2022 

(Left to right): Vincent Martinez, Yassin Chandran, Hannah Lilly, Kurt Butler, Pedro Vizzarro Vallejos, Ajmain Yamin, Eunice Ng, John Hemminger, Tahda Queer, Ajith Nair, Appu Panicker

Students & Projects

Below is a list of the student/mentor groups with the titles and abstracts of their final talks.

Mentors

Below is a list of the Spring 2022 mentors and their proposed projects. 

Research Interests:

Machine Learning, Dynamical Systems, Geometry, Probability Theory

Potential Projects:

1) Geometry and the Erlangen Program

For hundreds of years, the ideas of Euclid were the gold standard of geometric theory; however by the 1800s numerous non-Euclidean notions of geometry began to pop up, including hyperbolic, projective and inversion geometries. In 1872, Felix Klein showed how to unify many of these ideas under a group-theoretic framework. Klein's ideas were elegant, powerful and paved the way for many deep topics in topology, Lie theory, differential geometry and even computer graphics. In this project, we'll learn that this subject is not only beautiful and easy to understand, but also rather useful.

Reference: 

Prerequisites: Introductory linear algebra is necessary, but some calculus and group theory would be preferred. If you've seen some topology in Euclidean space, then that's even better.

2) Dynamical and Random Systems

Given a sequence of observations, we can hypothesize value ways in which the numbers that we observe may arise. Some models are dynamical, in that they specify a function that tells you how a system evolves. The other end of the spectrum is stochastic, where the future of the system is given by a probability distribution. The most interesting models are a mix of dynamical and stochastic, and in this project we will learn about some of the most popular models for analyzing such systems. 

Reference:

Prerequisites: Some calculus, linear algebra and probability theory is greatly preferred.

3) Information Theory and Machine Learning

Historically, information theory was developed in order to determine how to encode messages for communication. In modern times, information theory appears all over applied mathematics, and in this project we investigate why. We will introduce some classical information and coding theory, and we'll use that background to more deeply understand modern machine learning and statistical methods.

Reference: 

Prerequisites: Some calculus, linear algebra and probability theory is greatly preferred.

Bug on Notes of Thurston, by Jeffrey Brock and David Dumas.

Yassin Chandran

Research Interests:

Low Dimensional Topology, Geometry

Potential Projects:

1) Geometric Group Theory

We can study the symmetries of a geometric object abstractly as a group. We can in turn, turn this group of symmetries into a geometric object itself allowing us to use geometric techniques to study algebraic questions.

Reference: 

Prerequisites: No prerequisites, but at least one proof based course is recommended. 

2) Hyperbolic Surfaces & Teichmüller Theory

Not only can we study the shape of a space, we can in fact study an associated "space of shapes". This beautiful subject can be approached from a number of view points (analysis, low dimensional topology, algebraic geometry, etc.) and incorporates a rich variety of ideas. We'll approach our study from the perspectives of hyperbolic geometry and low dimensional topology.

Reference:

Prerequisites: Calculus and some complex analysis are preferred. Even better if you've taken some courses in abstract algebra. These are optional and we can approach this subject from all levels. 

3) Knot Theory!

Description: Despite being an old area of study, many of the most exciting results in knot theory have happened in the past 30 years. There are incredibly deep connections with geometry and topology in dimensions 2 and 3, as well as applications to physics and chemistry. Knot theory remains a very active field of study. Many arguments are visual, and a lot can be done without first establishing a lot of fancy technical machinery. 

Reference: 

Prerequisites: None.

Research Interests:

Differential Geometry, Category Theory 

Potential Projects:

Differential Geometry and Bundle Theory 

We will study some introductory differential geometry with a special emphasis on fiber bundle theory. Fiber bundles are very important objects in geometry and topology as well as other areas of math. In particular the study of principal bundles is central to understanding differential geometry and many areas of mathematical physics like gauge theory. 

References:

Prerequisites: Calculus sequence, knowledge on how to write proofs.

Category Theory 

Some knowledge of category theory is vital for many areas of modern mathematics. It is the "mathematics of mathematics" and unites many viewpoints and different areas. We will work through Emily Riehl's wonderful book and take a bird's eye view of math, seeing how many different areas are actually quite similar structurally. 

References: 

Prerequisites: Knowledge on how to write proofs, some knowledge of abstract algebra. 

