Description. You can get some inspiration from the list of past projects below.
Mentor. Ethan Cohen (ethan.cohen at yale dot edu)
Primary text. Ghys ‘Groups acting on the circle’ and/or Katok-Hasselblatt ‘Introduction to modern theory of dynamical systems’ chapters 10-12.
Description. This project will explore automorphisms of the circle. Although the circle is one of the simplest manifolds, its automorphisms and automorphism group can be quite complicated! This project will introduce several fundamental questions that appear throughout the field of dynamics. These include orbit classification, the conjugacy problem, the study of invariant measures, and the group-theoretic properties of the group of automorphisms. Also, the definition of ‘automorphism’ will change the answers to these questions.
Pre-requisites. Point-set topology is the only required pre-requisite. But basic knowledge of measure theory, real analysis, and group theory will each make the project more interesting!
Mentor. Max Steinberg (max.steinberg at yale dot edu)
Primary text. My differential forms notes :)
Description. You may remember that the curl of the gradient of a function is zero, as is the divergence of the curl of a function. Other than computational coincidence, why are both of these facts true? And why is the cross product so weird? Learn this and more in the framework of differential forms!
Pre-requisites. Multivariable calculus (120) and linear algebra (222/225/226)
Mentor. David Bai (david.bai at yale dot edu)
Description. Choose one of the past projects offered by David in previous semesters! (Twenty-Seven Lines, Sheaves and their cohomology, or Weil conjectures)
Pre-requisites. See the relevant project below
Description. You can get some inspiration from the list of past projects below.
Mentor. Dongryul Kim (dongryul.kim at yale dot edu)
Primary text. Luker, Approaches to Mostow rigidity in hyperbolic space; Thurston, Geometry and topology of three-manifolds
Description. It is a celebrated rigidity theorem of Mostow that a closed hyperbolic 3-manifold does not have any non-trivial deformation. In other words, its geometry is completely determined by its fundamental group. There are several known proofs, and this project aims to study some of them, including Mostow's original proofs. The idea of the proof comes from various fields of mathematics, algebra, geometry, topology, and dynamics. If time permits, we can discuss some generalizations of Mostow's rigidity, such as Sullivan's rigidity theorems or the Ending lamination theorem, which was conjectured by Thurston and proved by Brock, Canary, and Minsky.
Pre-requisites. Understanding of 2x2 matrices (Linear algebra) + Definition of group (abstract algebra) are prerequisites. Bits of knowledge about (algebraic) topology would be helpful. Background on hyperbolic geometry and Kleinian groups will be welcomed!!
Mentor. David Bai (david.bai at yale dot edu)
Primary text. Miles Reid, “Undergraduate Algebraic Geometry” (and possibly other texts, depending on the direction we are going towards)
Description. “There are precisely 27 lines on a smooth cubic surface.” This is the statement of an 1849 theorem by Cayley and Salmon, and is pretty much the start of enumerative algebraic geometry, a subject where one explores interesting phenomena associated with the problem of counting certain geometric objects. In this project, you will: 1) Learn some basic (classical) algebraic geometry so that you understand what this theorem is actually saying. 2) Understand at least one proof of the theorem. 3) Explore theories revolving around ideas used in the proof, such as moduli spaces and intersection theory.
Pre-requisites. This is a topic where the minimum prerequisite requirement is rather weak — basic understanding of algebra and some intuition about the geometry of calculus should be sufficient. That said, students with backgrounds in algebraic geometry, differential geometry, or even combinatorics, are more likely to get the most out of this project.
