All inquiries about our graduate program are warmly welcomed and answered daily:
All inquiries about our graduate program are warmly welcomed and answered daily:
This VIR class will concentrate on the interplay between group actions, group representations and physics. Symmetries in the physical world are often described by abstract groups in mathematics. For example, electrons on an orbit observe a certain symmetry defined by permutation group. We will focus on finite groups and continuous groups like the symplectic groups. It will cover topics that are interesting for undergraduate students, like the groups of permutations, and how one can determine their representations. For graduate students then there will be more advance topics like bounded symmetric domains, Geometric quantization, unitary representations of the symplectic groups and why those are interesting for questions in physics.
We will start with some introductory topics in physics and representations of the symmetric group and then slowly move into unsolved problems in mathematical physics. The topics are relevant in fields like analysis, number theory, and physics.
Over the last 20 years knot theory became one of the central areas in mathematics. One studies properties of knots, like the unknotting number which measures the easiest way to unknot a knot. Our interest will be in the surfaces on which knots project in some nice way, and what those projections tell us about the knot.
We will learn and apply methods in computer graphics, differential geometry, knot theory and other areas of mathematics. The course is intended for both undergraduate and graduate students.
The course provides a self contained and hands on introduction to the field of homogenization theory as well as a guide to the current research literature useful for understanding the mathematics and physics of complex heterogeneous media. The first part of the course introduces the variational tools and useful asymptotic techniques necessary for characterizing the macroscopic behavior of heterogeneous media. Next we explore methods for constructing solutions of field equations inside extreme microstructures such as the the space filling coated spheres construction of Hashin and Shtrikman and the confocal ellipsoid construction of Milton and Tartar. The third part of the course shows how to apply these tools and field constructions to recover new theorems that characterize extreme field behavior inside complex materials
in terms of the statistics of the random medium and the applied incident fields.
This course provides practical training in the teaching of mathematics at the pre-calculus level, how to write mathematics for publication, and treats other issues relating to mathematical exposition. Communicating Mathematics I and II are designed to provide training in all aspects of communicating mathematics. Their overall goal is to teach the students how to successfully teach, write, and talk about mathematics to a wide variety of audiences. In particular, the students will receive training in teaching both pre-calculus and calculus courses. They will also receive training in issues that relate to the presentation of research results by a professional mathematician. Classes tend to be structured to maximize discussion of the relevant issues. In particular, each student presentation is analyzed and evaluated by the class.
This is the first semester of the first year graduate algebra sequence. It will cover the basic notions of group, ring, and module theory. Topics will include symmetric and alternating groups, Cayley's theorem, group actions and the class equation, the Sylow theorems, finitely generated abelian groups, polynomial rings, Euclidean domains, principal ideal domains, unique factorization domains, finitely generated modules over PIDs and applications to linear algebra, field extensions, and finite fields.
The representation theory of Hopf algebras/Quantum groups has attracted a lot of attention since the mid 80's. This was stimulated by many interesting relations to combinatorics, topology, mathematical physics and integrable systems.
We will first go over the theory of Poisson Lie groups. Then we will study finite dimensional representations and crystal bases. The course will finish with representations in the case when a certain deformation parameter is a root of unity.
The notion of a module over a ring is one of the most basic concepts in algebra, but when the ring is not semisimple, understanding the possible structure of modules is remarkably difficult. This course will present a modern (post-1970) approach to this subject, using homological methods and the Auslander-Reiten theory of ``almost split sequences," and focusing on quiver representations as a source of motivation and examples. Later in the semester, I hope to discuss quasi-hereditary algebras, highest-weight categories, and the "abstract Kazhdan-Lusztig theory" developed by Cline-Parshall-Scott in the late 1980s.
This course will address the classical theory of real valued functions, measure, and integration.
Functional Analysis is the language of modern mathematics. The course provides an introduction to the general theory of normed and metric spaces, linear operators and linear functionals, Banach and Hilbert spaces, spectral theory, and other important topics.
This class introduces the basic nonlinear optimization theories and algorithms. The class (temporarily) includes the following topics:
The course will span a wide variety of topics in partial differential equations with a detailed understanding of illuminating examples. It will combine self-contained development of theories with exposure to broader developments through presentation of ideas and theorems from the literature.
