## Undergraduate Mathematics Seminar

The Johns Hopkins Undergraduate Mathematics Seminar hosts regular talks given by Hopkins undergraduates about advanced topics in mathematics. The Fall 2018 semester marks the second semester of the undergraduate seminar. The talks will be held in **Krieger 413** on **Thursdays at 7pm**.

If you are interested in giving a talk, please submit an abstract of your talk to JHU Math Club by email.

Below is a list of talks that will be given for the Fall 2018 semester. For a list of talks given in previous semester, please take a look at our archive of talks.

## Schedule of Talks - Fall 2018

**September 20th**

*Jin Lu*:

**Fermat's Last Theorem, Regular Primes, and Unique Factorization**

**Abstract**: Fermat's last theorem (abbreviated FLT), first conjectured by Pierre de Fermat in 1637, had been one of the most famous unsolved problems in mathematics before finally being proved by Andrew Wiles nearly 358 years later. This talk will focus on the attempts of mathematicians, especially Kummer, in attempts to prove FLT during the 19th century.

More specifically, we first observe that proving the theorem is equivalent to showing that $a^{n}+b^{n}=c^{n}$ has no integer solutions for any odd prime $p$, since having a solution for a given $n$ is equivalent to having a solution to any factor of $n$. Then using the fact that $t^{p-1}= (t-1)(t-w)(t-w^{2})(t-w^{p-1})$, $w$ being a primitive $p$th root of unity and
substituting $t$ for $\frac{a}{b}$, we obtain the nice equation $(a+b)(a+bw)(a+bw^{2})(a+bw^{p-1})=c^{p}$ from the original equation $a^{p}+b^{p}=c^{p}$. Thus we can consider solutions to Fermats last Theorem by working in the ring extension $\mathbb{Z}[w]$ over $\mathbb{Z}$. It turns out that if $\mathbb{Z}[w]$ is a UFD, then it is not difficult to show that FLT holds true. Unfortunately, the nice proper
ty that $\mathbb{Z}[w]$ possesses unique factorization doesnt hold for all $w$ (For example, when w is a primitive 23rd root unity). However, the remarkable discovery was made that $\mathbb{Z}[w
]$ always uniquely factorize as a product of prime ideals! Using this, Kummer was able to show that FLT holds true for certain primes called regular primes.
Unfortunately, while irregular primes exist, Kummers work was still an important stepping stone in the proof of FLT. Moreover, unique factorization plays a huge role in nu
mber theory, and Kummers work along with other mathematicians in the 19th century heavily influenced the development of algebraic number theory. Ultimately, this talk will
sketch the results of Kummers proof of FLT for regular primes, and will also illustrate the relevance of these results in number theory today.

**Prerequisites**: some basic knowledge of abstract algebra

**September 27th**

*Ross Dempsey*:

**Sphere Packing in Eight Dimensions**

**Abstract**:
Sphere packing - the problem of placing spheres in n-dimensional
space such that their density is maximized - is notoriously difficult. It
is trivial in one dimension, tricky in two dimensions, and took nearly 400
years to resolve in three dimensions. In higher dimensions, the optimal
packing was not known until 2016, when Maryna Viazovska proved the optimal
packings in both 8 and 24 dimensions. While other sphere packing results
have required detailed casework, becoming complicated enough to demand
automated proof-checking, these new results involve a clean and elegant
combination of harmonic analysis and the theory of modular forms. We will
discuss basic results in these fields, with a particular focus on modular
forms. Topics in harmonic analysis will include the Fourier inversion
theorem and the Poisson summation formula. Topics under the umbrella of
modular forms will include lattices, elliptic curves, and L-functions. By
the end, we will have a sketch of the proof of the optimal sphere packing
in eight dimensions.

**Prerequisites**: Linear Algebra, basic familiarity with complex functions

**October 4th**

*Sungwon Kim*: Randomized Game Trees

**Abstract**:
A Game Tree is a directed graph whose nodes are positions in a game and whose branches are moves. Many zero-sum two-play
er games could be represented by a game tree(e.g. Tic-tac-toe, chess, checkers, and nim). If one knows all the values of
the end node, then one could figure out the strategy using the minimax algorithm.
However, games like chess have too many leaves to use the minimax algorithm. Because a typical small piece of a ridiculo
usly enormous game looks random, assume the end nodes to either be one or zero with probability p. This assumption revea
ls interesting natures of a k,l tree where one player chooses from k options and other chooses from l options. For examp
le, in the k,l tree there exists a single fixed probability of fairness. Also, the relationship of it with that of the k
,l is exponential. Furthermore, we will prove that if the large game of k,l tree is in ones favor, one only need to kn
ow value of log(n) nodes (n being the total number of nodes) to win the game. We will also prove that if the end nodes f
ollow a random distribution of a function from 0 to 1, similar results hold as when end nodes are either one or zero. We
will end with a brief discussion on randomized algorithm, and calculation of its upper bound.

