The Directed Reading Program pairs undergraduate students with graduate student mentors to read and work through a mathematics text over the course of one semester. The pairs meet once each week for one hour, with the undergraduates expected to do about 4 hours of independent reading per week. At the end of the semester, undergraduates either give a talk to their peers or prepare a short exposition of some of the material from the semester. Undergraduates are expected to have a high level of mathematical maturity and eagerness to learn the topic.

There are 8 graduate student mentors and 15 projects offered (please note, however, that if there is more than one project per graduate student mentor, not all projects will be offered).

The deadline to apply is Friday, January 15, 4pm EST, and the application can be found here.

SPRING 2016 PROJECTS

For a more detailed description of the projects or any questions please contact The DRP Spring 2016 organizers pdf

Thomas Hille (thomas.hille@yale.edu)

Lam Pham (lam.pham@yale.edu)

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Project title: Exactly Solved Models in Statistical Mechanics

Graduate mentor: Efim Abrikosov

Book: Rodney J. Baxter, Exactly solved models in statistical mechanics

Description: Statistical Mechanics is a big branch of modern physics that studies properties of macroscopic systems, typically having large number of degrees of freedom, for example, gases or fluids. In such situation as one can suspect it is too difficult to give precise answers about microscopic behaviour of systems. How- ever, some macroscopic characteristics still can be investigated and that amounts to finding certain probability distributions.

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Project title: Introduction to Mathematical Physics

Graduate mentor: Dylan Allegretti

Book: John Baez and Javier P. Muniain, Gauge fields, knots and gravity

Description: Statistical Mechanics is a big branch of modern physics that studies properties of macroscopic systems, typically having large number of degrees of freedom, for example, gases or fluids. In such situation as one can suspect it is too difficult to give precise answers about microscopic behaviour of systems. How- ever, some macroscopic characteristics still can be investigated and that amounts to finding certain probability distributions.

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Project title: Introduction to Mathematical Physics

Graduate mentor: Dylan Allegretti

Book: John Baez and Javier P. Muniain, Gauge fields, knots and gravity

Description: The first few chapters of the book cover basic differential geometry, including the theory of manifolds, vector fields, and differential forms. These concepts are used to formulate Maxwell’s equations on arbitrary spacetime manifolds. The second part of the book presents the theory of vector bundles and connections and uses these concepts to discuss gauge theory and its relation to knots. The final part of the book explains Riemannian geometry and its applications in general relativity. Ideally, I would like to at least get to the section on knot theory, but in principle we could stop anywhere and it would still be a satisfying experience for the student. Actually, I think there’s a danger that we might finish the book too soon. If that happens, there are plenty of online materials and texts that I can share with the student.

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Project title: Arithmetic Geometry

Graduate mentor: Vesselin Dimitrov

Book: Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry

Description: This project is a rigorous introduction to modern arithmetic geometry. No previous exposure to algebraic geometry is required, although that would be helpful. We will start with the appendix on algebraic geometry and set up the Weil heights in the opening two chapters and then, depending on the student’s interests and background

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Project title: Introduction to Stochastic Calculus

Graduate mentor: Thomas Hille

Book: Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus

Description: The aim of this project is to become familiar with two of the main concepts in probability theory, namely Markov processes and martingales. Our main example of both concepts will be Brownian motion in Rd. One of the main applications of the notion of martingales is its connection to partial differential equations, which leads to the study of integration with respect to stochastic processes and in turn to the study of so-called stochastic differential equations.

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Project title: Introduction to Lie Groups and Lie Algebras

Graduate mentors: Thomas Hille and Lam Pham

Book: Anthony W. Knapp, Lie groups beyond an introduction

Description: We start with the basic definitions of Lie groups and Lie algebras. We then follow with basic representation theory of Lie groups and Lie algebras, and structure theory of Lie algebras and root systems. The goal is to build a good knowledge of general Lie groups

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Project title: SL2(R)

Graduate mentors: Thomas Hille

Book: Serge Lang, SL2(R)

Description: We will follow Lang's SL2(R) and it is mainly an introduction through SL2(R) to the infinite dimensional representation theory of semisimple Lie groups. We don’t need any knowledge of Lie theory here.

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Project title: Essentials of Stochastic Processes

Graduate mentors: Oanh Nguyen

Book: Serge Lang, SL2(R)

Description:Stochastic processes constitute an important subject in probability theory and have strong connection with ergodic theory, analysis, theoretical computer science, etc.

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Project title: Abstract Harmonic Analysis

Graduate mentors: Oanh Nguyen

Book: Richard Durrett, Essentials of stochastic processes

Description:Stochastic processes constitute an important subject in probability theory and have strong connection with ergodic theory, analysis, theoretical computer science, etc.

