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# Course Descriptions Summer 2006 - Spring 2007

## Contact

## Summer 2006

- Math 7290: Number Theory
- Instructor: Prof. Morales
- Prerequisite: Math 7200 or the equivalent.
- Text: Algebraic Number Theory by A. Froehlich and M. Taylor, Cambridge University Press, 1991.

This course is intended to be a bridge between undergraduate and graduate-level number theory and should appeal to a broad audience. Due to time constraints during the summer term, this is not intended to be a full-fledged research-level number theory course , but rather a complete introduction to the standard algebraic and analytic techniques that are used in more advanced courses.

We will develop in some detail algebraic tools such as the theory of Dedekind domains, discrete valuation rings, p-adic numbers, completions, etc., and also some analytic tools such as zeta functions. We will also discuss applications of analytic methods to classical results such as the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.

The "official" textbook is "Algebraic Number Theory" by A. Froehlich and M. Taylor (Cambridge University Press 1991), but we will also use other sources for some parts of the course.

## Fall 2006

- MATH 7001: Communicating Mathematics I
- Instructor: Prof. Oxley .
- Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.

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.

- MATH 7200: Geometric and Abstract Algebra
- Instructor: Prof. Adkins .
- Prerequisite: Math 4200 or the equivalent.
- Text: Algebra: an approach via module theory, by Adkins and Weintraub

This course will cover the basic material on groups, rings, fields and modules with special emphasis on linear algebra.

- MATH 7211: Representations of Finite & Compact Groups
- Instructor: Prof. Sage
- Prerequisite: Math 7200 or comparable familiarity with groups and linear algebra
- Text: Simon, Representations of finite and compact groups

Representation theory is the study of the ways in which a given group may act on vector spaces. Intuitively, it investigates ways in which an abstract group may be interpreted concretely as a group of matrices with matrix multiplication as the group operation. Group representations are ubiquitous in modern mathematics. Indeed, representation theory has significant applications throughout algebra, topology, analysis, and applied mathematics. It also is of fundamental importance in physics, chemistry, and material science. For example, it appears in quantum mechanics, crystallography, or any physical problem in which one studies how symmetries of a system affect the solutions.

This course is designed to give an introduction to the representation theory of finite and compact groups. It will start with the representations of finite groups over the complex numbers. In particular, we will discuss how finite-dimensional representations break up into sums of irreducible representations, Schur's lemma, the group algebra, character theory, induced representation and Frobenius reciprocity. We will give many examples, culminating in the representation theory of the symmetric groups and the remarkable combinatorics associated with it. We will then turn to the representation theory of compact groups, which is formally very similar to the finite group situation. The course will end with a discussion of the representation theory of the unitary groups and its close relationship with the representation theory of the symmetric groups.

- MATH 7280: Homological Algebra
- Instructor: Prof. Hoffman
- Prerequisites: Math 7200 or the equivalent.
- Text: An Introduction to Homological Algebra

by Charles A. Weibel, Cambridge University Press, ISBN: 0521559871

Homological Algebra is a tool that appears in almost all branches of mathematics, ranging from mathematical physics, PDEs, Lie groups and representation theory, through topology (where it originiated) and on to number theory and combinatorics. We will cover the basics of homology of chain complexes, elementary category language, derived functors, Ext and Tor, spectral sequences, simplicial methods. There are various further topics and directions which will depend on the available time and interests of the class. Each student will have a project to do - these can be chosen from an application of homological methods in the student's area of interest.

- Math 7290: Toric Geometry and Combinatorial Commutative Algebra
- Instructor: Prof. Madden.
- Prerequisite: Math 7210
- Text: Ezra Miller, Bernd Sturmfels: Combinatorial Commutative Algebra (Springer Graduate Texts in Mathematics, Paperback)

The theme of this course is the interplay between the combinatorial geometry of convex bodies in Rn with integral vertices and the algebraic geometry of toric varieties, algebraic varieties that are locally defined by differences of monomials. This is a very active area of current research, with a surprisingly diverse range of connections, including applications in computational algebra, statistical modeling, singularity theory and elsewhere. This course will cover the following topics:

- basic theory of commutative monoids, monoid algebras and affine toric varieties
- selected topics in toric geometry (selected from Ewald: Combinatorial Convexity and Algebraic Geometry)
- advanced topics, including resolutions and syzygies of binomial ideals, combinatorial descriptions of these objects, and applications (as in Part II of Miller and Sturmfels)

- MATH 7311: Real Analysis I
- Instructor: Prof. Fabec
- Prerequisite: Math 4032 or 4035 or the equivalent.
- Text: Real Analysis, Measure Theory, Integration, and Hilbert Spaces by E. M. Stein and Rami Shakarchi, Princeton University Press, 2005.

