Spring 2019

• Math 72: Topics in Geometry: An Introduction to Lie groups

  Instructor: Craig Sutton

  Description

  Created by Sophus Lie (1842-1899) for the purpose of developing a "Galois theory" of differential equations, Lie groups are a mathematically rigorous realization of our intuitive notion of "continuous transformation groups" and play a fundamental role in the study of geometry and physics. For example, the exceptional Lie group $E_8$ appears in string theory and the associated "$E_8$-lattice" was used by Freedman to construct an example of a four-dimensional topological manifold that does not admit a smooth structure. Technically speaking, a Lie group is a group $G$ equipped with the structure of a smooth manifold with respect to which the group operations (i.e., multiplication and inversion) are smooth. Our exploration of Lie groups will begin with the study of "matrix groups" (e.g., $\operatorname{SO}(n)$, $\operatorname{SU}(n)$ $\operatorname{Sp}(n)$ and $\operatorname{SL}_n(\mathbb{R})$). By focusing on this concrete class of examples, we will build our intuition and encounter many of the interesting themes that arise in the general theory of Lie groups.

  Audience and prerequisites: This course will be of interest to students who are curious about geometry, (theoretical) physics, robotics, computer animation and/or applications of geometric techniques. I will assume you have a working knowledge of multivariable calculus, (abstract) linear algebra and abstract algebra, or possess the willingness to fill in any gaps on the fly.

Spring 2017

• Math 118: Topics in Combinatorics: Language Theory and Applications to Enumerative Combinatorics

  Instructor: Jay Pantone

  Description

  The topic of this course is Language Theory and Applications to Enumerative Combinatorics. Among the topics we will cover are: finite state automata (deterministic and non-deterministic), regular languages, pushdown automata, context free grammars, Turing machines, finite state transducers, the pumping lemmas, and decidability. We will see many examples of combinatorial families that are in bijection with the objects above, allowing for automatic enumeration through the theory of generating functions.

  Prerequisites: An undergraduate combinatorics course (preferably, some familiarity with generating functions). Ask the instructor if in doubt.

Winter 2017

• Math 105: Topics in Number Theory

  Instructor: Naomi Tanabe

  Description

Fall 2016

• Math 108: Topics in Combinatorics: Permutations, partitions and lattice paths

  Instructor: Sergi Elizalde

  Description

  This course focuses on three important objects in combinatorics: permutations, partitions and lattice paths, as well as the connections between them. Although many of the topics that we will cover are well-known results in enumerative combinatorics, we will also get a glimpse of current research.

  This course is aimed at graduate students and strong undergraduate students who have taken some combinatorics course before.

Spring 2016

• Math 115: Elliptic Curves

  Instructor: John Voight

  Description

  Elliptic curves are almost too beautiful to exist: somehow, they manage to embody many different kinds of mathematical objects simultaneously. On the one hand, the set of points of an elliptic curve naturally forms a group, so you can add two points on the curve to make a third. This group law is provided by algebraic equations, but can also be described geometrically: three points sum to zero when they lie on a line. From a complex analytic point of view, an elliptic curve is just a torus. And on top of all of this, elliptic curves over finite fields are ideal for use in cryptography.

  This course will be a graduate-level introduction to elliptic curves in which we will cover the fundamentals of the subject. Topics may include: rudiments of algebraic curves, Weierstrass equations, isogenies and endomorphisms, elliptic curves over finite fields and applications to cryptography, elliptic curves over the complex numbers and complex multiplication, the Mordell-Weil theorem and applications to Diophantine equations, and other topics as time permits.

  The prerequisites for the course are one year of abstract algebra (groups, rings, fields), preferably at the graduate level, and some complex analysis. The course may be suitable for first-year graduate students--please see the instructor.

Winter 2016

• Math 118: Combinatorics

  Instructor: Jay Pantone

  Description

  This course will start by covering the symbolic method of Flajolet and Sedgewick and proceed to the asymptotic analysis of generating functions. After that we'll cover the theory of languages (e.g., regular languages, context-free grammars, etc). If there is time, then we'll finish the semester with a series of short special topics, including pattern-avoiding permutations.