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Project title: Abstract Harmonic Analysis

Graduate mentors: Lam Pham

Book: Gerald .B. Folland, A Course in Abstract Harmonic Analysis

Description: The primary goal is to become familiar with the non-commutative Fourier transform which is a very powerful tool. There are beautiful theories developed and we will see some of the special cases, in particular in the abelian case and the compact case. We will start with some review of topological groups and functional analysis (Banach algebras, spectral theory). Depending on the familiarity of the student, we can move quite quickly to unitary representations and functions of positive type before getting to our two main cases of study: analysis on locally compact abelian and compact groups.

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Project title: Kazhdan's Property (T)

Graduate mentors: Lam Pham

Book: Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T)

Description: D. Kazhdan introduced Prop- erty (T) in 1967. This property has been very useful and has been used among other things by G. Margulis to give the first explicit construction of expander graphs, and many famous lecture notes are available for supplementary reading.

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Project title: Probability on Locally Compact Groups and Compact Lie Groups

Graduate mentors: Lam Pham

Book: David Applebaum, Probability on compact Lie groups

Description: The goal is to become familiar with the anal- ysis tools available for treating probability on groups. We will follow Applebaum's book. Quickly reviewing some basics of Lie groups (Chapter 1), unitary representations and Peter-Weyl theory of compact Lie groups (Chapters 2 and 3), before moving on to the main part of the reading which is the study of probability measures on groups (Chapters 4-6).

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Project title: Additive Combinatorics

Graduate mentors: Lam Pham

Book: Terence Tao and Van H. Vu, Additive combinatorics

Description: Arithmetic Combinatorics (also formerly known as Additive Combinatorics) is a very popular area of research. In my opinion, what makes it extremely interesting is the very strong interaction between numerous fields such as: number theory, combinatorics, group theory, harmonic analysis, ergodic theory, and many more. There was tremendous progress in recent years, and both classical and modern viewpoints will be studied. Historically, additive combinatorics studies additive properties of subsets of the integers, or more generally, abelian groups. In the past 10 years, it has been partially extended to the non-commutative case, with many important applications in group theory and number theory. We will focus on purely additive combinatorics, studying the basic notions of the field, and some of the major results (Szemeredi’s Theorem, Erdos-Szemer ́edi conjecture, Green-Tao Theorem, Freiman’s Theorem, etc) and techniques involved in proving these (graph theoretic techniques, character theory and Fourier analysis, ergodic theory).

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Project title: Approximate Groups

Graduate mentors: Lam Pham

Book: [TV10] Terence Tao and Van H. Vu, Additive combinatorics, [Tao14] Hilbert’s fifth problem and related topics, [Tao08] Product set estimates for non-commutative groups.

Description: Arithmetic Combinatorics (also formerly known as Additive Combinatorics) is a very popular area of research. In my opinion, what makes it extremely interesting is the very strong interaction between numerous fields such as: number theory, combinatorics, group theory, harmonic analysis, ergodic theory, and many more. There was tremendous progress in recent years, and both classical and modern viewpoints will be studied. Historically, additive combinatorics studies additive properties of subsets of the integers, or more generally, abelian groups. In the past 10 years, it has been partially extended to the non-commutative case, with many important applications in group theory and number theory. We will start with a quick introduction to additive combinatorics using [TV10] and to move on directly to the non-commutative case using [Tao14] and to some extent, the seminal paper [Tao08]

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Project title: Expander Graphs

Graduate mentors: Lam Pham

Book:Terence Tao, Expansion in finite simple groups of Lie type

Description: We will follow Tao's book, based on a course given at UCLA. We will learn about the theory of expander graphs, isoperimetry in graphs, quasirandomness, probability on groups, arithmetic combinatorics, and applications to number theory.

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Project title: Introduction Matrix Lie Groups

Graduate mentors: Arseniy Sheydvasser

Book:Brian Hall, Lie groups, Lie algebras, and representations

Description: The theory of Lie groups and Lie algebras is not only beautiful – it is of great importance in physics, where the representation theory of these objects is used to study the symmetries of the laws of motion and similar.

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Project title: Naive Lie Theory

Graduate mentors: Gabriel Bergeron-Legros

Book: John Stillwell, Naive Lie theory

Description: Stillwell does an amazing job of introducing Lie theory using nothing more. He introduces the abstract group theory and the differential geometry that are needed for the book. Everything he does can easily be understood by following elementary computations.