This course deals with Lebesgue measure on Euclidean space and its analog in the more general setting of measure spaces. We will obtain all the basic properties of measure and integration, including convergence theorems, Fubini's Theorem, the Radon-Nikodymn Theorem, and the Caratheodory extension process. Banach spaces, particularly the Lp spaces will be discussed.

Time permitting, we will cover other topics important in real analysis, for which useful references may include Real Analysis by Royden and Real Analysis by Rudin.

- MATH 7360: Probability Theory
- Instructor: Prof. Sengupta
- Prerequisite: Math 7311.
- Text:

- Math 7380-1: Applied Stochastic Analysis
- Instructor: Prof. Sundar
- Prerequisites: Math 4031, Math 4032, and Math 3355.
- Text: Arbitrage theory in continuous time by T. Bjork

The aim of the course is to provide a mathematical account of the arbitrage theory of financial derivatives. A short treatment of stochastic differential equations and the It\^o calculus will be presented, and will include the Feynman-Kac formula, and the Kolmogorov equations. Risk neutral valuation formulas and martingale measures will be introduced through Feynman-Kac representations. The course will cover pricing and hedging problems in complete as well as incomplete markets. Barrier options, options on dividend-paying assets, as well as currency markets will be studied. Interest rate theory will include short rate models and the Heath-Jarrow-Morton approach to forward rate models. A self-contained treatment of stochastic optimal control theory will be used to study optimal consumption/investment problems.

- MATH 7380-2: Mathematical Topics in Systems Theory
- Instructor: Prof. Malisoff
- Prerequisite: Elementary differential equations and

linear algebra; some background in real analysis (e.g. Math 7311-7312) is suggested but not required - Text: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition by E.D. Sontag, Springer, New York, 1998. ISBN 0-387-984895

Systems and control theory is one of the most central and fast growing areas of applied mathematics and engineering. This course provides a basic introduction to the mathematics of finite-dimensional, continuous-time, deterministic control systems at the beginning graduate student level. The course is intended for PhD students in applied mathematics, and for engineering graduate students with a background in real analysis and nonlinear ordinary differential equations. It is designed to help students prepare for interdisciplinary research at the interface of applied mathematics and control engineering. This is a rigorous, proof-oriented systems theory course that goes beyond classical frequency-domain or more applied engineering courses. Emphasis will be placed on controllability and stabilization.

- MATH 7390-1: Applied Harmonic Analysis: Introduction to Radon transforms
- Instructor: Prof. Rubin
- Prerequisite: Math 7311 (Real Analysis-I) or equivalent.
- Text: Dr. Rubin will use his own notes for this course.

This is an introductory course in the theory of the Radon transform, one of the main objects in integral geometry and modern analysis. Topics to be studied include fractional integration and differentiation of functions of one and several variables, Radon transforms in the n-dimensional Euclidean space and on the unit sphere, some aspects of the Fourier analysis in the context of its application to integral geometry.

- MATH 7390-2: Scientific Computing
- Instructor: Prof. Aksoylu
- Prerequisite: Basic concepts of numerical analysis, basic knowledge of linear algebra, and basic programming skills in C, Matlab or other computer language. Preferably Math 4065, 4066 or equivalents.
- Text: Scientific Computing: An introductory survey by Michael T. Heath, McGraw Hill 2002, Lecture notes

This course serves as a foundation for analysis, design, and implementation of numerical methods. In particular, it provides an in-depth view of practical algorithms for solving large-scale linear systems of equations arising in the numerical implementation of various problems in mathematics, engineering and other applications.

