Spring 2023 Projects

Below is a list of the DRP projects from Spring 2023, sorted alphabetically by the mentee’s last name. Click on an entry to see the full project description.

Observe that Q is the field of fractions of Z. Consider an elliptic curve E defined over Q. One can transform E into a curve over Z by clearing denominators and using Weierstrass normal form to create an isomorphic elliptic curve over Z. If we endow Z with the Zariski topology then prime ideals of Z can parametrize the Weierstrass form of E together with the reductions of E in finite fields. Certain properties of E can be verified by confirming those properties across the reductions of finite fields. We plan to read on this method and its generalizations for other schemes and rings.

We're planning on exploring introductory topics in algebraic topology by reading Algebraic Topology by Allen Hatcher. We begin by familiarizing ourselves with basic concepts such as homotopy, fundamental group, covering spaces, and etc. By the end of the semester, we hope to be able to use these tools to find prove algebraic invariants that classify topological spaces. Besides following Hatcher's book, we'll supplement the textbook reading with Aaron Landesman's "Notes on the Fundamental Group" and Laurentiu Maxim's "Algebraic Topology: A Comprehensive Introduction."
Over the course of this semester, we aim to study some topics in Type Theory with a focus on applications in Formal Verification. Formal Verification languages like “Coq” have been used for writing the machine-checked proofs of major theorems in various areas of Math, as well as for formally proving the correctness of software, in particular, for mission-critical systems. The research therefore also overlaps with proof theory. We have started off the semester with reading the material for coursework in the Computer Science departments, in particular with CIS 341 and 500 (Compilers and Software Foundations) that touch on Type Theory and provable correctness, and will be following up with relevant papers once the exact topic is further specified.
Graph alignment algorithms seek to find a correspondence between different topological structures. The study of such algorithms span multiple subjects and are of great interest to fields like computational geometry and computer vision.
Our project aims at understanding the notion of “infinity” from logical and set-theoretical perspectives. We begin by exploring the notion of cardinality, and use Cantor-Schröder–Bernstein theorem to prove the ordering of infinity. Then, we will explore the notion of ordinal and cardinals to deal with larger infinities and their algebraic structures. The aim of this course is to learn basic set theory and formal logic by studying infinity.
We plan to understand how to formalise concepts from theoretical computer science using category theory, with Yanofsky’s “Theoretical Computer Science for the Working Category Theorist”. We start by learning how to translate different models of computation in categorical definitions, and how these models are linked together. We then hope to dive into computability and complexity theory.
Over the course of the semester, we hope to study the mathematical underpinnings of TDA by studying algebraic topology. We will use a more theoretical approach than last semester, looking at singular homology, CW complexes, homological algebra, cohomology, and Poincare duality. On the more applied side we will go over some specialized ideas in persistent homology. We plan to follow a combination of “Algebraic Topology” by Hatcher and MIT's Algebraic Topology I notes by Prof. Haynes Miller. The applied ideas will mostly come from Dr. Vidit Nanda's notes on Computational Applied Topology.
In this talk, we will give an overview of the mathematical tools needed to analyze quantum mechanical systems. Specifically, we will explore the properties of harmonic oscillators with Lie algebras and groups. We will delve into examples of these structures in building our model, which will include the Heisenberg algebra, Heisenberg group, and coherent states. Each of these examples provides information about our system in the form of position and momentum and their evolution over time — allowing physicists to effectively model the behavior of many larger physical systems.
We hope to study the field of causal inference over the semester. We will start with the two main approaches in causal inference, the potential outcome model and causal graph, which are used for estimation and identification, respectively. After studying the foundations, we will dive into methods for solving those two problems. We also aim to study at least one up-to-date topic, like machine learning in casual inference, or its real-world applications in tech companies. We plan to use Brady Neal’s lecture notes "Introduction to Causal Inference from a Machine Learning Perspective" as a reference, supplemented by some selected research papers.

During this semester, we will continue to study topics in symplectic geometry by reading and solving practice problems from the lecture notes by Ana Cannas da Silva. We will focus on fixed point theorems and recurrence theorems, especially the Poincaré Recurrence Theorem and the physical example of the game of billiards.

Over the semester, we hope to delve into understanding more about matroids and how they are used in combinatorial optimization. We will start by reading introductory papers about matroids as well as combinatorial optimization to gain a better understanding. We are still exploring what main references to use for the project.
Category theory generalizes various mathematical constructions and phenomena from different areas. Over the course of this semester we studied the fundamentals of category theory, including the yoneda lemma, limits/colimits, and adjunctions, and looked at how these concepts help distill the essential properties of many objects studied in abstract algebra, topology and graph theory. We followed "Category Theory in Context" by Emily Riehl.

See the final presentation slides.

The project will focus mainly on providing a summary of chaos theory, dynamics, and fractals. Then, real life examples ranging from fun and silly to significant and consequential will be touched upon. Thus, the project will focus on Chapters 1, 2, 5, 9, and 11 (subject to change) of the textbook: Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz.

Final presentation slides

The goal of this project is to get become familiar with the most important results in measure-theoretic probability theory such as the Law of Large Numbers and the Central Limit Theorem. We will follow Durrett’s “Probability: Theory and Examples” and Heil’s “Introduction to Real Analysis.”
The primary reading for this semester is Model Theory: An Introduction by David Marker. We hope to read about the foundations of Model Theory as a mathematical topic rather than as the bedrock of mathematical logic. We also plan on searching deeper into the pure side of model theory rather than exploring any of its applications, allowing us to begin working on open questions in the field. Our goal is to work on problems specific to model theory within the realm of mathematical logic and aim to make progress in an open problem.