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FALL 2015 PROJECTS

Project title: Analytic Algebraic Number Theory

Graduate mentor: Vesselin Dimitrov

Books: Elementary and Analytic Theory of Algebraic Numbers by W. Narkiewicz; Number Theory, Fourier Analysis, and Geometric Discrepancy by G. Travaglini. Additionally, some expository and research papers shall be assigned as reading.

Prerequisites: A solid grasp of elementary abstract algebra and elementary real and complex analysis. All further background will be developed in the course of the program. The pace will be determined depending on the student's background and interests.

Description: Traditionally the number theory curriculum has been divided into three main areas according to the methodology used to study them. Thus the elementary theory of numbers could be defined as the direct approach to the integers and the primes not involving particularly deep tools from other disciplines of mathematics; the algebraic theory of numbers begins with Kummer's invention of ideals and with the shift of focus to the Galois group, and has now largely merged with modern arithmetic algebraic geometry; whereas the analytic theory of numbers approaches elementary number theory questions with the methods of harmonic analysis.

Our reading project will merge elements from all three branches of this conditional division. With Travaglini's book we will develop some basic yet robust Fourier analysis and see how the Fourier duality allows to penetrate the famous circle problem of Gauss, on the size of the discrepancy between the area and number of lattice points inside of disk of growing radius, as well as the tightly analogous and equally important Dirichlet divisor problem. Then these problems could be extended to estimating the number of integral ideals with bounded norm from an algebraic number field. With Narkiewicz's book we will quickly build the theoretical minimum of algebra and complex analysis to see how the Fourier duality leads to the analytic continuation of zeta functions and to the duality of the primes with the latter's complex zeros, to reach several landmark results in algebraic number theory such as Landau's prime number (ideal) theorem, the Brauer-Siegel theorem, and the Chebotarev density theorem in its original logarithmic form not requiring Artin reciprocity. Time permitting, this could then be followed by an introduction to some currently hot topics such as the distribution of Euler-Kronecker invariants, the Tsfasman-Vladut theory extending Brauer-Siegel to infinite global fields, and a new approach to the distribution of primes via the distribution of algebraic numbers of small height.

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Project title: Cohomology and manifolds

Graduate mentor: Daping Weng

Book: Bott & Tu, Differential Forms in Algebraic Topology

Prerequisite: Calculus; some knowledge of smooth manifold will be helpful, but not necessary.

Project Description: We will learn about de Rham cohomology of a manifold, with Poincare duality as our goal. If time allows, we will also learn about sheaf (Cech) cohomology and spectral sequences.

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Title: Hurwitz theory

Graduate mentor: Dhruv Ranganathan

Book: We'll use a freely available copy of a textbook in preparation on undergraduate Hurwitz theory by Renzo Cavalieri and Eric Miles.

Prerequisites: Basic abstract algebra (e.g. the symmetric group, homomorphisms) and multivariable calculus/basic analysis.

Description: A common theme in mathematics is that objects, particularly geometric ones, aren't so fun in isolation -- geometry is a team sport. Maps between geometric objects (continuous maps, differentiable maps, or algebraic maps) are where the objects truly come alive. Hurwitz theory is the study of maps between Riemann surfaces (or algebraic curves). This old and remarkably rich theory intertwines between representation theory, topology, combinatorics, algebraic and complex geometry, and even physics. Our goal will be to learn the basics of Hurwitz theory, learning the relevant background along the way. As payoff for our hard work, we will be able to use Hurwitz theory as a lens through which to watch these subjects interact.

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Project title: Mapping class groups

Graduate mentor: Michael Landry

Textbook: A primer on mapping class groups, by Benson Farb and Dan Margalit

Prerequisites: Group theory, and ideally a basic course in algebraic topology (fundamental group, covering spaces). If necessary, we may read Hatcher's textbook for extra background.

Description: The mapping class group of a surface is its group of homeomorphisms modulo isotopy. According to Benson Farb, "the appearance of mapping class groups in mathematics is ubiquitous." Indeed, mapping class groups show up in fields including geometric group theory, symplectic geometry, algebraic geometry, and geometric and low-dimensional topology, just to name a few. This semester we will learn the basics of mapping class groups, and hopefully understand connections with the geometry and topology of 3-manifolds.

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Project Title: Toric Geometry

Graduate mentor: Jifeng (Tif) Shen

Prerequisites: Knowledge of basic abstract algebra: groups, rings, ideals, fields and vector spaces. Previous exposition to basic algebraic geometry is recommended. Knowlege of some basic point-set/algebraic topology is useful, though not necessary.

Book: Introduction to Toric Varieties, by William Fulton

Other references: Foundations of Algebraic Geometry, by Ravi Vakil; Toric Varieties, by David Cox, John Little, and Hal Schenck.

