Project suggestions
Fall 2023
Weil conjectures for curves
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.
Threshold circuits lower bounds
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.
Mathematical theory of machine learning
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.
Symmetry, representations, and invariants
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.
Conformal Field Theory
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.
Propose your own project! (PYOP)
Description. You can get some inspiration from the following list of books and books from past DRP projects in different universities.
Fall 2022
Knots and physics
Mentor. Sri Tata (sri.tata@yale.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.
Markov chains and mixing times
Mentor. Reuben Drogin and Haoyu Wang (reuben.drogin@yale.edu and haoyu.wang@yale.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.
Propose your own project! (PYOP)
Description. Remember you can always propose a project of your own. You can get some inspiration from the following list of books and books from past DRP projects in different universities.
Spring 2022
Partial differential equation's greatest hits
Mentor. Reuben Drogin (reuben.drogin@yale.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.
The probabilistic method and combinatorics
Mentor. Haoyu Wang (haoyu.wang@yale.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.
Random walks on groups
Mentor. Fernando Al Assal (fernando.alassal@yale.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.
Propose your own project! (PYOP)
Description. Remember you can always propose a project of your own. You can get some inspiration from the following list of books and books from past DRP projects in different universities.
Fall 2021
Plane algebraic curves
Mentor. Aaron Calderon (aaron.calderon@yale.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.
Random walks on groups
Mentor. Fernando Camacho Cadena (fernando.camachocadena@yale.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.)
Rational points on elliptic curves
Mentor. Aaron Calderon (aaron.calderon@yale.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.
Stochastic processes
Mentor. Fernando Al Assal (fernando.alassal@yale.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.
Propose your own project! (PYOP)
Description. Remember you can always propose a project of your own. You can get some inspiration from the following list of books and books from past DRP projects in different universities.
Spring 2021
All books should be available virtually.
Introduction to nonstandard analysis
Mentor. Yakir Forman (yakir.forman@yale.edu)
Primary text. Robert Goldblatt. Lectures on the hyperreals
Description. A century after Weierstrass put analysis on rigorous firm ground with the epsilon-delta definition of limit, Abraham Robinson developed an entirely different approach to analysis, with wide-ranging philosophical and pedagogical implications. Nonstandard analysis makes use of number systems which include infinite and infinitesimal numbers, and it (in some ways) accords better than standard analysis with our intuitions about how calculus works. In this project, we explore how to construct one such number system and how to develop calculus (and, time permitting, other areas of mathematics) using nonstandard analysis. We hope to gain an interesting new perspective on familiar areas of mathematics.
Pre-requisites. Some (standard) analysis would be helpful.
Large deviations
Mentor. Haoyu Wang (haoyu.wang@yale.edu)
Primary text. Amir Dembo and Ofer Zeitouni. Large Deviations: Techniques and Applications
Description. The theory of large deviations deals with the probability of events that happen rarely. The goal of the large deviation principle is to prove some exponential decay of the probability for such large events. Though originated from the study of statistical mechanics, it has proved to be a powerful tool in a broad range of problems. Starting from the definition of the large deviation principle, we will overview the basic concepts and results in finite-dimensional spaces. Then we will move on to the applications in various probability problems and generalize the results to a more abstract setting. We may also discuss the applications of large deviations in random matrix theory, theoretical statistics or statistical physics.
Pre-requisites. A solid background in real analysis and probability theory. Familiarity with stochastic processes and functional analysis would be helpful.
Lie groups and homogeneous spaces
Mentor. Aaron Calderon (aaron.calderon@yale.edu)
Primary text. Andreas Arvanitoyeorgos. An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
Description. Klein's famous Erlangen program characterized the (at the time) newly–discovered non-Euclidean geometries by looking at their symmetry groups. More generally, one can study all sorts of spaces which look the same at every point (a property called homogeneity) via the theory of Lie groups. This project will focus on Lie groups and Lie algebras from a geometric point of view, with the ultimate goal of understanding the geometry of their associated homogeneous spaces. No familiarity with Lie groups or algebras will be assumed.
Pre-requisites. Linear algebra, abstract algebra (groups, rings, ideally algebras and/or representations). Familiarity with basic differential geometry/topology concepts (smooth manifold, tangent space, Riemannian metric).
Modern signal processing
(Added Feb. 4th)
Mentor. Ben Ekeroth (ben.ekeroth@yale.edu)
Description. In this project, we will cover the math behind modern signal processing, including a programming component. The topics covered will depend on the background of the students, but should include applied Fourier analysis, image processing, signal processing with neural networks and possibly compressed sensing.
