**2013-14 GRADUATE COURSE DESCRIPTION**

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Measure theory and integration. Point set topology. Principles of functional analysis. Lp spaces. The Riesz representation theorem. Topics in real and functional analysis.

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Graduate level-matrix theory with introduction to matrix computations. SVDs, pseudoinverses, variational characterization of eigenvalues, perturbation theory, direct and interative methods for matrix computations.

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Linear multistep methods and Runge-Kutta methods for ordinary differential equations: stability, order and convergence. Stiffness. Differential algebraic equations. Numerical solution of boundary value problems.

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Finite difference methods for hyperbolic, parabolic and elliptic PDEs, with application to problems in science and engineering. Convergence, consistency, order and stability of finite difference methods. Dissipation and dispersion. Finite volume methods. Software design and adaptivity.

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Weighted residual and finite element methods for the solution of hyperbolic, parabolic and elliptical partial differential equations, with application to problems in science and engineering. Error estimates. Standard and discontinuous Galerkin methods.

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Group theory, ring and module theory, field theory, Galois theory, other topics.

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Metric spaces, topological spaces, continuity, Hausdorff condition, compactness, connectedness, product spaces, quotient spaces. Other topics as time allows.

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Homotopy groups, exact sequences, fiber spaces, covering spaces, van Kampen Theorem.

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Topological manifolds, differentiable manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefschetz fixed point theorem, vector fields.

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Mapping Class groups - I'll prove the Thurston-Nielsen classification of surface automorphisms.
Time permitted, I'll discuss applications in various areas of mathematics.

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Complex reflection groups - Abstract: A (real) reflection is a linear transformation of finite order which fixes a codimension one hyperplane. The finite groups generated by such reflections are classified by the well-known Dynkin diagrams and they belong to a class of groups known as Coxeter groups.
This course will focus on a variation of these results: reflections defined on complex vector spaces and the finite groups they generate. The main goal will be a complete proof of the Shepard-Todd classification theorem. As time permits we will focus on the topology of the complement of the hyperplanes involved.

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Quantum invariants - Since the early 1980s there has emerged a new mathematical theory. It is called quantum groups by algebraists and quantum topology or topological quantum field theory by topologists. This course will give an introduction to quantum topology, with an emphasis on knot theory, the Jones polynomial and other so called quantum invariants.

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Differentiable manifolds, definition and examples of lie groups, lie group-lie algebra correspondence, nilpotent and solvable lie algebras, classification of semi-simple lie algebras over the complexes, representations of lie groups and lie algebras, special topics.

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Singular homology and cohomology, exact sequences, Hurewicz theorem, Poincare duality.

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This will be a year-long course on algebraic geometry using schemes. We shall begin by studying algebraic varieties, and then we shall move on to schemes, cohomology of sheaves, and the theory of curves. A good idea of the contents of the first part of the course may be obtained by looking at the first four chapters of 'Algebraic Geometry', by Robin Hartshorne. We shall then go on to explore a number of different topics in the area of diophantine geometry.
The prerequisites for this course are a good knowledge of the material contained in Math 220ABC, and a level of mathematical maturity appropriate for an advanced graduate course.

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Topics include geometry of surfaces, manifolds, differential forms, Lie groups, Riemannian manifolds, Levi-Civita connection and curvature, curvature and topology, Hodge theory. Additional topics such as bundles and characteristic classes, spin structures Dirac operator, comparison theorems in Riemannian geometry.

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Geometry of Spaces with Ricci Curvature Lower Bound
Ricci curvature appears in many diverse fields such as Ricci flow, Einstein field equation, and Calabi-Yau manifolds. We will start with the classical story of geometry of manifolds with Ricci curvature lower bounds, going over celebrated results like the Cheeger-Gromoll splitting theorem, the gradient estimate, and the comparison goemetry. We will then move to the most recent development of Ricci curvature for metric measure spaces.

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Minimal Surfaces in Riemannian Manifolds
This course will develop calculus on manifolds which are modeled on Banach or Hilbert spaces. Except for the fact that we need to use some basic theorems from analysis, calculus on infinite-dimensional manifolds is mostly parallel to the finite-dimensional theory. And infinite-dimensional manifolds are useful for studying many of the nonlinear differential equations that arise in geometry, such as the Yang-Mills equation, the Seiberg-Witten equations and the equations for minimal surfaces in a Riemannian manifold, which is the main topic of the course.
For the theory of minimal surfaces, the examples of infinite-dimensional manifolds that are important are spaces of maps, such as the manifold Map(M,N) of maps f: M -> N, where M and N are finite-dimensional manifolds.
Smooth closed geodesics on a Riemannian manifold M can be regarded as critical points for the action function J : Map(C,M) -> R, where C is the unit circle. We will use Morse theory of J to give a generic version of a theorem of Gromoll and Meyer which shows that most compact manifold must have infinitely many geometrically distinct smooth closed geodesics for generic metrics.
This sets the stage for studying corresponding problems for minimal surfaces. We will develop Morse theory for the alpha-energy on Map(S,M), where S is a surface and use it to prove the theorem of Sacks and Uhlenbeck that any compact Riemannian manifold with finite fundamental group contains at least one minimal two-sphere, as well as theorems of Schoen and Yau on existence of minimal surfaces of higher genus. We will also explore the relationship between isotropic curvature and minimal surfaces.
Lecture notes: During the course, we will develop a set of lecture notes which will be made available in PDF format. The lecture notes will be revised as the course progresses.