**Prerequisites** High School Calculus

**October 11th**

*Julia Costacurta*: Control Theory

**Abstract**:
Brief Introduction to Mathematical Control Theory (and why you should care)
Abstract: Control theory is a subset of mathematics that deals with augmenting a dynamical system by introducing a controller into the system. It has been applied across all kinds of fields, from self-driving cars to the stock market to drug delivery. In this talk we will look at the strong mathematical foundation behind control theory, specifically the concept of controllability as it applies to linear autonomous systems. We will work through a few proofs relating to controllability, leading up to the proof of the controllability matrix, which allows us to tell which systems are totally controllable from their mathematical makeup. We will frame our investigation through the example of the rocket car, tying the concepts we think about to a concrete example.
Adaptation of DRP project completed with Patrick Martin

**Prerequisites**: Calculus

**October 18th - No Talk (Fall Break)**

**October 25th**

*Sydney Timmerman*: Creating Knots in 2+1 Dimensions: An Introduction to Topological Quantum Computing and Property $\textbf{F}$

**Abstract**:
Quantum computers store data in probabilistic quantum states, and use quantum mechanical phenomena like entanglement to perform computations on it. Quantum computing is good for two things: helping you work as many buzz words into a sentence as possible, and solving certain problems that are computationally difficult on classical computers, like the prime factorization of integers, in polynomial time. However, quantum computers are plagued by $\textit{decoherence}$ when the quantum states lose their crucial probabilistic nature before the computation is completed. Topological quantum computing provides one potential solution to the decoherence problem by storing data in $\textit{topological}$, or globally protected, quantum states of 2-dimensional quasi-particles (yes, fake particles) called $\textit{anyons}$. This talk will introduce the braiding of anyons as a method of computation, and modular categories as models for anyonic systems. It will provide intuition behind a proof that certain anyons can't be used to build a topological quantum computer because they have Property $\textbf{F}$, and discuss the strange physical consequences of this proof. Finally, it will demonstrate an application (who knew) of knot theory.
**Prerequisities** No physics! Basics of abstract algebra helpful, but not necessary.

**November 1st**

*Elvin Meng*: Limitless Calculus and the Foundation of Mathematics

**Abstract**:
This talk will give an introduction to an alternative formulation of analysis known as Inﬁnitesimal Analysis or Synthetic Diﬀerential Geometry (SDG). The idea is to introduce inﬁnitesimal objectsobjects that aligns with intuitive notions of the inﬁnitely smallinto the real line in such a way that, under constructivist logic, standard operations and constructions such that diﬀerentiation, integration, and Taylor series can be deﬁned (with their usual properties) without δ
formulations or limit computations. The talk will ﬁrst motivate the Kock-Lawvere Axiom that allows the existence of inﬁnitesimal objects, and then reconstruct roughly two semesters of calculus (hopefully in under 40 minutes) within this new framework. We will also provide a brief description of constructivist logic as an alternative mathematical paradigm. We will end by discussing manifolds in this framework and topos theory as the correct universe of discourse for models of this theory.
**Prerequisities** Two semesters of calculus. Knowledge of analysis, logic, and algebraic geometry is beneficial but by no means necessary.

**November 8th**

*Eliza Cohn*: The Use of Energy Storage in Low Inertia Power Systems

**Abstract**:
The current national energy consumption is dominated by fossil fuels, which are processed by generators. As we seek to include more renewable energy resources into our grid, the stability of the grid is no longer guaranteed. This lack of equilibrium is due to differences in the inertia generators being high inertia systems and renewables low to no inertia systems. This discrepancy causes frequency fluctuations that can lead to poor operations and power outages. My research project focuses on designing battery controlled inverters to be integrated into the grid to ensure stability. To show this, I begin by setting up the differential equations that govern the relationship between the output power of the generators and the frequency of the grid. I will examine the current solution in place (Droop Control) and show how its transient response can be improved. This is done by considering the proposal of two novel controllers and evaluating them in terms of efficiency and efficacy. To conclude, I will demonstrate how the chosen controller impacts the overall performance of the power grid and discuss further use of renewables in a large scale grid system.
**Prerequisities** High School Calculus

**November 15th**

*Eric Cochran*: A Brief Introduction to Category Theory - Putting the "Fun" into "Functor"

**Abstract**:
Category Theory is the unifying language of mathematics. It can be used to turn a topological problem into an algebraic one, or it can be used to understand the analogy between linear transformations between vector spaces and functions between sets. Well start off with some definitions: categories have objects which can be seen as the nouns of the category, and morphisms between the categories which are like verbs. Well then give some useful examples before quickly moving towards concepts such as isomorphisms, monomorphisms, epimorphisms, universal properties, functors, and natural transformations. Well also explore opposite categories and how every theorem in category have a duel theorem. Then well talk about (co)products, where well see a surprising relationship between the GCD of two numbers and the cartesian product of two sets. Other topics may be explored as time permits.
**Prerequisities** Knowledge of Abstract Algebra and Topology would be helpful, but not necessary

**November 29th**

*Raymond Weisbrot*: Proofs of Waring's Problem

**Abstract**:
In the year 1770, Lagrange proved the well-known four-square theorem, which states that each positive integer can be written as the sum of four squares. That same year, the British mathematician Edward Waring proposed a generalization of Lagrange's result, which states that for all natural numbers k, there is a corresponding positive integer g(k) such that every positive integer can be written as the sum of at most g(k) kth powers of natural numbers. Various cases for k were solved in the next century, and in 1909, Hilbert proved Waring's problem for all natural numbers k. Decades later, Hardy and Littlewood established bounds on g(k) and G(k), defined as the smallest positive integer such that every sufficiently large integer can be written as a sum of at most G(k) kth powers of natural numbers. This talk will be largely based on W.J. Ellison's paper on this topic, and will outline various approaches to the proof of Waring's problem, as well as similar problems in number theory.
**Prerequisities** Integral calculus, linear algebra and elementary number theory.