Mentor. Marco Pirazzini (marco.pirazzini at yale dot edu)
Primary text. An Introduction to Optimization on Smooth Manifolds - N. Boumal
Description. Optimization is a field of mathematics that has an enormous impact in society and our everyday life. It forms the building blocks of very powerful machine learning models, but it also considers algorithms for problems with mathematically rigorous theoretical guarantees. The field of optimization has its roots in the discovery of calculus, but it enjoyed a tremendous development with the invention of computers. The first wave of modern optimization algorithms focused on linear problems, while the second wave focused on convex problems. Convex programming algorithms are very powerful, but they do not tell the whole story because many interesting problems are not convex. In this project, we will learn about optimization on smooth manifolds, where we replace the Euclidean space with a smooth manifold as the domain of an optimization problem. This framework captures many interesting non-convex problems, and it is a very active area of research. We will focus on embedded submanifolds of R^n, and we will start from the basic ingredients needed to generalize optimization algorithms on Euclidean spaces. This will cover first order geometry of the manifold, such as tangent spaces and differentials of smooth maps, and second order geometry, such as Riemannian connections and Hessians. After that, we will use this structure to design first and second order manifold optimization algorithms, generalizing gradient descent and Newton's method. If time permits and based on the interest of the students, we can go through specific examples. We will follow chapters 3-6 of Boumal's book, and possibly some parts of the Absil, Mahony, Sepulchre book.
Pre-requisites. Familiarity with Linear Algebra (at the level of MATH 225) and basic Optimization algorithms (e.g. gradient descent). Prior knowledge of Smooth Manifolds is helpful but not required, as this project would nicely complement MATH 302.
Mentor. Christina Meng (christina.meng at yale dot edu)
Primary text. "Total Positivity: Tests and Parameterizations", Fomin and Zelevinsky; "Total Positivity, Grassmannians, and Networks", Postnikov
Description. As a starting point we will understand how certain networks parameterize cells of the Grassmannian. We will then explore connections to topics such as dimer models and cluster algebras, depending on interest and if time permits. This project is an opportunity to learn about interesting mathematical results from this century (!) in an accessible manner.
Pre-requisites. Linear algebra, and some abstract algebra and topology would be useful
Mentor. Max Steinberg (max.steinberg at yale dot edu)
Primary text. I have some notes on differential forms but Bott & Tu and Sherman (calculus & analysis) are the texts I like.
Description. You may remember that the curl of the gradient of a function is zero, as is the divergence of the curl of a function. Other than computational coincidence, why are both of these facts true? And why is the cross product so weird? Learn this and more in the framework of differential forms!
Pre-requisites. Multivariable calculus (math 120) and some basic linear algebra knowledge (225/226 not required but encouraged).
Mentor. Pedro Suarez (pedro.suarez at yale dot edu)
Primary text. Dale Rolfsen "Knots and Links", Alexandru Scorpan "The Wild World of 4-Manifolds"
Description. This project is an open-ended exploration of low-dimensional manifolds, following the student's interests. It also works as a fantastic setting to explore algebraic topology, either as a beginner or as a more seasoned student. Some of the topics that may be covered are the following: - An introduction to knot theory. - Surgery in 3-manifolds, and the Lickorish-Wallace theorem. - A survey on 4-manifolds: how to represent them using Kirby diagrams, and their algebraic-topological properties. - More advanced subjects dealing with differentiable structures in 4-manifolds, such as the construction of exotic pairs and the cork theorem.
Pre-requisites. Basic topology. Knowledge of algebraic topology is useful, but not required.
Mentor. Vladyslav Zveryk (vladyslav.zveryk at yale dot edu)
Primary text. D. Eisenbud, "Commutative algebra with a view towards algebraic geometry"; "Twenty-four hours of local cohomology" by 5 authors
Description. Commutative algebra is the foundation of algebraic geometry which is one of the most researched areas in modern mathematics. Statements in algebraic geometry usually break down into statements about modules and rings, which are central objects in commutative algebra. On the other hand, homological algebra is a collection of tools for producing and computing invariants of the studied objects. In this project, we will delve into the applications of homological algebra in commutative algebra, which may include the Tor and Ext functors, the theory of depth, local cohomology and finite free resolutions. The project is flexible in terms of your personal background and interests: the material could vary from the foundations of commutative algebra to powerful numerical invariants in algebraic geometry such as the Castelnuovo-Mumford regularity.
Pre-requisites. Standard courses in linear algebra and abstract algebra, understanding the concepts of commutative rings, ideals and modules, familiarity with the abstract concept of the tensor product over commutative rings. Understanding of basic homological algebra is helpful but not necessary
Mentor. Sam Panitch (sam.panitch at yale dot edu) and Danny Nackan (danny.nackan at yale dot edu)
Primary text. Knots and Physics by Louis Kauffman, or related sources based on interest.
Description. In this project, we explore surprising connections between knots, their invariants such as the Jones polynomial, and some concepts in physics.
