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.
Isabella Chittumuri (Mentor: Daniel Waxman)
From General Linear Models to Neural Networks
Khason Forde (Mentor: Kurt Butler)
Data Communication Channels and Noise
My talk will mainly be about channels, different types of channels, noise, different types of noise, and how channel coding (specifically block coding) can be used against noise.
Jakub Goclon (Mentor: Ajmain Yamin)
Euclidean Domain and Unique Factorization
"I will present about Algebraic number theory: in particular, ring theory and unique factorization in integral domains."
John Hemminger (Mentor: Kurt Butler)
A Probabilistic Introduction to Machine Learning
Hannah Lilly (Mentor: Yassin Chandran)
Knot Theory in Molecular Reactions
"Knot theory is field of topology that studies closed curves in 3D space. This presentation explores how knots are used to represent DNA and predict the functions of enzymes. Furthermore, the discussion illustrates how knot theory is applied to biological processes such as DNA replication, supercoiling, and enzymatic reactions."
Percy Martinez (Mentor: Daniel Waxman)
Fourier Analysis - The “Trick” That Keeps on Giving
"I will present an application of Fourier analysis and attempt to briefly show the theory of why each periodic function has a Fourier series."
Neena Noble (Mentor: Emilio Minichiello)
Category Theory and Electrical Circuits
"I will be talking about Category theory by presenting different types of categories. I will also relate the category GLA with electric circuits using string diagrams."
Jean Pulla (Mentor: Connor Stewart)
The Cauchy-Riemann Equations
Tahda Queer, Megan Trapanese (Mentor: Eunice Ng)
Gauss’s House Rules: An Intricate Analysis of Hypergeometric Series
Pedro Vizzarro Vallejos (Mentor: Yassin Chandran)
Basic Elements of Group Theory in Music
"A look at characteristics and sounds of two music-related groups: The T/I group and the PLR group."
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:
David Brannan, Matthew Esplen, Jeremy Gray, "Geometry"
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:
TBD
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:
David MacKay, "Information Theory, Inference and Learning Algorithms"
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:
Matt Clay, Dan Margalit, "Office Hours with a Geometric Group Theorist".
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:
Richard Schwarz, "Mostly Surfaces".
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:
Colin Adams, "The Knot Book"
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:
Loring W. Tu, "Introduction to Manifolds"
Stephen Bruce Sontz, "Principal Bundles: The Classical Case"
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:
Emily Riehl, "Category Theory in Context"
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:
Joseph H. Silverman, John Tate, "Rational Points on Elliptic Curves"
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:
Ernst Hairer, Gerhard Wanner, "Analysis by its History"
Walter Rudin. "Principles of Mathematical Analysis"
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:
Notes by Lasse Rempe
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):
Gerald B. Folland, “Fourier Analysis and Its Applications”, plus domain-specific texts based on the mentee’s interests.
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):
Andrew Gelman et. al., “Bayesian Data Analysis”
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:
Paul Pollack, "A Conversational Introduction to Algebraic Number Theory: Arithmetic Beyond Z"
Daniel A Marcus, "Number Fields"
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.