Description: For algebraic geometers, toric varieties expose the combinatorial aspects of algebraic geometry, and are key to the developments of tropical geometry. For combinatorialists, toric varieties made possible tackling hard problems about polyhedral geometry using cohomology theories on toric spaces.

In this reading project, we will use toric varieties as a starting point for algebraic geometry. We will mainly be following Bill Fulton's Toric Varieties. It is a clear and concise book, meant to reward those reading it with patience. Our hope is to reach chapter 3 of Fulton's book, where it will become clear that, in algebraic geometry, all roads lead to toric geometry.

Ravi's notes, and Toric Varities by Cox, Little and Schenck, will provide supplemental background readings, whenever needed. The mentor will provide futher contents and perspectives through discussions with the participant.

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Project title: Young tableaux

Graduate mentor: Dan Corey

Book: Young Tableaux by William Fulton

Prerequisites: Basic algebraic constructions (groups, rings, modules, fields). Exposure to representation theory and very basic algebraic geometry will be helpful for motivational purposes, but not absolutely necessary.

Description: Young tableaux are simple combinatorial gadgets that amount to putting numbers into an arrangement of boxes associated to partition. However, they prove to be a indispensable tool used to study the representation theory of S_n and GL(n,C). These techniques crop up in algebraic geometry while exploring the combinatorics of Grassmannians and flag varieties. In this directed reading, we will follow Fulton's beautiful exposition by carefully developing the combinatorics of Young tableaux and then applying our efforts to representation theory and algebraic geometry.

SPRING 2015 PROJECTS

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Project title: Algebraic Curves

Mentor: Shaked Koplewitz

Prerequisites: Basic familiarity with rings, ideals, and polynomials.

Description: I'd like to go through algebraic curves and get a good grasp on basic algebraic geometry, seen from the viewpoint of modern algebraic geometry while using it for curves to see its practical uses and applications. The planned text is Fulton's "Algebraic Curves".

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Project title: Analytic Number Theory

Mentor:Liyang Zhang

Prerequisites: basic algebra(linear algebra, groups, rings, fields), basics of real and complex analysis

Description: Analytic number theory is a branch of number theory that uses techniques from analysis to solve problems about the integers. It is well known for its results on prime numbers (for example the celebrated Prime Number Theorem states that the number of prime numbers less than N is about N/logN) and additive number theory (the recently proved Goldbach’s weak conjecture states that every odd number greater than 7 can be expressed as the sum of three odd primes). Depending on specific interests, we can read any one of the following texts:

(1) "An Introduction to Analytic Number Theory" by Tom Apostol. A proof of Prime Number Theorem using analytic properties of the zeta function.

(2) “Additive Number Theory” by Melvyn Nathanson. This book highlights two important methods in additive number theory: sieve methods and the circle method. Both of these methods are proved to be very effective in obtaining results towards the twin prime conjecture and the Goldbach conjecture.

(3) Multiplicative Number Theory I. Classical Theory by Montgomery and Vaughan. A more advanced introduction to the Prime Number Theorem and primes in arithmetic progressions.

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Project title: Category theory

Project title: Algebraic Curves

Mentor: Shaked Koplewitz

Prerequisites: Basic familiarity with rings, ideals, and polynomials.

Description: I'd like to go through algebraic curves and get a good grasp on basic algebraic geometry, seen from the viewpoint of modern algebraic geometry while using it for curves to see its practical uses and applications. The planned text is Fulton's "Algebraic Curves".

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Project title: Analytic Number Theory

Mentor:Liyang Zhang

Prerequisites: basic algebra(linear algebra, groups, rings, fields), basics of real and complex analysis

Description: Analytic number theory is a branch of number theory that uses techniques from analysis to solve problems about the integers. It is well known for its results on prime numbers (for example the celebrated Prime Number Theorem states that the number of prime numbers less than N is about N/logN) and additive number theory (the recently proved Goldbach’s weak conjecture states that every odd number greater than 7 can be expressed as the sum of three odd primes). Depending on specific interests, we can read any one of the following texts:

(1) "An Introduction to Analytic Number Theory" by Tom Apostol. A proof of Prime Number Theorem using analytic properties of the zeta function.

(2) “Additive Number Theory” by Melvyn Nathanson. This book highlights two important methods in additive number theory: sieve methods and the circle method. Both of these methods are proved to be very effective in obtaining results towards the twin prime conjecture and the Goldbach conjecture.

(3) Multiplicative Number Theory I. Classical Theory by Montgomery and Vaughan. A more advanced introduction to the Prime Number Theorem and primes in arithmetic progressions.