Pre-requisites. Linear algebra and analysis. Knowing what a Fourier series is would be good.
Plane algebraic curves
Mentor. Aaron Calderon (aaron.calderon@yale.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.
Undecidability in number theory
Mentor. Brian Tyrrell (brian.tyrrell@maths.ox.ac.uk)
Primary texts. We'll begin with Poonen's "Undecidability in Number Theory" (http://www-math.mit.edu/~poonen/papers/h10_notices.pdf) and Koenigsmann's "Undecidability in Number Theory" (https://arxiv.org/pdf/1309.0441.pdf). Further references will be supplied depending on the background and specific interests of the student.
Description. The concept of "decidability", although first formally defined in the early 20th century, has been with us for thousands of years. It has now become a commonplace question in mathematics, whether a theorem, construction or proof can be made "effective". The other side of the coin is, of course, when there does not exist an algorithm to solve some problem - when one can mathematically prove no algorithm can ever exist to solve this problem, even ignoring time, space and physical constraints. This is the realm of the "undecidable".
Such problems are particularly abundant in number theory, thanks to the work of Gödel, Tarski, Davis, Putnam, Robinson, and Matiyasevich (among many others). This project will begin by introducing the relevant model theory and computability theory needed to understand the fundamental results of the field; then we will see "Hilbert's Tenth Problem" and the writings that evolved from its resolution in 1970, topping up our number theory as we proceed. As the project progresses, the student's interests will guide the specific results we look into.
Pre-requisites. Some Mathematical Logic / Computability would be ideal, but not essential. There is a lot to say about this area, so the course can be adapted to suit the student's background.
Disclaimer. I am in the UK, so all meetings will necessarily be via Zoom, and be mid-morning/early afternoon (for you).
Fall 2020
Note for this semester: all books should be available virtually.
Braid groups and configuration spaces
Mentor. Aaron Calderon (aaron.calderon@yale.edu)
Primary texts: Chapter 18 of Office hours with a geometric group theorist https://press.princeton.edu/titles/11042.html
other references will be supplied depending on the interests of the student.
Description. Braid groups describe how points can move around in a space without hitting each other. This property is encoded mathematically in the concept of a configuration space, which describes the possible (non-colliding) configurations of points in that space. These concepts have numerous applications mathematically (representation theory, knot theory, mapping class groups) and in the “real world” (robotics, quantum computing). This project has two parts: in the first, the student will gain a familiarity with the relevant mathematical concepts (braid groups, fundamental groups, configuration spaces) while in the second she will pick one such application and focus on it.
Pre-requisites. Group theory and a comfortability with manifolds. A bit of algebraic topology (in particular cell complexes and the fundamental group of a space) would be ideal, but is not necessary. No familiarity with robotics or quantum computing is required (I certainly don’t have any!)
High dimensional probability: theory and applications
Mentor. George Linderman (george.linderman@yale.edu)
Primary text. Vershynin, Roman. High Dimensional Probability.
Description. A large number of statistical and computational techniques in “data science,” particularly those applied to high dimensional problems, have rich mathematical theory behind them. The goal of this reading course will be to explore the theory behind some of these methods. More details can be found in Vershynin’s book, which we will follow closely and is freely available online. Student(s) will be encouraged to perform numerical experiments for certain theorems, which can give crucial insight into how they work.
The goals can be very flexible. We can start from the beginning and do a few chapters, or if students have already worked with the text or are familiar with the material we can skip to the more advanced topics. We can also supplement the book with a more in-depth exploration of a specific topic, if the students would like.
Prerequisites. Strong background in probability theory and linear algebra. Some rudimentary functional analysis (e.g. some familiarity with Hilbert spaces).
Disclaimer. George is a very busy MD/PhD student applying for residencies, so students will need to be flexible to schedule the meetings.
Introduction to nonstandard analysis
Mentor. Yakir Forman (yakir.forman@yale.edu)
Primary text. Robert Goldblatt. Lectures on the hyperreals
Description. A century after Weierstrass put analysis on rigorous firm ground with the epsilon-delta definition of limit, Abraham Robinson developed an entirely different approach to analysis, with wide-ranging philosophical and pedagogical implications. Nonstandard analysis makes use of number systems which include infinite and infinitesimal numbers, and it (in some ways) accords better than standard analysis with our intuitions about how calculus works. In this project, we explore how to construct one such number system and how to develop calculus (and, time permitting, other areas of mathematics) using nonstandard analysis. We hope to gain an interesting new perspective on familiar areas of mathematics.