  Prerequisites: An undergraduate combinatorics course (in particular, familiarity with generating functions). Ask the instructor if in doubt.

• Math 123: Automorphic forms, representations and C*-algebras

  Instructor: Pierre Clare

  Description

  The representation theory of Lie groups is a vast subject, with ramifications ranging from harmonic analysis to number theory. The main objective of this course will be to explore some of its ties to the classical theory of automorphic forms. Along the way, we will discuss standard Lie-theoretic methods and the approach of non-commutative geometry to representation theory via group C*-algebras.

  Prerequisites: a good acquaintance with linear and general algebra (as provided for instance in Math 71) is necessary. Some exposure to complex and functional analysis is preferable. No prior knowledge of representation theory or C*-algebras will be assumed. Contact the instructor for more details.

Fall 2015

• Math 100—COSC 49/149
  An Introduction to Mathematics Beyond Calculus: Game Theory

  Instructor: Peter Winkler

  Description

  Game Theory is one of the great accomplishments of modern mathematics, with myriad applications to economics, computer science, political science, decision theory, warfare, and more. This course will tackle the key results in each of the main branches of game theory, using a variety of mathematical techniques.

  We will begin with the simplest and most famous game model: two-player, simultaneous, one-move games such as the “prisoners' dilemma,” and the notion of Nash equilibrium. (Sadly, John Nash, a “beautiful mind,” recently lost his life in a car crash on the NJ Turnpike.) From this we will branch out to multiple players, repeated games, auctions, voting, and mechanism design.

  Prerequisites: Mathematical background of a senior mathematics major or a beginning graduate student in mathematics or theoretical computer science; including a course in probability (e.g., MATH 20 or Math 60). If in doubt, please see the instructor.

• Math 112: Geometric Group Theory

  Instructor: Bjoern Muetzel

  Description

  By associating a group to one of its Cayley graphs, the properties of a group can be studied from a geometric point of view. The group itself acts on the Cayley graph via isometries which is reflected in the symmetries of the graph. The inherent beauty of a group can thus be visualized in its Cayley graph making these graphs a fascinating object of study.

  Geometric group theory examines the connection between geometric and algebraic invariants of a group. In order to obtain interesting invariants one usually restricts oneself to finitely generated groups and takes invariants from large scale geometry. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry and differential geometry and has numerous applications to problems in classical fields, like combinatorial group theory, graph theory and differential topology.

  Topics: Graphs and trees, Cayley graphs, free groups, hyperbolic groups, large scale geometry.

  Prerequisites: Math 71 and 101 and a solid background in topology (point set topology, fundamental group, covering space theory). This course aims at second year graduate students, but will be accessible to other students with the appropriate background.

Summer 2015

• Math 125: Geometry of Discrete Groups

  Instructor: John Voight

  Description

  This course will provide an introduction to the geometry of discrete groups. Specifically, we will study the action of discrete subgroups (lattices) inside groups of isometries of hyperbolic spaces, especially dimensions 2 and 3. An interesting class of such groups arises from arithmetic constructions, and the corresponding quotients give rise to orbifolds and manifolds of interest in number theory, geometry, spectral theory, combinatorics, and topology.

Spring 2015

• Math 116: Applied Mathematics: Mathematical Modeling in Biology

  Instructor: Olivia Prosper

  Description

  Biology presents complex problems requiring quantitative approaches to tackle them. This term, Math 116 will serve as an introduction to mathematical modeling in biology, with an emphasis in modeling disease dynamics. You will learn to construct, analyze, and simulate models and interpret your results within their biological context. The course will focus on two or three of the following topics: classical techniques for analyzing nonlinear ordinary differential equations, metapopulation models, delay-differential equations, age-structured and time-since infection (PDE) models, stochastic models, optimal control theory and its application to biological problems, and model validation. By the end of the term, you will have experience posing biological problems and using mathematics to elucidate them.