The course will cover the following material:

- Linear algebra and numerical linear algebra refresher, in particular, solutions to system of linear equations.
- Matrix factorizations.
- Linear least squares.
- Ortogonalization methods.
- Eigenvalue problems.
- Basic iterative methods.
- Krylov subspace methods.
- Preconditioning techniques.
- Multilevel methods such as multigrid.

The course will focus on both the theoretical aspects and the numerical implementation of these methods. Evaluation will be based upon both theoretical and numerical projects.

- MATH 7390-3: Mathematical Models of Superconductivity
- Instructor: Prof. Almog
- Prerequisite: Some functional analysis and basic PDE and ODE is necessary. Some of the background will be reviewed in class.
- Text: A list of articles for reading will be provided. The background material will be taught from: L. C. Evans - Partial Differential Equations and L. Nirenberg - Topics in nonlinear functional analysis.

At first I'll provide an introduction to the theory of Sobolev spaces, to elliptic regularity to direct methods of the calculus of variations, and to linear bifurcation theory. This should include about two thirds of the course. Then, I'll present the Ginzburg-Landau energy functional, prove existence of minimizers and discuss some general properties of the minimizer. I'll then cover some topics from the theory of superconductivity, including: Surface superconductivity, Abrikosov's lattices, radial vortices, self-duality and the Jaffe-Taubes assumption, thin rings, and if time allow the Bethuel-Brezis-Helein theorem and the weak field limit. (I probably won't be able to do that much so you can have influence on the material to be covered).

- MATH 7490: Matroid Theory
- Instructor: Prof. Oxley
- Prerequisite: Permission of the department.
- Text: J.G. Oxley, Matroid Theory, Oxford, 1992, reprinted in paperback with corrections, July, 2006.

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.

- MATH 7510: Topology I
- Instructor: Prof. Litherland
- Prerequisite: MATH 2057 or equivalent.
- Text: James R. Munkres, Topology, 2nd edition, Prentice Hall, 2000.

Topology contains at least three (overlapping) subbranches: general (or point-set) topology, geometric topology and algebraic topology. General topology grew out of the successful attempt to generalize some basic ideas and theorems (e.g., continuity, open and closed sets, the Intermediate Value and Bolzano-Weierstrass Theorems) from Euclidean spaces to more general spaces. The core of this course will be a thorough introduction to the central ideas of general topology (Chapters 2 - 5 of Munkres). This material is fundamental in much of modern mathematics. Time permitting, we will also look briefly at the ideas of homotopy and the fundamental group (Chapter 9), subjects that belong to the algebraic-geometric side of topology, and will be covered in greater depth in 7512.

- MATH 7590-1: Geometric Topology
- Instructor: Prof. Dasbach
- Prerequisite: MATH 7512 (Topology II) or equivalent.
- Text: Notes

The course will mainly focus on two aspects of Geometric Topology:

- Braid groups as central objects in topology: Representations of braid groups, Linearity of braid groups, knots as closed braids, subgroups of braid groups, solutions to word problems and conjugacy problems, generalization of braid groups
- Three manifolds and their groups: Dehn surgery on knots and links

If time permits we will cover related topics in Geometric Topology.

- MATH 7590-2: Riemannian Geometry (connections and Gauge theory)
- Instructor: Prof. Baldridge
- Prerequisite: Math 7510 (Topology I) or the equivalent.
- References: Carmo, Riemannian Geometry (Birkhauser, ISBN: 0-8176-3490-8) and Introduction to Symplectic Geometry by Dusa McDuff and Dietmar Salamon (Oxford University Press, ISBN: 0-1985-0451-9 ) will be useful.

Riemannian Geometry (connections and Gauge theory). This course is an introduction to Riemannian geometry: manifolds, metrics, Levi-Civita connections, and curvature. Riemannian geometry is key to understanding Einstein′s general relativity and plays an important role in gauge theory and invariants of smooth 4-manifolds. We will use this technology to introduce and investigate symplectic geometry, another important topic in mathematics that also comes from physics.

Students do not need to be enrolled in the Spring 2006 differential geometry course (Math 7550) to take Math 7590-2 in the fall.

- Instructor: Prof. Vertigan
- Prerequisite: Permission of the instructor.
- Readings to be selected after discussion with the students.

This reading seminar will meet weekly at at a time to be announced. If you are interested in this seminar, please contact Prof. Vertigan before you register.

