or the curve at its singular point. It's a completely harmless operation with no reported casualties so far. Just come to my talk on Friday and I'll clear everything up. I'm sure we'll all be laughing about this soon enough.

### On Resurgent Analysis of Explicit Formulae in Fractal Geometry

This work concerns the intermixing of fractal geometry and resurgent asymptotics. We focus on explicit formulae in fractal geometry that describe geometric or spectral quantities through singularities of an associated fractal zeta function. These formulae are known to contain information about the asymptotics and oscillations of the fractal's geometry, but may not be complete descriptions for certain fractals or generalized fractal strings having singular behavior. Namely, it is expected that some of these expansions will contain exponentially small or non-perturbative effects. We discuss the use of Borel-Écalle resummation and transseriation to recover such information in this context.

### Correspondence Between the Spectrum of Jacobi Operator and Dominated Splitting of its Cocycle Map

In this talk we show that one can identify the spectrum of Jacobi operators by those energies, whose cocycle map does not admit dominated splitting. This result extends a well-known Johnson's theorem for Schrodinger operators, which identifies the spectrum of Schrodinger operators by all energy values, such that the cocycle map is not uniformly hyperbolic. Jacobi operator is a natural extension to Schrodinger operator, and dominated splitting in some sense generalizes the notion of uniform hyperbolicity. The difference lies in the fact that for Jacobi operators, the cocycle map can be singular, i.e. have zero determinant.

### An Introduction to Machine Learning and Neural Networks

Artificial intelligence, machine learning, neural networks, deep learning -- so many buzzwords! But what do they all mean?! When I search online for answers (and not just on math stack exchange) I often find that the majority of explanations either give you a bunch of deep theory without any realistic applications or it's all about the code and fails to provide any understanding. In this talk I'll attempt to give a fairly basic introduction to ML and neural nets that lies somewhere in between, and probably somehow manage to give you neither application nor theory.

## Winter 2021

### F_lling _n Mis_ing D_ta: Reed-Solomon Error Correction & Coding Theory

It’s a classic sitcom trope: the telephone is cutting out, one character says something, another mishears, and hijinks ensue. Though comedic, the problem is very real. In the modern world, the vast quantities of information being communicated at all hours of the day are naturally susceptible to errors introduced by the environment. Luckily, we can construct safeguards against the sorts of mistakes that plague our favorite television characters. In this talk we discuss the applications of coding theory to our everyday lives and examine a particular class of algebraic codes devised by Reed and Solomon.

### Problem Solving Without Ansibles: An Introduction to Communication Complexity

In the world of Science Fiction, an "ansible" is a device that allows for faster-than-light communication. Without ansibles, interstellar travel puts an interesting constraint on computation. If two planets want to collaborate on solving a problem, the obstruction will likely not be the computation that either planet does individually. Instead, what matters is the *Communication Complexity* which tracks the amount of messages the planets have to send each other to solve their problem. In this talk we will solve a prototypical problem in communication complexity. But be warned: the answer may surprise you!

### An Introduction to Orderable Groups

We call a group (left-)orderable when it admits a left-invariant total order. This simple structure leads to a rich theory with ties to dynamics and topology. In this talk, I'll discuss some examples, useful machinery, and the space of left-orderings.

### MathPeople Study Guide Suite

For the latter half of my graduate school career I have been playing around with the mechanics of building websites. I also had the fortune to lead the complex qual prep seminar over the past two summers. These two endeavors unavoidably overlapped and I made a website to host the material in the seminar. Naturally I wanted to abstract this project and it has since turned itself into a full fledged (though perhaps not yet finished) web API for integrating study guides into websites. I want to share this project with you fellow educators in the hopes that it may be of interest if you ever want to compile, for yourself or your students, an electronic version of a stack of notecards which has built-in organizational tools and practice test generation. The project is on github at https://github.com/MathPeople/MathPeople.github.io and I will be introducing it and showing how it works in GSS this week.

### Delta Sets of Nonminimally Generated Numerical Monoids

A numerical monoid is a subset of $$(\mathbb{N}_0,+)$$ that is closed under addition, and we can factor elements of a numerical monoid into a monoid's generating elements, which form a generating set. These factorizations are not unique, and invariants such as elasticity and the delta set of these factorizations of numerical monoids have been previously studied. When a non-irreducible element is included in the generating set, the structure of factorizations and factorization invariants may be altered. In minimal geometric generating sets of the form $$\{a^n, a^{n-1}b, ..., ab^{n-1}, b^n\}$$, the delta set is simply $$\{b - a\}$$. We study the case when a non-irreducible element, labeled $$s$$, is added to the generating set. In this specialized instance, we prove several basic properties of the delta set and further characterize the delta sets for particular values of $$s$$ and $$b - a$$.

### Combatting Academic Distancing during the times of Physical Distance (and normal times too)

We will explore what we personally deem to be participation from students, and why we desperately need it as educators. After getting to know what we want and what we don't by way of our classrooms, we can think about various tools to add more of what we like to our everyday discussions. As a talk on participation, this talk will be given via jamboard be prepared to be interactive!

### Why Think? Letting Computers Do Math For Us

There are a number of results in logic of the form "If your question can be asked a certain way, then there is an algorithm which tells you the answer". These are called Decidability Theorems, because an algorithm can Decide if a proposition is true or false. In this talk we will survey some results in this area, discuss a few techniques for proving these results, and discuss some open problems are only a few computer cycles away from being solved.