Pre-requisites. Some (standard) analysis would be helpful.
Lie groups and homogeneous spaces
Mentor. Aaron Calderon (aaron.calderon@yale.edu)
Primary text. Andreas Arvanitoyeorgos. An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
Description. Klein's famous Erlangen program characterized the (at the time) newly–discovered non-Euclidean geometries by looking at their symmetry groups. More generally, one can study all sorts of spaces which look the same at every point (a property called homogeneity) via the theory of Lie groups. This project will focus on Lie groups and Lie algebras from a geometric point of view, with the ultimate goal of understanding the geometry of their associated homogeneous spaces. No familiarity with Lie groups or algebras will be assumed.
Pre-requisites. Linear algebra, abstract algebra (groups, rings, ideally algebras and/or representations). Familiarity with basic differential geometry/topology concepts (smooth manifold, tangent space, Riemannian metric).
Stochastic processes
Mentor. Fernando Al Assal (fernando.alassal@yale.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.
Spring 2020
Braid groups and configuration spaces
Pre-requisites. Complex analysis, maybe some topology.
Mentor. Aaron Calderon (aaron.calderon@yale.edu)
Primary texts: Chapter 18 of “Office hours with a geometric group theorist” https://press.princeton.edu/titles/11042.html
other references will be supplied depending on the interests of the student.
Description. Braid groups describe how points can move around in a space without hitting each other. This property is encoded mathematically in the concept of a configuration space, which describes the possible (non-colliding) configurations of points in that space. These concepts have numerous applications mathematically (representation theory, knot theory, mapping class groups) and in the “real world” (robotics, quantum computing). This project has two parts: in the first, the student will gain a familiarity with the relevant mathematical concepts (braid groups, fundamental groups, configuration spaces) while in the second she will pick one such application and focus on it.
Pre-requisites. Group theory and a comfortability with manifolds. A bit of algebraic topology (in particular cell complexes and the fundamental group of a space) would be ideal, but is not necessary. No familiarity with robotics or quantum computing is required (I certainly don’t have any!)
Complex dynamics
Mentor. Elijah Fromm (elijah.fromm@yale.edu)
Primary texts. Beardon, Alan. Iteration of Rational Functions
Milnor, John. Dynamics in One Complex Variable
Description. In case you haven’t noticed all the decorations around the department, Benoit Mandelbrot really liked fractals. Some of his favorite fractals, including his eponymous set, come from complex dynamics!
As stated in Milnor’s preface, we “will study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere.” I’m flexible on which of the two texts we rely on primarily.
Pre-requisites. Complex analysis, maybe some topology.
Foliations of 3-manifolds
Mentor. Alex Rasmussen (alexander.rasmussen@yale.edu)
Primary texts. Calegari - Foliations and the Geometry of 3-Manifolds. We might also consult: Candel-Conlon - Foliations, Fathi-Laudenbach-Poénaru - Thurston's Work on Surfaces, Farb-Margalit - A Primer on Mapping Class Groups.
Description. A foliation is a particularly nice decomposition of a manifold into lower-dimensional manifolds. They are ubiquitous throughout mathematics - arising from vector fields, subgroups of Lie groups, self-homeomorphisms of manifolds, and from many other things. We will study 2-dimensional foliations of 3-manifolds. Our main motivating examples will come from self-homeomorphisms of surfaces and in our project we will try to generalize the machinery that comes with them. An eventual goal might be to understand Thurston's "universal circle" construction which associates an action on a circle to any 3-manifold with a particularly nice foliation. In any case, we will learn about a lot of cool machinery along the way - branched surfaces, the Thurston norm, pseudo-Anosov homeomorphisms, and more.
Pre-requisites. Algebraic topology
Lie groups and homogeneous spaces
Mentor. Aaron Calderon (aaron.calderon@yale.edu)
Primary text. Andreas Arvanitoyeorgos. An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
Description. Klein's famous Erlangen program characterized the (at the time) newly–discovered non-Euclidean geometries by looking at their symmetry groups. More generally, one can study all sorts of spaces which look the same at every point (a property called homogeneity) via the theory of Lie groups. This project will focus on Lie groups and Lie algebras from a geometric point of view, with the ultimate goal of understanding the geometry of their associated homogeneous spaces. No familiarity with Lie groups or algebras will be assumed.
Pre-requisites. Linear algebra, abstract algebra (groups, rings, ideally algebras and/or representations). Familiarity with basic differential geometry/topology concepts (smooth manifold, tangent space, Riemannian metric).
Stochastic processes
Mentor. Fernando Al Assal
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.