Fall 2014

• Math 100: Topics in Probability: Large Networks and Graph Limits

  Instructor: Peter Winkler

  Description

  Large networks are everywhere these days and in this course we will study a whole new branch of mathematics designed to help us understand these beasts. Our text (and about 250 research papers) constitute the literature of the field. The idea is that by introducing limit structures, one can state and prove theorems about graphical structures so large that they can be appreciated only via statistical tests.

  Prerequisites: A solid background in mathematics, including calculus and at least one course in probability. Exposure to graph theory or measure theory will be handy but won't be assumed. Graduate students and advanced undergraduates studying mathematics or the theory of computing will most likely have adequate mathematical sophistication, but fair warning: this stuff is at the frontier of research; it isn't easy!

• Math 105: Algebraic number theory

  Instructor: John Voight

  Description

  This course will be a graduate-level introduction to algebraic number theory, in which we will cover the fundamentals of the subject. Topics may include: rings of integers, Dedekind domains, factorization of prime ideals, Galois theory in number fields, geometry of numbers and Minkowski's theorem, finiteness of the class number, Dirichlet's unit theorem, selected topics from analytic number theory, quadratic and cyclotomic fields, localization and local rings, valuations (i.e. p-adic) and completions, an introduction to class field theory, application to Diophantine equations, and other topics as time permits.

  The prerequisite for the course is one year of abstract algebra (groups, rings, fields) at the advanced undergraduate or graduate level.

Spring 2014

• Math 96: Mathematical Finance II

  Instructor: Sutton

  Description

  In the study of ordinary differential equations the parameters and coefficients of the equations are assumed to be deterministic. In contrast, a stochastic differential equation is a differential equation in which one or more of its terms is governed by a stochastic (or random) process. As one might imagine, stochastic differential equations (SDEs) arise naturally in the study of finance, engineering, economics, physics and mathematics. This term Math 96 will serve as an introduction to the theory and applications of SDEs with an eye towards finance. Topics may include some of the following:

  1. Probability spaces, Stochastic processes & Brownian Motion
  2. Ito Integrals, the Ito formula and the Martingale Representation Theorem
  3. Existence & Uniqueness of Solutions
  4. Basic Properties of Diffusions
  5. Feynman-Kac Theorem & the Girsanov Theorem
  6. Optimal Stopping Time
  7. Applications to financial derivatives on equities and fixed-income securities
  8. Applications to Boundary-Value Problems & Control Theory

  Prerequisites:

  1. Math 86 or Permission of the instructor

  2. Some knowledge of analysis at the level of Math 63/35 or 103 will be useful

• Math 112: Geometry — The Structure and Representation Theory of Compact Lie Groups

  Instructor: Sutton

  Description

  The theory of Lie groups has its roots in Sophus Lie's quest to develop a Galois theory for differential equations and Klein's “Erlanger Programm” which places symmetry groups at the center of the study of geometry. During the ensuing 120 plus years, Lie groups have become an indispensable part of the study of modern geometry and physics. In this course we will focus on the structure and representation theory of compact Lie groups with an eye towards applications in geometry. Topics will include some of the following:

  1. Lie Groups and Lie Algebra Basics: Lie groups; the adjoint representation, the Lie bracket and Lie algebras; left-invariant vector fields, one-parameter subgroups and the exponential map; logarithmic coordinates and Dynkin's formula; the Killing form.
  2. Structure of Compact Lie groups: the theorem on maximal tori; roots, root systems, Dynkin diagrams and their classifications; the co-root, central and integral lattices; the center & fundamental group of a compact Lie group; Structure.
  3. Representation theory of Compact Lie groups: Schur's lemma, the Peter-Weyl Theorem, induced representations, Weyl's character and dimension formulae, representations of the classical Lie groups.
  4. Introduction to Symmetric Spaces

  Prerequisites: familiarity/comfort with manifolds (e.g. Math 124) & a solid background in linear algebra (e.g., Math 24) and groups (e.g., Math 71).

• Math 116: Applied Mathematics — Great Papers in Numerical Analysis (Alex Barnett)

  Prereqs: programming (eg CS1 or Math26), Math 63, Math 23, Math 22/24. Graduate analysis (73/103) will help.