## Spring 2007

- MATH 7002: Communicating Mathematics II
- Instructor: Prof. Oxley .
- Prerequisite: Consent of department. This course is required for all incoming graduate students. It is a lab course meeting for an average of two hours per week throughout the semester for one-semester-hour's credit.

This course provides practical training in the teaching of calculus, how to write mathematics for publication, how to give a mathematical talk, 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.

- MATH 7210: Algebra - I
- Instructor: Prof. Nobile
- Prerequisite: Math 7200: Geometric and Abstract Algebra.
- Text: A Course in Galois Theory by D. J. H. Garling (Cambridge University Press) and Fields and Galois Theory - notes by J. S. Milne, available on-line.

This is primarily a course on Theory of Fields. Topics include: review of results from Group and Ring Theory, algebraic field extensions, ruler and compass constructions, separability, Galois Theory, algebraic equations, finite fields, transcendental extensions. If time permits, some applications to Algebraic geometry and /or Number Theory will be discussed.

- MATH 7280: Applications of Homological Algebra
- Instructor: Prof. Achar
- Prerequisite: Some knowledge of homological algebra, such as Prof. Hoffman's Fall 2006 Math 7280 course on homological algebra.
- Text: no required text; notes will be made available as appropriate, and references will be given during the semester to books, such as M. Kashiwara and P. Schapira, Sheaves on manifolds or A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers in Asterisque.

The theory of "perverse sheaves" was introduced around 1980 as a tool in algebraic topology for constructing a particular kind of homology theory. Since then, it has turned out to be an incredibly powerful framework with far-reaching consequences, especially in representation theory. A number of major results since then (e.g., the proof of the Kazhdan-Lusztig conjectures, the whole theory of character sheaves, and much work related to the geometric Langlands program) could not have gotten off the ground without perverse sheaves.

The goal of this course will be to develop the basics of the theory of perverse sheaves and look at a couple of applications. Perverse sheaves are defined in terms of the derived category of ordinary sheaves. No prior knowledge of sheaves or derived categories is necessary; these topics will be covered in the course. However, it is a good idea to have some familiarity with abelian categories and derived functors. The applications will likely come from representation theory, but no prior knowledge of representation theory will be assumed.

- MATH 7290: Number Theory in Function Fields
- Instructor: Prof. Morales.
- Prerequisite: Prerequisites: MATH 7210 or equivalent. A graduate-level course in

classical algebraic number theory would be helpful but is not indispensable. Please feel free to contact Prof. Morales if you have any questions. - Text: Number Theory in Function Fields by Michael Rosen, Springer-Verlag 2002.

Classically, number theory is concerned with properties of the ring of integers Z, its field of fractions Q and its finite extensions (number fields). In this course we will do number theory over the ring A = F[t], instead of Z, where F is a finite field, and study the arithmetic properties of finite extensions of F(t) (function fields).

We will discuss function field analogues of the Riemann zeta function and prove analogues of classical theorems such as quadratic reciprocity, Dirichlet’s theorem on primes in arithmetic progressions, etc. We will also discuss the relation with algebraic curves over F and the Riemann-Roch Theorem. We will see how the Riemann Conjecture (which is a theorem in the function field case) yields estimates of the number of rational points on algebraic curves. This course can be seen as an introduction to both number theory and the theory of algebraic curves.

- MATH 7320: Ordinary Differential Equations
- Instructor: Prof. Estrada
- Prerequisite: Advanced Calculus and Linear Algebra (preferably Math 7311 and 7200)
- Text: W.Walter, Ordinary Differential Equations, Springer, 1998.

This is a first graduate course in ordinary differential equations. The following topics will be covered. Existence and uniqueness results for first order equations and systems, and for higher order equations. Linear equations and systems. Complex linear systems. Boundary value and eigenvalue problems. Stability.

- MATH 7330: Functional Analysis
- Instructor: Prof. Davidson
- Prerequisite: Math 7312
- Text: Functional Analysis by George Bachman and Lawrence Narici, (1966, 2000) now published by Dover.

Banach spaces and their generalizations; Baire category, Banach-Steinhaus, open mapping, closed graph, and Hahn-Banach theorems; duality in Banach spaces, weak topologies; Hilbert Spaces, other topics such as commutative Banach algebras, spectral theory, and Gelfand theory.