Pre-requisites. Linear algebra and some abstract algebra. Willingness to imagine things in 3D.
Mentor. Mikey Chow (mikey.chow at yale dot edu)
Primary text. TBD - depends on area of interest and background level
Description. The focus will likely be learning fundamentals from a staple textbook in group actions, geometry, and/or dynamics. Looking at a research paper in parallel may be possible.
Pre-requisites. Abstract algebra and real analysis
Mentor: David Bai (david.bai at yale dot edu)
Primary texts: Robin Hartshorne "Algebraic Geometry", Torsten Wedhorn "Manifolds, Sheaves, and Cohomology"
Description: Sheaves are ubiquitous objects that encode the interactions between local information contained in e.g. a manifold. They have tremendous applications especially in algebraic geometry.
In this project, you will work on understanding (a subset of): 1. The definition of sheaves and their basic properties. 2. Examples of sheaves. 3. Derived functors and cohomology of sheaves. 4. Cech Cohomology. 5. Classical applications, such as the proof of de Rham’s Theorem. 6. [if time allows] Sheaves in algebraic geometry: Grothendieck’s theorems, Serre’s theorems, and so on.
We might explore other topics depending on the background and interests of the student.
Pre-requisites: Topology and abstract algebra are essential. The language of category theory will be needed from time to time, but it should not take long to learn it on-the-fly. Previous exposure to algebraic geometry, differential geometry, algebraic topology will be helpful as they will provide a wealth of examples.
Mentor: David Bai (david.bai at yale dot edu)
Primary texts: The Arithmetic of Elliptic Curves, by Joseph H. Silverman; Algebraic Geometry, by Robin Hartshorne; assorted online notes.
Description: One of the biggest advancement in algebraic geometry in the 20th century is the proof of the Weil conjectures, which establish a monumental connection between the algebraic topology of a projective variety and its zeta function over a finite field.
In this project, you will try to understand the proof of Weil conjectures in dimension one. We will first discuss the classical theory of algebraic curves. We will then study genus 1 curves in depth, which will allow us to reach a simple proof of Weil conjectures in this occasion (this was actually what motivated Weil to consider these problems in general). The project is expected to end with the proof of Weil conjectures, including the "Riemann hypothesis", for curves of all genus.
If time permits, we might talk about the proof of Weil conjectures in general, their modern applications, open problems, and so on.
Pre-requisites: General knowledge of classical algebraic geometry; some scheme theory; and the courage to learn more algebraic geometry when needed. Previous exposure to algebraic topology, Riemann surfaces and number theory are preferred but not required.
Mentor: Gal Yehuda (gal.yehuda at yale dot edu)
Primary texts: Mostly papers (we might use some textbooks, depending on your background)
Description: What is the most efficient way to "compute" a function? Can "deep learning" solve everything? What can we prove in this context?
Complexity theory aims at separating the class of functions which can be "efficiently computed" from the class of functions which are unfeasible to compute. In this project we focus on a specific type of computational model: threshold circuits (also known as "feed forward neural networks"). We will try to better understand their computational power, and (hopefully) prove new results. We will mostly use tools from combinatorics, probability and discrepancy theory. No prior knowledge in complexity is required.
Pre-requisites: Being comfortable with probability and combinatorics. Prior knowledge of circuit complexity can help but is not required.
Mentor: Sasha Cui (sasha.cui at yale dot edu)
Primary texts: Barron 1994, Approximation and Estimation Bounds for Artificial Neural Networks; Neal 1994, Bayesian Learning for Neural Networks; Zhang et al 2017, Understanding Deep Learning Requires Rethinking Generalization; Hastie et al, Surprises in High-Dimensional Ridgeless Least Squares Interpolation
Description: Identifying the right mathematical framework for understanding Machine Learning is among the most important open problems in STEM research today.
Pre-requisites: Real analysis & linear algebra.
Mentor: Do Kien Hoang (dokien.hoang at yale dot edu)
Primary texts: Symmetry, representations, and invariants, by Roe Goodman
Description: In this project, we will study basic invariant theory and representation theory of classical Lie group based on the first ten chapters of Goodman's Symmetry, representations, and invariants
Pre-requisites: Linear algebra and abstract algebra.
