## Undergraduate Mathematics Seminar

The Johns Hopkins Undergraduate Mathematics Seminar hosts regular talks given by Hopkins undergraduates about advanced topics in mathematics. The Fall 2019 semester marks the fourth 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 2019 semester. For a list of talks given in previous semester, please take a look at our archive of talks.

## Schedule of Talks - Spring 2020

**February 20th**

William Berardoni:**Classifying 3D Rotations in N-Dimensions using Representation Theory**

**Abstract:**A quick intro to the Representation Theory of Lie Groups and Lie Algebras, we will walk through what a Representation is and how we can classify all of the irreducible representations of the Lie Algebra so(3,C). If you've ever wondered how a 3-Dimensional complex spherical symmetry could appear in 11-dimensions, this is your talk! Knowledge of basic group theory and linear algebra will be assumed, but otherwise most concepts will be introduced as necessary.

**February 27th**

Casey Weiner:**Exploring Solutions to Higher Order PDEs**

**Abstract:**About 100 years after calculus was conceived in the 17th century, mathematicians around the world began to formulate solutions to many of the famous partial differential equations (PDEs) we know of today. Describing every kind of phenomena in the natural world, from light and sound to blood flow and earthquakes, the beautiful relations we have unearthed speak a great deal about the process of solving PDEs as well as the nature of mathematical progress today. The strategies we have created for solvable PDEs led to insight in understanding differential equations we cannot find a closed form or even general solution for. This talk will discuss methods for solving some higher order PDEs, the nature of those solutions, and how these relate to PDEs we are tackling in the present.

**Prerequisites**: Calculus of a single variable, ordinary differential equations, multivariable calculus would also be helpful

**March 12th**

Sonny Yi: **Hilbert Class Field of Imaginary Quadratic Fields with Class Number One**

**Abstract: **The Hilbert class field of a number field $K$ is the maximal abelian unramified extension of $K$. It is considered as the extension of $K$ with ``nicest" properties and algebraic implications.
In this talk, I will introduce the simplest Hilbert class field, which is Hilbert class field of imaginary quadratic field $K$ with class number one. There are in total only 13 of such $K$. We will introduce an analytical method for constructing the Hilbert class field for them.
The goal of this talk is to give the audience a general idea of Hilbert class field and a very naive introduction to class field theory. I will start the talk from scratch. Very explicit and approachable examples are to be provided throughout the talk, so please don't worry if you haven't taken Galois theory or Algebraic Number Theory yet.

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

## Schedule of Talks - Fall 2019

**September 26th**

*Elvin Meng*:

**Compositional Distributional Semantics: Modeling Natural language Meaning with Vector Spaces and Categories**

**Abstract:**This talk will introduce the theory of compositional distributional semantics, which hopes to use linguistic theory to combine the meaning of individual words into the meaning of full sentences in a computational setting. We will begin by introducing a simple theory of syntax using pregroups, and then, in steps, upgrade it to a strictly monodical category equipped with a Frobenius algebra. The category theory will be approached through string diagrams. Moving into a categorical setting allows us to construct the Transfer functor that carries syntactic data into the category of real vector spaces, so that syntactic derivations of a sentence become semantic computations. I will try to limit my examples to the English language. No knowledge of linguistics, natural language processing, or category theory is required, although knowledge of tensors, or at least vector spaces, will be helpful.

**Prerequisites**: some basic knowledge of vector spaces and tensors would be helpful.

**October 3rd**

*Julia Costacurta*:

**Partial Differential Equations (PDEs)**

**Abstract:**Partial Differential Equations (PDEs) are very important in explaining and modeling natural phenomena, but it's often difficult to produce analytical solutions for these problems. Thus, we are often forced to rely on numerical approximations to solutions of PDEs.
The two most common numerical methods, Finite Element and Finite Difference, can be ineffective on irregular domains which are common in biological problems.
The goal of this work is to implement an alternative numerical method for solving PDEs that overcomes the difficulty introduced by irregular boundaries by involving stochastic processes. In addition, we have involved machine learning methods, like temporal difference learning, to increase the performance of our code.
No prior experience in PDEs/ML needed!
Joint work with: Cameron Martin (Univ of Toronto), Hongyuan Zhang (Grinnell College)
Advisors: Adam Stinchcombe and Mihai Nica (Univ of Toronto)

**October 10th**

*Emily Quinan*:

**Graph Colorings**

**Abstract:** Mathematical graphs, made up of vertices connected by edges, can be used to model networks in many disciplines. Eulers paper on the Seven Bridges of Königsberg in 1736 is considered the first graph theory paper, and in the last century the subject has expanded in many different directions. This talk will focus on graph colorings, a way of labeling vertices and edges according to a variety of rules. In applications, colorings can help solve scheduling problems, register allocation, and even Sudoku puzzles. Starting from the basics of graph theory, this talk will introduce several notions of graph coloring. We will begin by defining a proper coloring of vertices and edges, and introduce some interesting graphs such as the Kneser Graph and the Mycielski Construction. In addition, we will explore several different graph colorings, including list coloring and fractional coloring. We will develop enough tools to prove the Five-Color Theorem, the precursor to the more famous Four-Color Theorem of Planar Graphs. No prior knowledge of graph theory is necessary.
**Prerequisites**: None

**October 24th**

*Phillip Yoon*:

**Philosophy of Mathematics: What IS Mathematics?**

**Abstract:** Philosophy of mathematics is a branch of philosophy which poses the basic questions about mathematics of which no mathematicians consider in their day-to-day research: What are numbers? How are mathematical truths necessarily true? What is the relationship between numbers and sentences (propositions)? It is the study of implications, foundations, and assumptions we make about mathematics when doing mathematics. There are lots of philosophers who contributed to this topic, but we will cover some of the most crucial figures in philosophy of mathematics and their key ideas in philosophy of mathematics. Figures we will cover include Plato, Aristotle, Immanuel Kant, Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein. To make the talk less technical and more engaging, I will omit discussing figures such as Gödel, as his theorems cannot be discussed without the discussion of mathematical logic. The talks aim is to give a historical survey of the topic matter and basic familiarity with philosophical theory regarding the nature of mathematics.
**Prerequisites**: No prior knowledge in philosophy is necessary. Some basic understanding and familiarity with high school mathematics should suffice. Some familiarity in logic will be beneficial.

**October 31st**

*Zecheng Yi*:

**Isogeny Graphs and Their Compositions**

**Abstract: ** Quantum computers represent an existential threat to current techniques in cryptography.
Methods relying on supersingular isogenies are potentially resistant to quantum attacks.
The difficulty of this isogeny problem is analogous to finding a path between two vertices of an isogeny graph.
The graph over $\overline{\mathbb{F}_p}$ is a directed multi-graph, with vertices as isomorphism classes of elliptic curves and edges as isogenies with a fixed degree.
Using properties about isogenies, we prove relations between the adjacency matrices corresponding to prime degree graphs and composite degree graphs.
Composite degree isogenies are significantly harder to compute.
So, this provides a useful method for finding isogenies.
Moreover, we use a result by C. Delfs and S. Galbraith to determine the cycle types within supersingular isogeny graphs over $\mathbb{F}_p$.
Specifically, we show that isogeny graph has exactly the same shape for curves with endomorphism ring $\mathbb{Z}[\sqrt{-p}]$ and $\mathbb{Z}[\frac{1+\sqrt{-p}}{2}]$.
Using this result, we also prove that the adjacency matrices corresponding to a 2-degree and odd prime degree graph are commutative.