Ajith Nair

Research Interests:

Number theory, Representation Theory, Geometry

Potential Project:

Rational Points on Elliptic Curves

Rational points on Elliptic Curves is a fascinating topic which involves a beautiful interplay of geometry, algebra and number theory. It is part of the larger subject of Diophantine equations i.e. polynomial equations with integer solutions. Elliptic curves form an essential ingredient in the proof of Fermat's last theorem.

Briefly, an elliptic curve is an equation of the form y^2=x^3+ax+b, where a and b are coefficients in a field, say rational numbers. We are interested in finding the integer/rational solutions to this equation. As it turns out, one can consider the plane curve defined by the above equation and in this setting our goal is to find rational points on these elliptic curves. It is a remarkable fact that one can define a natural binary operation on the set of rational points which gives it an abelian group structure. 

In our project, we will try to understand the geometry and the group structure of elliptic curves and further explore how this is tied to questions in number theory.

References: 

Prerequisites: Group theory; some familiarity with abstract algebra (field theory, for example) might be useful.

Eunice Ng

Research Interests:

Analysis, Geometry

Potential Project:

Analysis by its History 

A typical calculus sequence progresses in the following way: sets, functions → limits, continuity → derivatives → integration. But historically, these subjects were discovered in reverse order*. In this project, we'll learn about the original problems that motivated the development of integral/differential calculus and differential equations. We'll also be discussing proofs of fundamental results in analysis.

References: 

Prerequisites: Calculus. Background in analysis is helpful but not necessary.

*Archimedes (~200 BCE) studied volume and surface area, Newton and Leibniz (1600s) independently invented differential calculus, Cauchy and Weierstrass (early 1800s) formalized the concept of a limit, and Cantor and Dedekind (late 1800s) established set theory

Connor Stewart 

Research Interests:

Algebra, Number Theory

Potential Project:

Intro to Complex Dynamics 

Complex dynamics studies the orbits of points in the complex plane under iterates of holomorphic maps. We will introduce the Julia and Fatou sets of a map, as well as the famous Mandelbrot set, and study their properties. If time permits, we will read a short expository paper showing that the exponential map is chaotic. 

References: 

Prerequisites: Complex analysis and a proof-based analysis course; some basic topology is useful 

Research Interests:

Machine Learning, Statistics

Potential Projects:

Fourier Analysis and Applications

Fourier analysis is a tremendously useful subject with fundamental applications in many different fields, from the study of partial differential equations, to various problems in physics, to many important functions in electrical engineering, and much much more. In this project, we’ll read through Folland’s book on the subject, while branching out towards specific applications in subjects which interest the mentee.

Reference(s): 

Prerequisites: Multivariable calculus and ordinary differential equations. Real analysis is a plus.

Bayesian Data Analysis

In classical approaches to statistics, we often create some model to represent data, then find a point estimate of its parameters, interpreting uncertainty by making various assumptions about their distributions. In Bayesian statistics, we instead find estimates of the parameters’ distributions directly. In this project, we will take a principled approach to the foundations of Bayesian inference, as well as perform data analysis ourselves in a statistical computing language.

Reference(s): 

Prerequisites: A course in statistics and probability, calculus, and linear algebra. Experience programming is a plus.

Ajmain Yamin 

Research Interests:

Number Theory

Potential Project:

Arithmetic Beyond Z 

This is a project about Algebraic Number Theory. More specifically it's about number systems that go beyond the usual integers (denoted as Z). To give an example of such an "extended number system", consider the Gaussian integers (denoted as Z[i]). It contains all numbers of the form a+bi where a,b are usual integers, and "i" is a new number whose square is equal to -1. Something interesting happens when we pass from the usual integers to the Gaussian integers. The number 5 stops being prime! See: 5 = (2+i)(2-i). This immediately raises several questions: which prime numbers in Z stop being prime in the Gaussian integers? What about other number systems such as Z[sqrt(2)]? Studying questions like these will be a major theme of this project, and the answers have important applications to natural questions about the usual integers. 

References: 

Required Prerequisites: Linear Algebra, some Abstract Algebra including basic familiarity with rings and fields. 

Optional Prerequisites: Galois Theory*, Elementary Number Theory.
*Galois Theory is essential for this reading project, but don't worry if you are unfamiliar with it. As long as you know what a field is, we can build up the necessary Galois theory as we progress. 

Organizers

Yassin Chandran, Vincent Martinez, Eunice Ng, and Ajmain Yamin.