### Resurgence & Fractals

In this talk, we shall introduce two modern disciplines of mathematics and discuss current work in intermixing them. Namely, we will give an introduction to the subjects of resurgent asymptotics and to explicit formulae in fractal geometry. The first is a study of asymptotic analysis arising from strengthening Borel resummation. Next, the explicit formulae in fractal geometry are akin to Riemann's explicit formula for the prime counting function, describing geometric and spectral properties of applicable fractals. Our aim is the better understand such formulae, including analytic continuation and studing how their asymptotics change in the complex plane.

## Fall 2020

#### 4 December 2020

Indexing is used throughout math and science -- it's how we organize things in order to analyze or operate on them. There are two ways of thinking about an I-indexed set: a function I -> Set, or a function A -> I. In the latter case, we use preimage to partition the domain into "fibers" over the elements of I. We can think of such a function as a simple example of a "dependent type": for each i in I, there is a type A_i which depends on it.

In this talk, we generalize this to categories! For each ring R, you get a whole category of R-modules: this forms an "indexed category" Mod : Ring^op -> Cat. A functor B^op -> Cat is equivalent to a fibration p : E -> B, a functor with a good notion of preimage. The fiber over each b in B are the types that depend on b, such as R-modules. But now these are categories rather than just sets, so we have morphisms which allow the fibers to interact.

We focus on the example where the fiber over each set is its power set -- this is the canonical model of higher-order predicate logic. The goal is to show that fibrations model foundations: the base category B provides the type theory, and the total category E provides a logic "over" this type theory. (Jacobs' Categorical Logic and Type Theory) I will incorporate how I am using these ideas in my research to generate logics for programming languages.

### The Milnor-Schwartz Lemma and You: Why we care about Geometric Group Theory

The Milnor-Schwartz Lemma roughly states that when a finitely generated group G acts in a nice way on a geodesic metric space X, then the group G is essentially the same as X. The relationship between the generating set of G and the metric on X provides a strong tool for asking, for example, questions about word representations of elements in G, and about the self-isometries on X. In this talk, I will describe what it means for a group to act nicely on a metric space, what it means for a group and a metric space to be essentially the same, and I'll sketch a proof of the Milnor-Schwartz Lemma.

### What does Noble even do?

Most of you have probably heard rumors that I do math. Many of you may have even heard rumors that I work on algebraic geometry. This may have raised some concerns given the well-established fact that algebraic geometry is incomprehensible. Between now and when I give this talk I will attempt to figure out what it is that I do and once I do that I will attempt to figure out how to explain it to others. Wish me luck.

### Uniqueness of Kähler structure for $$CP^n$$

In this talk, we will focus on many of the cohomological properties that Kähler manifolds enjoy and discuss exactly how formal they are. Three main examples of Kähler manifolds will be emphasized and we will primarily pay attention to complex projective space $$CP^n$$. We will also mention important uniqueness results for $$CP^n$$ by Hirzebruch-Kodaira-Yau, Kobayashi-Ochiai-Michelsohn, along with Libgober-Wood. Some proofs when n is at most 6 will be included.

### HOMFLY Braid Closures in the Annulus Become Symmetric Functions

Skein algebras are made up of equivalence classes of link diagrams on a surface where multiplication is given by stacking diagrams. Studying the skein algebra of the annulus is a good first step towards studying other skein algebras; the annulus typically embeds into other surfaces in interesting and sometimes exhaustive ways. To study the annulus, we first study HOMFLY braids and see what their closures look like. Turaev showed that the set of braid closures is isomorphic to the ring of symmetric functions. Furthermore, linear combinations of HOMFLY braids are elements of the (type A) Hecke algebra, which admits a certain family of minimal idempotents. The fact I will discuss is how closures of these idempotents correspond to Schur functions under the Turaev isomorphism. This is the topic of Sascha Lukac's PhD thesis from the early 2000's, which you can find online.

### The Polynomial Method over Finite Fields

The polynomial method is an umbrella term for using polynomials to bound the possible behavior of finite collections of geometric objects such as points, lines, and other hyperplanes with respect to some geometric relation. To understand the polynomial method over finite fields we will give a brief background to the capset theorem on $$\mathbb{F}_3^4$$ and prove it in the method of Dvir. We will also introduce the finite field kakeya theorem and prove it in the method of Ellenberg and Gijswijt.

### Fractals and Taylor Series Expansion

In this talk, I will present some preliminary results regarding the error term in the generalized Taylor series expansions for distributions or generalized functions. We will first uncover what happens when our generalized function is an actual function that is continuous up to order $$k$$. The error term in this particular case should coincide with the error term in the normal Taylor series expansion of the function.

#### 9 October 2020

Model Theory is the branch of mathematical logic that deals with "truth" in axiomatic structures. Perhaps surprisingly, Model Theory is extremely insensitive to the notion of cardinality, and we can leverage this insensitivity to prove infinite theorems by considering only finite objects. In this talk we will discuss the fundamentals of Model Theory, as well as the Logical Compactness theorem, which we will use to prove theorems in combinatorics and algebra. Time permitting, we will state the Lowenheim-Skolem Theorem, and outline some potential applications.

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

Maranda Smith, President

Jonathan Alcaraz, Vice President

Noble Williamson, Secretary

Alec Martin, Treasurer