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Existence and stability of solutions, Floquet theory, Poincare-Bendixson theorem, invariant manifolds, existence and stability of periodic solutions, Bifurcation theory and normal forms, hyperbolic structure and chaos, Feigenbaum period-doubling cascade, Ruelle-Takens cascade.

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This is the first of three courses on the mathematical foundations of topological quantum computation (TQC)---an interdisciplinary field at the triple juncture of mathematics, physics, and computer science. The goal of TQC is the construction of a large scale quantum computer based on braiding non-abelian anyons---the central part of a futuristic field anyonics broadly defined as the science and technology that cover the development, behavior, and application of anyonic devices. The emphasis will be on the detailed treatment of two important mathematical structures: topological quantum field theory (TQFT) and modular tensor category (MTC). An (n+1)-TQFT is mathematically defined as a “quantization” functor from the category of n-manifolds (spaces) and (n+1)-bordisms (space-times) to the category of finite dimensional vector spaces and linear maps. TQFTs arise as low energy effective theories of topological phases of matter, whose elementary excitations in two spatial dimensions are anyons---particles with statistics more general than bosons and fermions. The algebraic models of anyons are unitary MTCs, which are also the algebraic data for unitary (2+1)-TQFTs. Subsequent courses will cover topological phases of matter and anyonic quantum computing models.
The main prerequisite for the course is linear algebra, and a familiarity with basic category theory such as Chapter 1 of Category Theory for the Working Mathematician by S. Mac Lane, manifold topology such as the book by J. Lee: Introduction to Topological Manifolds, and representation theory such as parts I and II of Representation Theory: a First Course by W. Fulton and J. Harris. Some knowledge about quantum mechanics and quantum field theory will be very helpful, though not required. There are no textbooks available, so there will be references and lecture notes on the instructor's website during the course. Basically, the course will cover in details some of the topics in Chapters 1, 4, and 5 of the instructor's CBMS monograph Topological Quantum Computation.

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Topics in algebra, analysis, applied mathematics, combinatorial mathematics, functional analysis, geometry, statistics, topology, by means of lectures and informal conferences with members of faculty.

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Abstract: High frequency wave propagation is an important and interesting topic in many research fields - acoustic/elastic waves in seismic migration, inversion and earthquake models; acoustics waves in underwater communication, sonar detection and ultrasound therapy; electromagnetic waves in wireless communications, to name a few. The simulation of high frequency wave propagation brings up difficult problems in applied and computational mathematics. One of the major challenges is that direct numerical simulation is prohibitively expensive due to the fact that the wave length is comparatively small to the computational domain size. This class aims to systematically introduce numerical tools to compute such waves based on the semiclassical limit of the wave equations, which can achieve high accuracy with coarse mesh, hence improve the efficiency of computation.

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Many important equations in physics such as Maxwell, Ginzburg-Landau, Yang-Mills and others have the essential feature of the gauge invariance. The gauge invariance plays also an important role in modern analysis of PDEs. It is closely related to the geometric nature of the equations. The purpose of the course is to derive and analyze the main equations listed above from such geometric point of view. The necessary tools to be developed include differentiation in vector bundles, Lie group actions, differential forms and exterior algebra.

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This will be an expository class on the relationship between curvature and topology. One of the most fundamental topological invariants is the fundamental group. This course will focus on the fundamental
groups of Riemannian manifolds with curvature bounds.

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One familiar example is Synge's theorem from Math 240: Every closed, orientable, even-dimensional Riemannian manifold with positive sectional curvature has trivial fundamental group. The Bonnet-Myers theorem provides another example: If a complete Riemannian manifold has Ricci curvature bounded from below by a positive constant, then the fundamental group is finite. We will study the fundamental groups of manifolds with positive, nonnegative, and almost nonnegative curvatures.

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Graduate students in geometry, topology, and related areas are encouraged to enroll, as are advanced undergraduate students.

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Surface groups have beautiful and flexible representation theory into a variety of target groups. This course will focus on the representations of surface groups into a variety of geometric settings. Possible topics include representations into finite groups (i.e. residual finiteness), hyperbolic structures on surfaces, convex projective structures, and various aspects of higher teichmuller theory.

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The objects of the title are algebras based on quivers (= finite directed graphs), whose representation theory is an interesting mix of algebra, combinatorics, and geometry. In introducing them, we will place particular emphasis on the finite dimensional case. In particular, we will show how frequently finite dimensional path algebras modulo relations occur in mathematical nature: Over an algebraically closed field $, any finite dimensional $-algebra is -- in a (strong) sense, to be specified -- equivalent to a path algebra modulo relations. We will partly prove, partly sketch, some classical results in this theory and apply them to the exploration of specific algebras. As we will find, this will lead us to classification problems for vector space diagrams (not typically covered in Linear Algebra) and elegant solutions thereof.

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The only prerequisite is Math 220 or equivalent. In particular, it is not necessary to have taken my two introductory quarters on the representation theory of finite groups and finite dimensional algebras (F12/S13). For students who did take the latter course, the proposed one will complete a 3-quarter sequence.

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Consideration of ideas about the process of learning mathematics and discussion of approaches to teaching.