Mentor: Sri Tata (sri.tata at yale dot edu) & Sam Panitch (sam.panitch at yale dot edu)
Primary texts: Curriculum is open ended, could use the Big Yellow Book [di Francesco et al, Conformal Field Theory] or various online resources.
Description: Conformal field theory (CFT) is the theory of quantized holomorphic functions on a surface. In two dimensions conformal transformation is an angle preserving map, which are represented by holomorphic functions. This theory has broad applications and deeply tied to much of mathematics [quantum groups, knot invariants, modular forms, etc] and physics [critical phenomena, string theory, quantum hall effect, etc.].
Feel free to contact sri.tata@yale.edu or sam.panitch@yale.edu for a consultation.
Pre-requisites: May need some decent Calculus, Complex Analysis, Fourier Analysis chops or alternatively Lie Algebra chops or a Physics Background depending on the route you’d want to take. We’ll be a bit flexible with the background, but we’re also trying to learn CFT so ideally this would be a kind of joint reading course where we all learn and synergize.
Mentor. Sri Tata (sri.tata at yale dot edu) and Danny Nackan (danny.nackan at yale dot edu)
Primary text. Knots and physics, by Louis Kauffman.
Description. In this project, we explore surprising connections between knots, and their invariants such as the Jones polynomial, and some concepts in physics.
Pre-requisites. Linear algebra and some abstract algebra. Willingness to imagine things in 3D.
Mentor. Reuben Drogin and Haoyu Wang (reuben.drogin at yale dot edu and haoyu.wang at yale dot edu)
Primary text. Markov chains and mixing times, by David Levin and Yuval Peres.
Description. Mixing times quantify the rate at which a Markov Chain converges to its stationary state, i.e. how long it takes for a Markov chain to “randomize”. This concept comes up naturally and is important in many areas of mathematics, computer science, and machine learning. For instance “How many shuffles does it take to mix up a deck of cards?” is a mixing time question. We will start from the basics (definitions, classic examples) and cover most of the first part of the text by Levin and Peres.
Pre-requisites. Being comfortable with probability and combinatorics.
Mentor. Reuben Drogin (reuben.drogin at yale dot edu)
Primary text. Partial differential equations by Fritz John, possibly supplemented by Lectures on partial differential equations by Vladimir Arnold and Partial differential equations by Lawrence Evans .
Description. A Partial Differential Equation is an equation involving an unknown function and its derivatives. Given the open-endedness of this definition, it is not surprising that PDE's are ubiquitous, appearing in differential geometry, dynamical systems, probability, physics, etc. This reading course would serve as an introduction to Partial Differential Equations centered around some of the "greatest hits". As PDE is such a broad field the precise topic selection and depth of study can be catered to a students interests and background, but a few examples are:
- The Laplace Equation and Harmonic Functions
- A bit of Harmonic Analysis and the Heat equation
- The Wave Equation and Maxwell's Equations
- Heisenberg Uncertainty Principle
- Calculus of Variations
- Wave-Particle Duality and the Hamilton-Jacobi Equations
Along the way students can also learn about Sobolev spaces, Maximal functions, topics in functional analysis and applications to PDE, aspects of dynamical systems, and other more technical or tangential aspects.
Pre-requisites. Measure theory and familiarity with functional analysis. Math 320 would suffice. This would also be a nice complement to Math 325.
Mentor. Haoyu Wang (haoyu.wang at yale dot edu)
Primary text. The Probabilistic Mehtod by Noga Alon and Joel Spencer.
Description. The probabilistic method is a fundamental and powerful technique in combinatorics. Besides being mathematically interesting itself, it is widely used in theoretical computer science and statistics. One of the essences of the approach is the following: To show that some combinatorial object exists, we prove that a certain random construction works with positive probability. We will focus on various methods in probability and combinatorics, as well as algorithmic techniques with applications.
Pre-requisites. Being comfortable with probability, combinatorics, and real analysis.
Mentor. Fernando Al Assal (fernando.alassal at yale dot edu)
Primary text. Lecture notes by Giulio Tiozzo and other references as needed.