  Description

  We will overview key background in numerical analysis and computation, and then students, with instructor help, will read and present from a set of a dozen or so classic influential papers that have shaped modern numerical methods. The course will include coding up some of the algorithms discussed.

Winter 2014

• Math 128: Current Problems in Symmetric Functions (Rosa Orellana)

  Prerequisite: An algebra course ( e.g., M71 or M101) and a basic combinatorics course (M28) as well as a desire to learn and solve problems. If you have not had a course in combinatorics and would like to take the course, talk to Rosa.

  Description

  The first 2–3 weeks will be a crash course on symmetric functions. Then we will spend the rest of the term surveying open problems related to Kostka numbers, Littlewood-Richardson coefficients and Kronecker coefficients. We will also survey the literature for recent results in the area.

  Students will learn how to use SAGE to generate data and look for patterns in the data. We will make conjectures and work on them throughout the term and beyond if there is enough interest.

  Homework: Students will be expected to write simple programs and read articles and present them to the class. There will be no exams.

• Math 123: Geometry and Quantization (Erik Van Erp)

  Description

  I will start with explaining the formalism of Hamiltonian mechanics. First we develop it in Euclidean space, and then generalize to the setting of symplectic manifolds. We study Hamiltonian flow, Poisson brackets, moment maps, Darboux's theorem, and perhaps Noether's theorem on the relation between symmetry and conservation laws. We then discuss the relation between Hamiltonian mechanics and quantum mechanics, and study mathematically precise ways in which classical mechanics is a limit of quantum mechanics (the “correspondence principle”). We then cover various topics from geometric quantization theory, and/or contact topology, depending on time and interest.

• SPECIAL TOPICS COURSE: Homogeneous Ricci flow and solitons

  Instructor: Jorge Lauret (visiting from University of Cordoba, Argentina). Preparation lectures: Carolyn Gordon

  Prerequisite: Differential topology. Students should have had some exposure to Riemannian geometry. However, if you are interested in the course and have not had a course in Riemannian geometry, we can include an introduction in January. Please discuss your background in advance with Carolyn Gordon.

  Description

  The Ricci flow on Riemannian manifolds has been a subject of intense activity in recent years, especially since it played a central role in the solution of the Poincare Conjecture. Jorge Lauret has done groundbreaking working on Ricci flow and Ricci solitons in the setting of homogeneous spaces.

  Lauret will give twelve 90-minute lectures during the month of February. The first six lectures will address flows on homogeneous spaces, in particular, the Ricci flow. The second half of the lecture series will address homogeneous Ricci solitons; these are metrics that essentially remain stable under the Ricci flow.

  In preparation for Lauret's lectures, we will hold regular 65-minute classes in January to address background material on Lie groups and homogeneous spaces. The exact content of the January lectures will depend on the backgrounds of the students.

Spring 2013

• Math 17: Imaginary numbers are real! Complex numbers are simple! (Doyle)

  We will survey the role of complex numbers across the mathematical spectrum, from the central limit theorem of probability, to the distribution of prime numbers, to hyperbolic geometry, to the mathematical apparatus of quantum mechanics.

• Math 102: Topics in Geometry (Doyle)

  Description

  This course will be a general introduction to Fuchsian groups, with emphasis on spectral geometry and arithmetic groups. The subject of Fuchsian groups lies at the intersection between geometry, topology, algebra, number theory, and analysis, so it should be of interest to a wide range of graduate students, including first-year students. The course should also be accessible to undergraduates who have taken Math 63 and 81 (or in exceptional case, 63 and 71).

  The primary course text will be Svetlana's Katok's book ‘Fuchsian groups’. A good part of the course will be devoted to developing the techniques to compute explicit examples, using Mathematica, Sage, and Magma.