1. Basic linear equations and separation of variables.
2. Fourier transform methods.
3. Boundary-integral equations and their connection to complex variables and wave scattering.
4. The method of characteristics for first-order equations.
5. Calculus of variations, the Hamilton-Jacobi equation, the iconal equation in geometric optics.
6. Quantization and the Schrödinger equation.
7. Unitary groups, and the role of self-adjoint extensions of symmetric operators in boundary-value problems.
8. Nonlinear evolution equations: hyperbolic systems and shocks; the Korteweg-deVries equation.
9. Asymptotic analysis: the method of stationary phase and WKB theory; application to the vanishing-dispersion limit of the KdV equation.
10. Special relativity: differential geometry and the special role of the wave equation.
The purpose of this course is to survey the entire theory of graph-minors. We will emphasis more on the breath of the subject, while the depth will be secondary. Topics to be covered in this course include the following.
1. Classical results of Kuratowski, Wagner, Dirac, and Tutte.
2. Connectivity and splitter theorems.
3. The graph-minors project.
This course is a preparation course for the Core I examination in topology. The course will cover General (Point Set) Topology, Basic Homotopy Theory and Fundamental Group Theory. We will also introduce examples from simplicial complexes and manifolds.
This course continues the study of algebraic topology begun in MATH 7510 and MATH 7512. The basic idea of this subject is to associate algebraic objects to a topological space (e.g., the fundamental group in MATH 7510, the homology groups in MATH 7512) in such a way that topologically equivalent spaces get assigned equivalent objects (e.g., isomorphic groups). Such algebraic objects are invariants of the space, and provide a means for distinguishing between topological spaces: two spaces with inequivalent invariants cannot be topologically equivalent.
The focus of this course will be on cohomology theory, dual to the homology theory developed in MATH 7512. One reason to pursue cohomology theory is that the cohomology of a space may be given a natural ring structure. This additional algebraic structure provides another topological invariant. In developing this structure, we will study several products relating homology and cohomology. These considerations will be used to study the topology of manifolds, yielding a number of duality theorems also relating homology and cohomology, with a variety of applications.
In addition to its importance within topology, cohomology theory also provides connections between topology and other subjects, including algebra and geometry. Depending on the interests of the audience, we may pursue some of these connections, such as cohomology of groups or the De Rham theorem.
The study of fixed embeddings of graphs in surfaces is known by many names: dessin d'enfant, rotation systems, combinatorial maps, ribbon graphs, inter alia. The course will develop the equivalences between the following objects:
For more information see the MATH 7590 course web page
On a rotating hollow sphere, there are exactly two points that don't move at all (we might call them the "north and south poles"). The sphere belongs to a large class of important topological spaces with two key features: (1) they have some (perhaps more than one) kind of rotational symmetry, and (2) they have finitely many points that don't move when the space is rotated. "Equivariant cohomology," the subject of this seminar, is a powerful tool for studying the geometry of such spaces. Using equivariant cohomology, we will explore remarkable connections to combinatorics and group theory.
This semester, we will discuss an important result from current research in representation theory called the geometric Satake isomorphism, restricting ourselves to the simplest possible case of the 2 by 2 invertible complex matrices GL2(C). Roughly speaking, this result states that representations of GL2(C) (i.e., group homomorphisms from GL2(C) to GLn(C) for any n) can be understood in terms of the equivariant cohomology of a topological space called the affine Grassmannian. The goal of this course is to make sense of this isomorphism as explicitly as possible and to come up with a new simple proof in this case. (No background in representation theory is assumed.)
Discontinuous Galerkin (DG) methods are numerical methods for solving differential equations. We will review DG methods developed in recent years and discuss open problems. Graduate students can develop DG methods for various applications. Undergraduate students with background in multivariable calculus, linear algebra and programming will have the opportunity to participate in computational projects.
In this course, we will discuss the connections between analysis, group theory and physics. The students will read articles and give presentations. The selection of topics will be discussed at the organizational meeting at the beginning of the semester. Topics can include, but not limited to, geometric quantization, uncertainty principle, lowest-energy representations, reflection positive and duality theory.
