## Upcoming Talk

### 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.

## Scheduled Talks

### 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.

## Fall 2020

### 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