## Upcoming Talk

### 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.

## Winter 2021

### 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