This is an introductory course on the theory of algebraic curves. Our viewpoint will be very classical. We will start with the basic geometry of algebraic curves in P2, developing all the needed tools from elementary algebraic geometry. Then we will study the topological properties of curves over the complex numbers and the interplay between topological invariants and geometric invariants such as the degree-genus formula. Finally, we will study curves from the Riemann surface point of view and prove the Riemann-Roch theorem. Time permitting, we will study curves over non-algebraically closed fields (in particular number fields and finite fields) and give applications to some classical Diophantine equations.
Homological algebra is a branch of algebra that developed in the mid-twentieth century as a way to systematize and abstract techniques from algebraic topology and module theory involving homology--a procedure in which a sequence of abelian groups or modules is associated to each object in a given category. Its influence has expanded far beyond its primarily topological origins, and it is now a fundamental tool in such far-flung branches of mathematics as representation theory, algebraic geometry, number theory, complex analysis, partial differential equations, functional analysis, and mathematical physics.
This class will provide an introduction to homological algebra. We will discuss some category theory, chain and cochain complexes, derived functors, Tor and Ext, spectral sequences, and derived categories. We will also consider various applications.
A standard first year graduate course in complex analysis. Topics include holomorphic functions, covering spaces and the monodromy theorem, winding numbers, residues, Runge's theorem, Riemann mapping theorem, harmonic functions.
The flagship series of the American Mathematical Society, Colloquium Publications, contains volumes that provide definitive treatments of some of the most significant and enduring results in mathematics. Published first in 1957, the all time bestselling volume in this series is the treatise Functional Analysis and Semi-groups by Einar Hille and R. S. Phillips. The material covered in this class represents modern variants of central parts of this jewel of a book, including more recent results and research developments.Course Outline:
The theory of variational inequalities treats optimization problems over convex sets. In this course we study the existence, uniqueness and regularity of the solution of a variational inequality. Applications and numerical methods will also be discussed.
The aim of the course is to provide a simple and precise account of Brownian motion and its applications. The course will begin with the construction of a Brownian motion, and an analysis of its path properties. Next, the construction of stochastic integrals, and the proof of the Ito formula will be presented with full details. This will lead us to a fruitful theory of stochastic differential equations. The theory will be illustrated by several examples and applications.
The course will focus on the applications of stochastic calculus to mathematical finance and partial differential equations. In particular, the course will provide a detailed account of (i) pricing and hedging problems in complete markets,and (ii) probabilistic representations of solutions to certain partial differential equations.
This course focuses on the modern theory of materials design for optimal structural performance. The modern theory of optimal design is a blend of the calculus of variations, the basic theory of elliptic PDE, and scientific computation. The first 4 weeks of the course provides an introduction to the basic existence and uniqueness theory for second order elliptic PDE associated with heterogeneous media. The next three weeks introduce the theory of homogenization of solution operators for elliptic PDE. We then trace the approaches of Spagnolo and Murat and Tartar to exhibit the connection between homogenization convergence and the notion of effective material properties that were used and developed by a significant community of scientists including J.C. Maxwell and A. Einstein. The following 7 weeks shows how the theory of homogenization together with the calculus of variations is used to develop explicit numerical schemes for optimal material design. The course concludes by developing a numerical algorithm for the design of an optimal thermal lens.
What is the essence of the similarity between forests in a graph and linearly independent sets of columns in a matrix? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular? Matroid theory examines and answers questions like these.
This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.
This course will introduce the homology theory of topological spaces. To each space X and nonnegative integer k, there is assigned an abelian group the kth homology group of X. We will learn to calculate these groups, and use them to prove topological results such as the Brouwer Fixed point theorem (in all dimensions) and generalizations of the Jordan curve theorem. Homology theory is important in many parts of modern mathematics. Depending on what was covered in 7510, we may also discuss some other topics related to the fundamental group.
This course gives an introduction to smooth manifolds, which are spaces which locally resemble Euclidean space and have enough structure to support the basic concepts of calculus. We shall cover such fundamental ideas as submanifolds, tangent and cotangent vectors and bundles, smooth mappings and their derivatives, vector fields, differential forms and Stokes's theorem. If time permits, we shall also cover the de Rham Theorem.