Propose your own project! (PYOP)
Description. Remember you can always propose a project of your own. You can get some inspiration from the following list of books recommended by the DRP network.
Fall 2019
Complex dynamics
Mentor. Elijah Fromm (elijah.fromm@yale.edu)
Primary texts. Beardon, Alan. Iteration of Rational Functions
Milnor, John. Dynamics in One Complex Variable
Description. In case you haven’t noticed all the decorations around the department, Benoit Mandelbrot really liked fractals. Some of his favorite fractals, including his eponymous set, come from complex dynamics!
As stated in Milnor’s preface, we “will study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere.” I’m flexible on which of the two texts we rely on primarily.
Pre-requisites. Complex analysis, maybe some topology.
Differential geometry of curves and surfaces
Mentor. Fernando Al Assal
Primary text. Do Carmo, Differential Geometry of Curves and Surfaces
Description. Differential geometry is an old and important area of math, which studies shapes (Riemannian manifolds) in which you can measure angles and distances. We will focus manifolds of dimension 1 and 2 (curves and surfaces) and concretely understand their geometry, with plenty of computations and interesting theorems.
Pre-requisites. Linear algebra and multivariable calculus (including the implicit function theorem etc).
Disclaimer. I will be away from Oct 20 to Nov 23, and the following week is Thanksgiving break. So I will only be around for six weeks in which we can have face to face conversations. So the project might be short.
High dimensional probability: theory and applications
Mentor. George Linderman (george.linderman@yale.edu)
Primary text. Vershynin, Roman. High Dimensional Probability.
Description. A large number of statistical and computational techniques in “data science,” particularly those applied to high dimensional problems, have rich mathematical theory behind them. The goal of this reading course will be to explore the theory behind some of these methods. More details can be found in Vershynin’s book, which we will follow closely and is freely available online. Student(s) will be encouraged to perform numerical experiments for certain theorems, which can give crucial insight into how they work.
The goals can be very flexible. We can start from the beginning and do a few chapters, or if students have already worked with the text or are familiar with the material we can skip to the more advanced topics. We can also supplement the book with a more in-depth exploration of a specific topic, if the students would like.
Prerequisites. Strong background in probability theory and linear algebra. Some rudimentary functional analysis (e.g. some familiarity with Hilbert spaces).
Disclaimer. George is an MD/PhD student (PhD in Applied Math) and will be at the hospital during usual business hours, so meetings would take place on weekends or evenings.
Lattices in semisimple Lie groups
Mentor. Min Ju Lee (minju.lee@yale.edu)
Primary text. Morris, David. Introduction to Arithmetic Groups
Description. Study of flows on homogeneous space G/Γ where G is a Lie group and Γ is its discrete subgroup had been influential and is still an active area of research. It had been intensively studied especially when G/Γ has a "finite volume", in which case Γ is called a lattice in G.
We will study properties of lattices and its fundamental examples, following a few chapters of the textbook. Our first goal is to obtain preliminaries and understand the statement, but not the proof, of one important theorem.
Pre-requisites. Lie groups, topology, measure theory.
Lie groups and homogeneous spaces
Mentor. Aaron Calderon (aaron.calderon@yale.edu)
Primary text. Andreas Arvanitoyeorgos. An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
Description. Klein's famous Erlangen program characterized the (at the time) newly–discovered non-Euclidean geometries by looking at their symmetry groups. More generally, one can study all sorts of spaces which look the same at every point (a property called homogeneity) via the theory of Lie groups. This project will focus on Lie groups and Lie algebras from a geometric point of view, with the ultimate goal of understanding the geometry of their associated homogeneous spaces. No familiarity with Lie groups or algebras will be assumed.
Pre-requisites. Linear algebra, abstract algebra (groups, rings, ideally algebras and/or representations). Familiarity with basic differential geometry/topology concepts (smooth manifold, tangent space, Riemannian metric).
Stochastic process modeling
Mentor. Tal Malinovitch (tal.malinovitch@yale.edu)
Primary texts. Panik, Michael. Stochastic Differential Equations. Lawler, G. Introduction to Stochastic Processes. Lemieux, C. Introduction to Monte Carlo and Quasi-Monte Carlo Sampling. Dunn, L and Shultis, K. Exploring Monte Carlo Methods.
Description. This project has two parts. The first is a theoretical part: we will learn to how analyze random processes in discrete time such as Markov Chains, discrete time martingales, and random walks. The second is a practical part in which we will try and model a simple system, and examine it's behavior using a Monte Carlo simulation, this will require learning about proper sampling, building a Monte Carlo simulation, and doing basic performance analysis on it, comparing it to the analytic results and so on.