Description. Imagine yourself as a person randomly moving through an infinite group. Will you come back to where you started your walk, and perhaps visit every place in the group, or perhaps you will move away towards infinity in some sense? We will explore examples such as the integer lattice, SL_2 R, hyperbolic groups, amenable groups among others and see how properties of the random walk change in each of these.
Pre-requisites. Analysis (measure theory), linear algebra. Functional analysis would be good too. Probability theory is helpful but not necessary. Knowing what a group is.
Mentor. Aaron Calderon (aaron.calderon at yale dot edu)
Primary text. Keith Kendig's A guide to plane algebraic curves (can be found through the Yale Library)
Description. Complex algebraic curves (which sometimes masquerade as Riemann surfaces, field extensions, or hyperbolic surfaces) are a fundamental object in mathematics, lying at the interface of algebra, analysis, and topology. This project will study perhaps the most classical examples of these, those arising as the zero set of a polynomial in two variables. The goal of this project is to get some hands-on experience working with plane curves and understand how some important notions of algebraic geometry (Bezout’s theorem, resolution of singularities, function fields, etc.) look in this setting. The book does not contain many complete proofs, so it will be the student’s job each week to flesh out details and work examples.
Pre-requisites. Complex analysis, rudimentary topology, abstract algebra (rings, ideals, field extensions). The student is not expected to have any familiarity with commutative algebra or algebraic geometry.
Mentor. Fernando Camacho Cadena (fernando.camachocadena at yale dot edu)
Primary text. Random Walks on Reductive Groups by Yves Benoist and Jean-François Quint. Further references: An Introduction to Random Walks on Homogeneous Spaces by Benoist and Quint and An Introduction to Random Walks on Groups by Quint. Further references will be provided depending on the student's background and interest. All texts are freely available online.
Description. Say you are in some ambient space and are equipped with a group whose elements tell you how to move within it. By randomly picking elements from the group, you walk randomly through the space. The paths heavily depend on properties of the group and certain groups could lead to wild paths! One may then ask questions with the following flavours. After randomly walking for a long time,
do I keep returning to the place where I started (recurrence)?
do I visit most places in the ambient space equally often (equidistribution)?
if I somehow average my distance walked over time, can I say anything about these averages (limit theorems)?
Those are very interesting questions in their own. But they also have powerful application in various areas of math (which we will probably not get to). Some examples include homogeneous dynamics, which has strong connections to number theory; and the celebrated work of Maryam Mirzakhani on the geometry of surfaces.
The goal of the project is to understand these random walks on groups and explore some of the above questions. There are a lot of different directions to go in this area, so we can discuss and choose the one that interests you the most!
Pre-requisites. Linear algebra, measure theory, probability theory could be helpful but is not necessary. (Although groups are essential objects in this topic, knowing what they are is enough as we will mostly be dealing with familiar groups of matrices.)
Mentor. Aaron Calderon (aaron.calderon at yale dot edu)
Primary text. Rational Points on Elliptic Curves by Silverman and Tate (can be found through the Yale Library)
Description. From the viewpoint of algebraic geometry, Pythagorean triples are just rational solutions to the equation x^2 + y^2 = 1. Given an equation like x^2 + y^3 = 1, you could also ask for its rational (or integer) solutions; it turns out that this problem is much harder, and in general we can't find all of them! These types of problems are called *Diophantine,* and in this project we'll look at some of the simplest (yet still incredibly rich!) Diophantine problems, those associated with cubic equations.
Pre-requisites. Calculus and basic abstract algebra (groups, rings). Complex analysis would be nice but not necessary.
Mentor. Fernando Al Assal (fernando.alassal at yale dot edu)
Primary Texts. Introduction to Stochastic Processes by Lawler; An Introduction to Stochastic Differential Equations by Evans; Online lecture notes by Lalley.
Description. In this project we study random processes evolving with time! These objects are important in pure as well as applied math. On the one hand, we can study processes that evolve with a tick of a discrete clock, such as Markov chains — random processes whose evolution only depends on the previous instant. Examples include random walks and some ecological models. On the other hand, with more analytical machinery (or faith in such machinery) we can study processes that evolve continuously, such as Brownian motion, which models things such as the price of a stock and can also be used to prove theorems in complex analysis (!).
Pre-requisites. Multivariable calculus and linear algebra. Measure theory relevant but optional.