Final presentation slides

Over the semester, we hope to learn the basics of optimal control theory and use dynamic programming to solve the problem. We will first begin with linear time optimal control and talk about the Pontryagin maximum principle. Then, we will learn dynamic programming – a widely used approach to solve both linear and nonlinear control. Some interesting applications in financial economics will be introduced if time permits.

Final presentation slides

We are going through Lee’s book for Riemannian Manifolds and are going to eventually go more in depth into a topic as we progress. Riemannian manifolds and their relation to topology is one direction.
This semester, we will be diving into a study of harmonic analysis, transforms, and their applications in electrical engineering and robotics. We will start with reading several chapters in “Applied Analysis” by Cornelius Lanczos, and then we will move onto a more involved text within a more specific topic.

We plan to study some fundamental topics in algebraic geometry with a particular emphasis on varieties. We are beginning by studying affine varieties, their inherent connections to both Algebra and Geometry via the Nullstellensatz, and the Zariski Topology defined on them. From there we will spend some time studying the sheaf of regular functions and morphisms between varieties, before defining varieties generally separate from an affine context. Time permitting, we plan to study specific varieties including projective varieties and Grassmanians. Additional topics include birational maps and an introduction to schemes. The fundamental goal is to understand varieties as objects and the type of morphisms relating them before diving into motivated examples and specific uses.

 

We plan to use Andreas Gathmann's Notes on Algebraic Geometry as a guide, with other textbooks such as Algebraic Geometry by Hartshorne and Commutative Algebra by Atiyah and Macdonald as supplementary reading.

Final presentation slides

We are going to look at linear algebraic groups and look at their structure theory from Humphrey’s and Springer’s books.

Final presentation slides

According to Martin Eichler, there are five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. We will begin with studying some basics of L-functions, modular forms, and Hecke operators. Then, we will read some elliptic curves, and finally spend the rest of the semester on examining the connections between modular forms, elliptic curves, and number theory. We plan to follow Chapter 7 of Serre’s “A Course In Arithmetic” and Silverman’s “The Arithmetic of Elliptic Curves.”
Throughout the semester, I will be reading Algebraic Curves by William Fulton, with the goal of exploring the Riemann-Roch Theorem.
Throughout this semester, we hope to study the building blocks of information theory and their specific uses in applications of quantum entanglement. We will start with basic concepts in information theory, such as entropy, statistical mechanics, and conservation of information. As I develop a solid understanding of this field, we will then dive into its interface with quantum mechanics, with the goal of investigating the transmission of mutual information between entangled particles. Finally, we will look into the applications of quantum entanglement in communications, computations, and radar systems.
Throughout the semester, we will study Hopper's book "The Ricci Flow in Riemannian Geometry". We will first cover the basics of Ricci flow, including the background material in Riemannian geometry, and then explore the short-time existence and uniqueness of solutions to the flow equation, as well as the stability of solutions under perturbations. In the rest of the semester, we hope to investigate the geometric and topological consequences of the Ricci flow, including the behavior of the curvature tensor, the formation of singularities, and the implications for the topology of the manifold. If we have time, we also hope to discuss the relationship between Ricci Flow and other curvature flows such as mean curvature flow. Final presentation slides.
In this project, we will cover the foundational material in point-set topology to begin talking about smooth manifolds. Roughly aiming to work through the first few chapters and Appendix A of John M. Lee’s “Introduction to Smooth Manifolds,” we intend to cover some of the theorems traditionally stated in calculus, and see them in the more general setting of smooth manifolds. The principal aim is to see how one can treat spaces that are locally familiar, i.e. locally Euclidean, in a way similar to the way one thinks about multivariable calculus.

This semester, we plan to focus primarily on the textbook "Deep Learning" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville. 

In recent years, Deep Learning-based algorithms have been widely used in numerous life-
critical applications, such as autonomous driving and medical imaging. Therefore, it is important
for us to quantify the uncertainties of these algorithms in a provable way. Conformal Prediction
has emerged as a powerful tool to generate, instead of a single prediction, highly probable
output sets with coverage guarantees. In my presentation, I will explain the basic intuitions and
workflows of Conformal Prediction, outline the proof for its theoretical guarantees, and briefly
discuss variants and applications of this method to different settings.

This semester, we will start with the classical story of the Milnor number and its role in understanding singularities of complex hypersurfaces. While the classical Milnor number at a singularity of a complex hypersurface could be expressed as the topological degree of the gradient of the defining polynomial, there is a similar story in the world of A1-homotopy theory: the Milnor form of a polynomial defined on an affine space at a singularity point is exactly given by the local motivic degree of the gradient of this polynomial. We will walk through these theories carefully and work on some fun problems if time allows.

Final presentation slides

The goal of this project is to see the widespread use of generating functions in counting particularly difficult sequences that may not have a closed form. In many cases, the generating functions captures a lot of information. We will begin the semester by studying general combinatorial principles and tools using “The Art of Counting” by Bruce Sagan. Afterwards, we will focus more on generating functions and specific examples of their use in combinatorics, such as rook polynomials. We seek to understand generating functions more rigorously and how to manipulate them. After a few chapters, we will switch to select sections of Enumerative Combinatorics, Vol. 1 by Richard Stanley.
This is a presentation on the zeros of recurrent sequences using p-adic analysis. We will begin with a discussion on valuations and complete fields, and move on to defining the p-adic numbers. Then, we will discuss tools such as Hansel’s lemma and Strassmann’s theorem, and apply them to recurrent sequences.