A Variety-Focused View of Algebraic Geometry

Ethan Soloway

Avik Chakravarty

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.

A1 Milnor Numbers

Zhong Zhang

Thomas Brazelton

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.

Algebraic Topology

Angela Cai

Marielle Ong

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."

Algebraic Topology and the Mathematical Basis of Topological Data Analysis

Elena Isasi Theus

Maxine Calle

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

Causal Inference and Its Applications

Chenxi Leng

Miaoqing Yu

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.

Chaos, Fractals, Dynamics and their Real Life Applications

Tise Ogunmesa

Yi Wang

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.

Cohomology of Sheaves

Eric Myzelev

Marc Muhleisen

A cohomology theory is a way of functorially assigning algebraic invariants to spaces, which among other uses can help identify when two spaces fail to be isomorphic. In this project, we will study examples of cohomology groups, give applications to enumerative geometry, discuss a relationship between cohomology and line bundles, and/or prove a recognition theorem for affine schemes. We will start with the basics of commutative algebra and category theory by going through some of "Commutative Algebra" by Atiyah and MacDonald and "Category Theory in Context" by Emily Riehl. We will then move to "Algebraic Geometry" by Hartshorne.

Computation and Complexity with Category Theory

Ruxandra Icleanu

Julian Gould

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.

Concepts and Theories of Ricci Flow and its Consequences

Ling Xu

Jacob Van Hook

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.

Deep Learning

Boya Zeng

Leonardo Ferreira Guilhoto

This semester, we plan to focus primarily on the textbook "Deep Learning" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville. We will closely read Part II (Modern Practical Deep Networks) to gain a deeper understanding of the structure, mechanisms, and mathematical foundations of modern neural networks. After finishing Part III (Deep Learning Research) of the book, we will select a specific topic of interest to explore in depth. Tentative topics include large-scale transformer-based language models and text-to-image diffusion models.

Exploring Algebraic Geometry

Max Wang

Deependra Singh

Throughout the semester, I will be reading Algebraic Curves by William Fulton, with the goal of exploring the Riemann-Roch Theorem.

Formal Methods of Verification in the Context of Type Theory

Kaan Erdogmus

Oualid Merzouga

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.

Foundations of Differential Geometry and Topology

Eric Yu

Benjamin Keigwin

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.

Harmonic Analysis and Transforms

Ekaterina Skorniakova

Travis Leadbetter

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.

Information Theory with Focus on Quantum Entanglement

Shuyi Wang

Christopher Bailey

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.

Introduction to Model Theory

Elan Roth

Jin Wei

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.

Linear Algebraic Groups

Santiago Velazquez Iannuzzelli

Yidi Wang

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

Matroids & Combinatorial Optimization

Vibha Makam

Zoe Cooperband

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.

Modular Forms

Chenglu Wang

Souparna Purohit

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."

Optimal control theory and dynamic programming

Ayanav Roy

Hangjun He

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.

Probability Theory

Evan Qiang

Jae Choi

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."

Reductions of Elliptic Curves in Finite Fields

Quincy Alston

Jianing Yang

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.

Riemannian Manifolds

Arjun Shah

Elijah Gunther

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.

Understanding Generating Functions

Darren Zheng

Xinxuan (Jennifer) Zhang

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.

Understanding Optimization Algorithms for Graph Alignment Problem

Yuntong Fu

Xinrui Yu

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.

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Lex Giglio

Xiangrui Luo

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Mike Zhou

Andrew Kwon

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Sophie Kadan

Christopher Bailey

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Zakaria Sines

Miguel Lopez

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Yiyang Liu

Tianyue Liu

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