PhD Proposal April 2025 -- Talk Notes

PhD Proposal April 2025 -- Written Materials

Wednesday (Graduate Geometry) Seminar Talks-- 

Fall 2023 -- Representation Theory 1 -- Reducibility of Representations of Finite Groups

Fall 2023 -- Representation Theory 2 -- Tensor Representations

Spring 2024 -- Elliptic Curves 1 -- Group Structure of Elliptic Curves

Figure -- associativity of group law -- https://www.desmos.com/calculator/eaiddpk5ue

Spring 2024 -- Elliptic Curves 2 -- Moduli Space of Elliptic Curves 1

Spring 2024 -- Elliptic Curves 3 -- Moduli Space of Elliptic Curves 2

Fall 2024 -- Fermat's Christmas Theorem

Spring 2025 -- Kauffman Bracket Skein Module

Spring 2026 -- Symplectic Vector Spaces

WWCC (MSU Graduate Student Seminar) Talks--

Spring 2022 -- Pay (no) Attention to the Math Behind the Curtain 1: Where do Divisibility Rules Come From?

First, a quick intro to/review of modular arithmetic and some simple number theory and group theory notions. Then, an exploration of where the divisibility rules we all know from elementary school come from. We will prove the validity of several of the well-known divisibility tests, and finally, develop divisibility rules for some “less popular” numbers. Bring a pencil and paper, the audience will have work to do!

Fall 2022 -- The "Russian Peasant" Method for Multiplying

There are almost always multiple methods or strategies to answer a mathematical question. Sometimes, it might seem like a given strategy makes a question more complicated than it needs to be, even if it does give a valid result. This phenomenon certainly occurs in higher math, but it can even appear in simple arithmetic. A demonstration of a seemingly ridiculous method of multiplying two numbers, and a proof of its validity.

Spring 2023 -- Pay (no) Attention to the Math Behind the Curtain 2

Taking a look at some math ideas that we probably all know, but we may not have thought deeply about the why or how. We learned the properties of logarithms and the Pythagorean theorem during middle or high school, but may not have thought a lot about why these things are true or how they could be used. We will explore an application of the rules for logarithms that gives us a simple algorithm to calculate logarithms with only a four-function calculator (or a very ambitious person with pencil-and-paper). If time allows, a survey of several proofs of the Pythagorean theorem ranging in time from antiquity to the last century.

Fall 2023 -- Pay (no) Attention to the Math Behind the Curtain 3: RSA encryption

RSA encryption was introduced in 1977, and it is still widely used to secure data transmission today. This encryption scheme has proven to be quite secure, despite its rather simple formulation and publicly available mechanism. We will review some basic notions in number theory before exploring the RSA encryption algorithm and a bit about possible attacks against it.

Fall 2023 -- Wreath Product of Groups and the Rubik’s Cube Group

We will take a look at the wreath product of groups, a construction using the ideas of group actions and semi-direct products. Then we will explore the group of valid positions of a Rubik’s Cube and finally compute its order.

Spring 2024 -- Willans' Formula and What Counts as a Formula

In 1964, Willans introduced a formula for the n-th prime number. We will take a look at this formula and explain how it works. Then, a discussion of why we might not consider this to be a formula for the n-th prime number.

Spring 2024 -- The Wirtinger Presentation of Knot Groups

A knot invariant is an assignment of some kind (polynomial(s), yes/no, number, group…) to each knot which is invariant under ambient isotopy. In other words, a knot invariant is a function on the set of isotopy classes of knots. One such invariant is the knot group, the fundamental group of the complement of a knot embedded in S3. In this talk, we will construct this group for a general knot using the Seifert-Van Kampen theorem, then compute a few nice examples and observe a few simple consequences of this construction.

Fall 2024 -- Rational Points and Pythagorean Triples

Two classical problems in math are finding the rational points on a curve and finding Pythagorean triples. As it turns out, when the curve is the unit circle, these are actually the same problem. We will look at how we can find rational points on the unit circle, and how these points specify every possible Pythagorean Triple. If time allows: a fun proof of the Pythagorean Theorem.

Spring 2025 -- Nakayama’s Lemma

This talk will introduce several theorems in commutative algebra that are known (collectively or individually) as Nakayama’s Lemma. These are a sequence of results which govern the interaction of the Jacobsen radical of a ring and the finitely generated modules over that ring. We will prove these results and discuss a few corollaries which (hopefully) illustrate the utility of these results in commutative algebra and algebraic geometry.

Spring 2025 -- Proposal Prep 

In preparation for my thesis proposal talk next week, some background on algebraic geometry. Some key vocab that will appear in the talk including Spec, prime and radical ideals, vanishing sets, affine varieties, and a quick look at Hilbert’s Nullstellensatz.

Fall 2025 -- Kaprekar's Constant

Kaprekar’s Routine is given by reordering the digits of a number in two particular ways, and then subtracting the resulting numbers. Repeating this routine suggests a simple question: does this sequence of numbers converge? We will explore this question in a few small cases, and see some more general results.

Spring 2026 -- Skein Bimodules

A description of a certain skein theoretic construction for 3-manifolds with boundary and a theorem about gluing. On the way- what does it mean to take the tensor product of two functors?