• Math 118: Topics in Combinatorics (Elizalde)

  Prerequisites: An "advanced" undergraduate combinatorics course

  Description

  I've chosen topics that I haven't covered in my more recent graduate courses, so they should be mostly new for everyone. Possible topics include sieve methods (Gessel-Viennot formula, involutions), partitions (Euler, Jacobi and Rogers-Ramanujan identities), P-partitions, partially ordered sets, plane partitions, tilings, reduced decompositions of permutations, parking functions, and an introduction to polytopes (Dehn-Somerville equations, associahedron, permutahedron).First paragraph

• Math 125: Quadratic forms and spaces (Shemanske)

  Prerequisites: Everyone should have adequate algebra by spring, and while it would be nice to be acquainted with number fields and their (p-adic) completions, the essentials can be picked up with reasonable ease.

  Description

  In the mid 1930's Witt dramatically changed the perspective by which mathematicians viewed the theory of quadratic forms, from a study of homogeneous polynomials of degree 2 to the study of vector spaces endowed with a symmetric bilinear form. The theory then became one of determining canonical decompositions and of classification of these spaces in terms of arithmetic invariants.

  The course will be distinctly algebraic in flavor, and in general we will be interested in the theory over number fields and Dedekind rings, meaning the focus will also be influenced by questions in algebraic number theory.

Winter 2013

• Math 100: Topics in Probability [Random Walk on a Graph] (Winkler)

  Prerequisites: Basic probability (e.g. Math 20 or 60), and some experience with proofs; graph theory or combinatorics will be useful but not necessary. Graduate students at all levels in math and in computer science are welcome, as are advanced undergrad majors.

  Description

  When a token moves randomly from vertex to vertex via the edges of a graph, it is taking a random walk. This simple process is as useful as it is elegant, with multiple analogies in electrical networks (cf. Doyle and Snell's Random Walks and Electrical Networks , which will be one of our sources) and applications to the theory of computing. There have been remarkable new advances in this now-classical subject and we will spend much time with recent work and unsolved problems.

• Math 112: Topics in Geometry [Introduction to Riemannian Geometry] (Sutton)

  Description

  Manifolds are one way in which mathematicians deal with the concept of "space," and the presence of a Riemannian metric on a manifold provides us with a framework and a set of tools with which we may explore the geometric and topological nature of the space in question. Among the tools available to us, curvature is perhaps the most fundamental. The three primary flavors of curvature are sectional curvature, Ricci curvature and scalar curvature, and each provides us with a local characterization of how much a given space deviates from being Euclidean. After laying the basic foundations of Riemannian geometry, one of our main objectives will be to examine how (global) constraints on curvature influence the topological and geometric structure of the space. In particular, we will prove the topological sphere theorem and---as time and interest permits---we will discuss the differentiable sphere theorem of Brendle and Schoen from 2009.

  Topics: connections, Riemannian metrics, volume, curvature and isometries; geodesics, Jacobi fields, the energy functional, first & second variations of energy; spaces of constant curvature; (locally) symmetric spaces; isoperimetirc inequalities; spectral geometry; sphere theorems.

• Math 126: Topics in Applied Mathematics (Gillman)

  Prerequisites: Graduate students at all levels in math and in computer science are welcome, as are advanced undergrad majors who have taken Math 23.
  Suggested background: Some coding experience (Matlab, Fortran, or C), Math 46, Math 63

  Description

  Partial differential equations (PDEs) are essential for the modelling of physical phenomena appearing in a variety of fields from geophysics and fluid dynamics to geometry. In this course, we will study three major topics one should understand when modelling with PDEs. The topics are:
  
  (i) the theory (e.g. existence and uniqueness of solutions)>
  (ii) when and how can solutions be found analytically>
  (iii) classic numerical techniques (e.g. finite difference and finite element methods) and how to determine if the method is stable and convergent.

  In addition, we will discuss the limitations of existing solution techniques in the context of open research questions.

Fall 2012

• Math 105: Topics in Number Theory (Pomerance)

  Prerequisites: A knowledge of elementary number theory and some abstract algebra.

  This class has been scheduled for the 10A period.

  Description

  This course will be an introduction to Diophantine Equations and Diophantine Approximation. The first topic involves determining if a polynomial equation in several variables has any integer or rational solutions, and if it does, how to find them. The second topic refers to the approximation of real numbers with rational numbers. We shall see that the two topics are closely intertwined. Some of the more specific topics will be the Pell equation (quadratic in two variables) and the Thue equation (a homogeneous polynomial in two variables equal to a constant). We shall see how to prove that e and pi are transcendental numbers, and we'll discuss some other pretty constants.