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Project title: Category theory

Mentor: Dan Corey

Prerequisites: basic algebraic structures (groups, rings, modules, etc.) and some algebraic topology.

Description: At a first glance category theory provides a unifying language among all the different areas of mathematics. The basic philosophy is that one should shift their focus from mathematical objects themselves to the "morphisms" between them. Following Saunders MacLane's "Categories for the working mathematician," we will begin with the study of categories, functors, natural transformations, limits, universal properties, and adjoints. The main focus will be on how these constructions manifest themselves in the categories already known by the student. After the basics, we can proceed into deeper areas of the subject depending on the student's background and interests.

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Project title: Concentration inequalities

Mentor: Oanh Nguyen

Text: Concentration Inequalities: A Nonasymptotic Theory of Independence, by Stéphane Boucheron, Gábor Lugosi, and Pascal Massart.

Prerequisites: Measure-theoretic Probability.

Description: Roughly speaking, concentration inequalities assert that a random variable stays around its mean with high probability. Markov’s inequality and Chebyshev’s inequality are two of the simplest examples. This book investigates the concentration of functions of independent random variables and offers a lot of applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena, among others. Students in many areas such as machine learning, statistics, discrete mathematics, and high-dimensional geometry will find the book helpful. The text is very nicely written with both intuition and rigorous, yet accessible proofs.

We will read the first few chapters and learn about beautiful/useful/powerful results and concepts such as Chernoff’s, Hoeffding’s inequalites, and Efron-Stein inequality, Log Sobolev inequality with Herbst’s argument, Johnson-Lindenstrauss Lemma, and Entropy.

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Project Title: Enumerative Combinatorics

Mentor: Daniel Montealegre

Prerequisites: Basic discrete math class, the student should be familiar with combinatorial proofs, abstract and linear algebra.

The goal of enumerative combinatorics is to count the number of objects in a finite set. Usually we have an infinite collection of finite sets indexed by the natural numbers (S_0, S_1,S_2,...) and we wish to know information about |S_i|. In this project we will look at different techniques that exploit the algebraic and geometric richness of the objects in order to obtain a satisfactory solution for our problem.

We will be using R. Stanley's Enumerative Combinatorics vol 1, as well as some notes of F. Ardila.

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Project title: Geometric group theory

Mentor: Tarik Aougab

Prerequisites: basics of algebraic topology (fundamental group, quotient spaces).

Description: The main aim of geometric group theory is to understand an infinite group by studying geometric objects on which the group acts. This fascinating subject ties together areas of geometry/topology, probability theory, complex analysis, combinatorics and representation theory. Depending on specific interests, we can read any one of the following texts, or jump around between them:

(1) "Primer on mapping class groups", by Farb and Margalit: a study of the mapping class group of a surface, one of the most fundamentally important groups in low-dimensional topology.

(2) "Notes on notes of Thurston", by Canary, Epstein & Marden: a summarized version of Thurston's famous "notes" on hyperbolic geometry and 3-manifolds.

(3) "Trees", by Serre, and "Topological methods in group theory", by Scott and Wall: these readings summarize tools used to study groups which act on contractible graphs.

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Project Title: Topics in Probability

Mentor:Kyle Luh

Prerequisites: A basic understanding of probability, discrete math, and linear algebra

Description: It's a strange (subjective) phenomenon in mathematics that everyday, mundane mathematical objects become intriguing and vibrant entities simply by appending the word "random" to them. Potential topics for this reading are random matrices and random graphs.

For the first, we would read Tao's book on Random Matrix Theory and for the second, Bollobas' Random Graphs. RMT is an active field of research with many surprising connections to other areas of mathematics. Tao's book guides the reader from basic probability to some of the classic results in the field (semi-circular law, least singular value, circular law, etc.) Bollobas' Random Graphs develops the necessary probability before taking a tour of the major problems in the area (Various models, threshold phenomenon, Ramsey theory, etc.).

Another option is to read the Probabilistic Method by Alon and Spencer. This book touches on the two topics above but mainly focuses on the use of probability in combinatorics and number theory. It will blow your mind.

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Project title: 3-manifolds

Mentor: Michael Landry

Prerequisites: abstract algebra, point set topology, ideally some basic algebraic topology (fundamental group, covering spaces)

Project description: We will try to read through Dale Rolfsen's beautiful book "Knots and Links," which is about knots and links but also much more. One theorem I would like us to cover is the result of Gordon and Luecke that a knot is determined by the homeomorphism type of its complement in the 3-sphere. The book is fairly advanced, so we may need to take some detours to learn background material.

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The Directed Reading Program is inspired by similar programs at UC Berkeley, University of Chicago, UConn, University of Maryland, and Rutgers.