- MATH 7350: Complex Analysis. This course will serve as a Core-2 course for the current year.
- Instructor: Prof. Lawson
- Prerequisite: Math 7311
- Text: Function Theory of One Complex Variable (3rd Edition): Robert E. Greene, Steven G. Krantz, AMS Publication, 2006.

This is a first rigorous course in the theory of functions of one complex variable. Topics include holomorphic (or complex analytic) functions; power series; complex line integrals; Cauchy's integral formula, and some of its applications; singularities of holomorphic functions; Laurent series, and computation of definite integrals by residues; the maximum principle and Schwarz's lemma; conformal mapping; and harmonic functions (if time permits). The text and the course emphasize connections with multidimensional calculus.

Note: Math 7350 is being offered this year as a Core-2 Course. Students taking this course will have the option of taking a Complex Analysis Core-2 PhD Qualifying Examination.

- MATH 7370: Lie Groups and Representation Theory
- Instructor: Prof. He
- Prerequisites: Real Analysis and Abstract Algebra
- Text: Non-Abelian Harmonic Analysis, Howe, Roger and Tan, Eng Chye. Springer Verlag Universitext Series, 1992. Softcover ISBN-10: 0-387-97768-6 or ISBN-13: 978-0-387-97768-3.
- Optional Reference: SL(2,R) by Serge Lang, Addison-Wesley, 1975. ISBN: 0201042487.

Topics will include Lie groups and Lie Algebras, Linear Semisimple Groups, Finite and Infinite Dimensional Representations, Unitary dual, Character formula,Matrix coefficients and applications. The emphasis will be on the very important example of SL(2,R) and its representations.

- MATH 7380: Partial Differential Equations
- Instructor: Prof. Tom
- Prerequisites: Math 4032 or Math 4340 or Math 4038.
- Text: Partial Differential Equations by Lawrence C. Evans.

This course will be suitable to mathematics graduate students with good backgrounds in analysis but with no previous course on PDEs. It will also be suitable to Physics and Engineering students who are minoring in mathematics but have taken either Math 4340 or Math 4038. The course will concentrate on representation formulas for solutions of various partial differential equations. Topics that will be covered include the transport equation, Laplace equation, the heat equation, the wave equation, nonlinear first order PDEs, other ways to represent solutions; similarity solutions, Fourier and Laplace transforms, converting nonlinear PDEs into linear PDEs Hopf-Cole transformation, etc.

For more information, contact Michael M. Tom at 310 Lockett Hall or call Prof. Tom at 578-1613 or email Prof. Tom.

- MATH 7390-1: Stochastic Analysis
- Instructor: Prof. Kuo
- Prerequisites: Math 7311 (Real Analysis I) or the equivalent.
- Text: 1. Kuo, H.-H.: Introductory Stochastic Integration. Universitext, Springer 2006. 2. Kuo, H.-H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math., Vol. 463, Springer-Verlag, 1975 (New Printing by BookSurge, Oct 2006.) 3. Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996

This course contains three parts: (1) stochastic integration,(2) infinite dimensional integration theory, and (3) white noise theory. Here are some topics to be covered in this course: Brownian motion, Ito integrals, Ito's formula, Girsanov theorem, Stochastic differential equations, Applications to finance, Abstract Wiener space, Gaussian processes, Transformation of measures, Potentai theory on Hilbert space, Gel'fand triple, White noise space, Test and generalized functions, Characterization theorems, Operators on white nose functions, Anticipating stochastic integrals

- MATH 7390-3: Finite Element Method: Analysis and Implementation
- Instructor: Prof. Brenner
- Prerequisites: MATH 7311 (Real Analysis I) or the equivalent.
- Text: The Mathematical Theory of Finite Element Methods, Second Edition, by S.C. Brenner and L.R. Scott, Springer-Verlag, 2002

In this course we will develop the basic mathematical theory of the finite element method for elliptic boundary value problems, which provides a foundation for further study and research in the area of finite elements. Topics include background material for Hilbert spaces and Sobolev spaces, variational formulations, constructions of finite element spaces, interpolation error estimates, and discretization error estimates. We will cover Chapters 0--5 of the text. Additional material will be covered if time permits.