  We will start out using the book Lecture notes on Diophantine analysis by Umberto Zannier. This book is available from amazon.com for about $30 including shipping, but I'm not sure how robust the supply is. It would be good to order a copy soon. Partway into the term we'll branch out using some other material, which I'll make available.

  There will be some written homework assignments, and students will be expected to occasionally present topics to the class, possibly a homework solution or a new topic. There will be no formal examinations.

• Math 109: Topics in Logic (Groszek)

  Prerequisites: no prerequisites for graduate students.

  Description

  The course will be on the general topic of mathematical logic and arithmetic. Depending on the interests of the students, we will begin with a discussion of Godel's Incompleteness Theorem, and then move on to reverse mathematics (using logic to analyze the relative strength of theorems of classical mathematics), nonstandard analysis (calculus with infinitesimals), the unsolvability of Hilbert's Tenth Problem (maybe my favorite choice), or a related topic of interest.

  For students who have begun thesis work in another area, attendance will be the major part of the grade. Students who are (possibly) preparing for a logic qual should talk to me about targeted homework assignments.

Spring 2012

• Math 121: Current problems in algebra (Webb)

• Math 125: Reflection and Coxeter groups; Buildings and Classical Groups (Shemanske)

  Prerequisites: 101, 111 (suitable for first year students).
  Any number theory needed (not much) will be developed.

  Details

  This is bits of algebra, geometry, topology and a little combinatorics all rolled into one course. The goal is to say something about the theory of buildings, with slightly more interest in their combinatorial/topological structure and number theoretic applications than their original purpose, which was to understand the structure and classify classical p-adic groups (using the building as a representation space), in analogy to semi-simple Lie theory.

• Math 128: Topics in Combinatorics (Elizalde)

  Prerequisites: Math 118. If you have had an advanced undergraduate course in combinatorics and are interested, talk to Sergi about your preparation.

  Details

  The course will cover pattern-avoiding permutations, permutations in dynamical systems, bijections, counting methods, generating trees, and perhaps asymptotic analysis of generating functions, among other topics to be decided.

Winter 2012

• Math 17: An Introduction to Mathematics Beyond Calculus (Shemanske)

  Prerequisite: Math 8, or placment into Math 11.

  Details: This year's offering will be “From Caculus to Elliptic Curve Cryptography in ten weeks”
  See the web site.

• Math 89: Set theory (Weber)

  Prerequisite: Math 39 or Math 69 or familiarity with the language of first-order logic and readiness for an upper level math course.

  Details

  Math 89 satisfies the culminating experience requirement for mathematics majors, and is appropriate for any graduate student who wants to take a course in logic and doesn't yet know a lot of set theory.

  We will study the axioms of set theory, what they tell us about the universe of sets, and how we can use the universe of sets to study the universe of mathematics.

• Math 112: Geometry (Gordon)

  Prerequisite: A course in differential topology, including vector fields and their flows. (The course that used to be called Math 124 and is called Math 102 this fall is ideal.)

  Details

  Lie groups and Lie algebras are ubiquitous throughout mathematics, e.g. in geometry, number theory, combinatorics, analysis and dynamics. The first part of the course will address the basic notions and results elucidating the Lie group -- Lie algebra correspondence. We will then introduce various types of Lie groups and Lie algebras (semisimple, solvable, nilpotent, compact). There are various directions that we could pursue after that, depending on time and the interests of the class.

• Math 122: The Atiyah-Singer index theorem and the heat kernel proof (van Erp)

  Prerequisites: the introductory graduate level analysis and topology sequences (103/113, 124/114).

• Math 126: Numerical analysis for PDEs and wave scattering (Barnett)

  Prerequisite: some programming experience (preferred: Matlab/octave, C, or fortran; esp. the first).