Pre-requisites. Multivariable calculus and linear algebra. It is helpful to have some background with ODE and basic measure theory. Basic programming (i.e. MATLAB type) will be helpful but not necessary.
Disclaimer. As you can see this is not just a reading project, it requires a bit more (and sometimes programming can be time consuming), please take this into account when applying.
Submodular optimization
Mentor. Chris Harshaw (christopher.harshaw@yale.edu)
Primary texts. Reading Materials include the following two surveys, along with Chris' previously prepared notes, and selected research papers.
"Submodular Function Maximization" Krause & Golovin. 2012. link
"Learning with Submodular Functions: A Convex Optimization Perspective" Bach. 2011. link
Description. Submodular optimization is a bustling area of modern theoretical computer science. Submodular functions are those which exhibit a diminishing returns property and they arise in a variety of domains, from economic game theory to sensor placement. Submodular optimization is the design and study of algorithms for maximizing or minimizing submodular functions, potentially under constraints. Aside from the real applications, the theory of submodular functions is mathematically elegant and beautiful in its own right, intersecting with fields such as convex analysis, matroid theory, and probability. In this course, we will learn about the properties and characterizations of submodular functions, discrete greedy algorithms for constrained submodular maximization, matroids and independent set systems, and the multilinear relaxation of a submodular function and the continuous greedy algorithm. If time permits, we can cover either submodular minimization and the Lovasz extension or alternative computational models such as streaming, distributed, or parallel computation.
Pre-requisites. Prerequisites include a discrete math course and some programming experience. Prior exposure to probability and convex analysis / optimization are helpful, but not assumed.
Propose your own project! (PYOP)
Description. Remember you can always propose a project of your own. You can get some inspiration from the following list of books recommended by the DRP network.
Spring 2019
Riemann surfaces and dessins d'enfants
Mentor. Aaron Calderon (aaron.calderon@yale.edu)
Primary text. Introduction to Compact Riemann Surfaces and Dessins d'Enfants
Description. Riemann surfaces (which equivalently masquerade as algebraic curves, 1-D complex manifolds, hyperbolic surfaces, or Fuchsian groups of finite type) are one of the most fundamental objects of all of mathematics. Sitting at the intersection of algebraic geometry, complex geometry, geometric topology, number theory, and Galois theory, they provide a glimpse into many different fields, and their relatively low complexity allows for fascinating connections between all of these fields. This course will study the development of the theory of Riemann surfaces up through Grothendieck’s dessins d’enfants, or “childs’ drawings.” These combinatorial objects allow one to encode the algebraic structure of a Riemann surface defined over a number field in a very small amount of data, and by a theorem of Belyi, every such structure arises from this sort of data. Along the way to understanding this theorem, we will trace through the development of the theory of Riemann surfaces, with a particular emphasis on understanding their connections to other areas of mathematics.
Syllabus. We will begin by understand the equivalences between Riemann surfaces, (complex) algebraic curves, and hyperbolic surfaces, with particular focus on the examples in the book. We will also spend some time investigating the Galois action on a (branched) cover of a Riemann surface. From there we will cover Belyi’s theorem, and if time permits, begin to understand the action of Gal(\bar{Q}/Q) on the set of dessins.
Primary goals. The goal of the project is to understand exactly what a dessin d’enfant is. The main mathematical content of the project is to prove Belyi’s theorem relating dessins d’enfants and branched covers of the Riemann sphere.
Expected outputs. By the end of the project, I hope that the students will have produced an expository article written at the advanced undergraduate level (think Math Magazine).
Pre-requisites. Complex analysis, abstract algebra (groups, fields and field extensions). Ideally the student(s) will also have some knowledge of topology (fundamental groups, covering spaces) and Galois theory. Some basic notions from hyperbolic and/or algebraic geometry would come in handy, but neither are required.
Stochastic processes
Mentor. Fernando Al Assal
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 (!).
Primary Goals. Track 1: Markov chains, discrete-time martingales, optimal stopping problems and a satisfying glimpse of Brownian motion and stochastic differential equations. Track 2: Discrete and continuous-time martingales, construction of Brownian motion, its relation to harmonic functions and complex analysis, stochastic differential equations and Ito’s formula. Long-term goals could include the Girsanov theorem, Lyons’ theory of rough paths etc.
Prerequisites. Track 1: Multivariable calculus and linear algebra. Track 2: Real and functional analysis; probability theory helpful but strictly optional