- MATH 7400: Graph Theory
- Instructor: Prof. Oporowski
- Prerequisite: The prerequisites for the course are very modest---all graduate students in Mathematics should be able to follow the lectures.
- Recommended References: Graph Theory by Reinhard Diestel, Third Edition, Springer, 2006, which is available both as a paperback (for about $43 + shipping from various online stores), or as a free download at http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/GraphTheoryIII.pdf.

Another good book on the subject is Introduction to Graph Theory by Douglas B. West, Prentice Hall, 1996.

The main theme of this course will be graph theory. We will discuss a wide range of topics, including spanning trees, Eulerian trails, matching theory, connectivity, hamiltonian cycles, coloring, planarity, integer flows, and graph minors.

There are many books on graph theory. Prof. Oporowski recommends the references listed above, but they are not absolutely necessaary. He will present the lectures with his own notes, which will be available for download, and which should make good study material. However, the amount of detail in the lecture notes is less than that of either of the mentioned books. If your interest in the subject is anything more than superficial, you would be well advised to get at least one of those books - especially the first one listed.

Grades for the course will be based 60% on homework and 40% on two exams (midterm and final). Decisions in borderline cases will be made on the basis of class participation. There will be the total of over twenty problems given as homework, and the two lowest problem scores will be dropped.

If you have any questions about the course, do not hesitate to contact Prof. Oporowski.

- MATH 7512: Topology - II
- Instructor: Prof. Gilmer
- Prerequisite: Topology 1 Math 7510
- Text: Algebraic Topology by Allen Hatcher,Cambridge University Press (ISBN 0-521-79540-0) online version is available for free download at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

This is an introduction to algebraic topology. In algebraic topology, topological questions are related to algebraic questions. Sometimes this allows one to answer the topological questions using algebra. We will discuss homotopy, homotopy type, the fundamental group, the Jordan curve theorem, Brouwer fixed point theorem, covering spaces, and time permitting elementary aspects of higher homotopy groups. We will study chapters 0 and 1 and perhaps the elementary parts of chapter 4. Munkres's Topology will be an alternative source.

- MATH 7520: Algebraic Topology
- Instructor: Prof. Owens
- Prerequisite: Math 7200 and 7510; or equivalent.
- Text: Allen Hatcher, Algebraic Topology, Cambridge University Press, downloadable free from http://www.math.cornell.edu/~hatcher/#ATI

This course is an introduction to algebraic topology, with an emphasis on homology and cohomology. The goal will be to cover as much of the material in chapters 2 and 3 of Hatcher's book as time permits. This includes simplicial, singular and cellular homology, excision, the Mayer-Vietoris sequence, cohomology groups, the Universal Coefficient Theorem, the cup product, and Poincare duality.

- MATH 7550: Differential Geometry
- Instructor: Prof. Cohen
- Prerequisites: MATH 7200 and 7510 (and undergraduate analysis).
- Text: F. Warner, Foundations of Differentiable Manifolds and Lie Groups,

Springer-Verlag, GTM 94.

This course provides an introduction to smooth manifolds, roughly speaking, spaces which locally resemble Euclidean space and have enough structure to support the basic concepts of calculus, as well as related constructions. These include submanifolds, tangent vectors and bundles, smooth mappings and their effect on tangent vectors, implicit and inverse function theorems, vector fields, differential forms, Stokes' theorem, etc.

- MATH 7590: Mapping Class Groups
- Instructor: Prof. Brendle
- Prerequisite: Math 7510 (Topology-I or equivalent)
- Text: Farb and Margalit, A Primer on Mapping Class Groups (manuscript in preparation)

This course will serve as an introduction to mapping class groups of surfaces. Mapping class groups are a fundamental object of study in topology, as the automorphism groups of 2-manifolds, but also arise naturally in many other fields, such as complex analysis and algebraic geometry. We will survey some basics such as generators and relations for the mapping class group, subgroups important in 3-manifold theory such as the Torelli group and the handlebody subgroup, and representations of mapping class groups. As time permits, and according to the interests of the class, we will also discuss related groups in geometric group theory, Teichmuller theory, and associated combinatorial structures such as the curve complex, among other topics.