  Recommended background: some PDEs (could be at undergrad level, eg Math 46) and real analysis (Math 63 and some graduate-level functional analysis). However the background is flexible: a motivated advanced undergrad or other science/engineering/CS student (undergrad or grad) could pick up enough to learn a lot and do well.

  Details

  This will cover modern integral-equation-based methods for solving piecewise-constant coefficient PDEs (focusing on those arising in electrostatics and time-harmonic waves), including the theory, analysis, and coding experience required for proficiency. Much content will overlap with previous Math 116's offered by Barnett.

  We start with an introduction to numerical analysis, conditioning and stability, numerical interpolation and integration, followed by potential theory, Laplace's equation and Helmholtz equation. The latter leads to scattering problems, a flavor of fast algorithms (FFT, fast multipole), and final projects. Homework will be dominated by coding and implementing the algorithms, but also include some proofs. Projects can be coding-based or a deeper study of numerical analysis results.

Fall 2011

• Math 102: Foundations of Smooth Manifolds (Sutton)

  Prerequisites: Linear algebra (Math 24), point-set topology (math 54) and multivariable analysis (Math 73). It will also help to be familiar with covering spaces and the fundamental group.

  Description

  Manifolds provide mathematicians and other scientists with a way of grappling with the concept of "space." The space occupied by an object. The space that we inhabit. Or, perhaps, the space of configurations of a mechanical system. While manifolds are center stage in the study of geometry and topology, they also provide an appropriate framework in which to explore aspects of mathematical physics, dynamics, control theory, medical imaging, econometrics and robotics, to name just a few. This course will serve as an introduction to the basics of manifold theory and lay the foundations needed to explore problems in which "space" plays a fundamental role.

  Topics will include some of the following: manifolds & tangent bundles; vector fields & vector bundles; smooth maps and the inverse function theorem; cotangent bundle & differential forms; densities, integration on manifolds and the generalized theorem of Stokes; Whitney's imbedding theorem; Lie groups, group actions & homogeneous spaces; Tensor Fields & Riemannian metrics; Differential forms & orientation; integral curves & the existence and uniqueness theorem for ODEs; Distributions and Frobenius' Theorem.

• Math 105: Primes and polynomials (Pomerance)

  Prerequisites: An undergrad number theory course, as well as some abstract algebra. I'll be happy to try and fill in gaps for motivated students.

  Description/Grading

  This course will develop elementary estimates on the distribution of prime numbers, including elementary sieve methods. A recurrent theme will be the prime factorization of integer-polynomial values. We will follow the first 3 chapters of Pollack's "Not always buried deep", as well as Chapter 6, and also Chapter 3 of Montgomery & Vaughan's "Multiplicative number theory". I recommend acquiring Pollack's book, the second book is optional.

  Grading: There will be weekly written assignments, plus some class presentations. There will be no formal exams. Enrolled graduate students who have been admitted to candidacy will be excused from the written assignments.

• Math 108: Combinatorial Representation Theory (Orellana)

  Prerequisites: Linear algebra and algebra (Math 31, 71, or 101). No prior knowledge of combinatorics or representation theory is expected.

  Details

  This is an introduction to algebraic combinatorics, specifically symmetric function theory and its connections to representation theory, algebraic geometry, and tableaux combinatorics.
  Full description.

Winter 2011

• Math 17: Beyond Calculus

  
  W11 topic: Random Walks and Electric Networks
  
  

• Math 102: Topics in Geometry

  
  W11 topic: The soccer ball and the solution of equations of the fifth degree
  
  

Winter 2010

• Math 17 (An Introduction to Mathematics Beyond Calculus)

  
  W10 topic: Applications of algebraic structures in number theory and geometry
  
  

Winter 2008

• Math 17 (An Introduction to Mathematics Beyond Calculus)

  
  W08 topic: The Isoperimetric Problem
  

Fall 2007

• Math 53

  (Chaos!)

Winter 2007

• Math 17 (An Introduction to Mathematics Beyond Calculus)

  
  W07 topic: The Geometry of the Fourth Dimension

Spring 2006

• Math 112 (Introduction to Riemannian Geometry)

Winter 2006

• Math 116 (Boundary Methods and Wave Asymptotics)