Mathematics Department - Graduate Course Descriptions - Spring 2010

### Theory of Functions of a Real Variable II

Text: We will continue to use the 501 text, Real Analysis: Modern Techniques and Their Applications, by J. Folland (Wiley-Interscience; 2nd edition, 1999. ISBN-10: 0471317160). We will supplement the main text by material from other sources. Particularly worthy of special recommendation is a forthcoming book of Professor Brezis, "Functional analysis, Sobolev spaces and PDEs".

Prerequisites: 640:501 or permission of Instructor

Description:

This course is a continuation of 640:501 from Fall 2009. The goal is to give an introduction to core topics in real and functional analysis that every professional mathematician should know.

The choice of topics will be somewhat influenced by what is covered in the 501 course in fall 09, but will focus on concepts and techniques that have broad applications to different areas of mathematics. More concretely, we will discuss different modes of convergences and their applications (including weak convergence); manifestations of completeness from different perspectives ( including some Banach space theory and basic theorems involving bounded linear operations); compactness and applications; and some elementary aspects of spectral theory of (compact) linear operators. All the general ideas will be illustrated in some concrete contexts of applications, which include Fourier series and transforms, ODE's, integral and partial differential equations , and probability.

Although many of the topics to be covered are not on the syllabus of the written qualifying exam, they provide ample space for students to witness the applications of the ideas and tools learned in 501 in a variety of contexts, and to practice problem solving skills.

### Theory of Functions of a Complex Variable II

Text: Green and Krantz: Function Theory of One Complex Variable, AMS.

Prerequisites: Math 503

Description:

This will be a continuation of Math 503. We will emphasis on the relationship between classical complex analysis and other related fields (algebraic geometry, geometry, and analysis) through Riemann surfaces.

The theory of Riemann surface is a pillar in 20th century mathematics. It appears in such seemingly diverse areas as integrable systems, number theory, algebraic geometry, and string theory. We would like to concentrate on the interaction between the complex analytic, classical geometric, and algebraic geometry points of view.

The following two parts will be covered: (1) Classical Complex Analysis and (2) Riemann surfaces.

Part 1. Analytic continuation, the monodromy theorem, normal families and Riemann mapping theorem, Picard theorems, harmonic functions and elliptic functions.

Part 2. Introduction to Riemann surfaces and algebraic curves. Hyperbolic geometry and uniformization theorem. Riemann-Roch theorem, Abel and Jacobi theorems.

The reference for the first part:

[1] Green and Krantz: Function Theory of One Complex Variable, AMS.

[2] Ahlfors, Lars V. Complex analysis. McGraw-Hill Book Co.

The reference for the second part:

[3] Farkas, H & Kra, I.: Riemann Surfaces (2nd ed.), Springer-Verlag

[4] Forster, O.: Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991

[5] Narasimhan, R.: Compact Riemann surfaces. Birkhäuser Verlag, Basel, 1992.

[6] Griffiths, P.: Introduction to Algebraic Curves, American Mathematical Society,

### Selected Topics in Analysis

Text: Though the course has no textbook, students can look at C.E. Guttierez's book, The Monge Ampere Equation ( Birkhauser ) to get an idea of what topics will be covered.

Prerequisites: Math 501, Math 502

Description: The course will be an introduction to the Real Monge-Ampere Equation. First I will show how to construct the Generalized solution of Alexandrov to the Monge Ampere equation. Next I will develop the Caffarelli theory of sections, hopefully leading upto the $W^{2,p}$ estimates and regularity. If time permits I will indicate applications. The tools in many places come from Harmonic Analysis.

### Ordinary Differential Equations

Text: Ordinary Differential Equations with Applications by Carmen Chicone (Springer Texts in Applied Mathematics, vol. 34, second edition; Springer, 2006; ISBN-13: 978-0387-30769-5). We expect to cover much of the material in Chapters 1-3 of the book.

Prerequisites: An undergraduate course on ordinary differential equations, linear algebra, advanced calculus and some basic results from analysis, e.g., the definition of a Banach space. An attempt will be made to keep the course self-contained. Thus, while we shall use the implicit function theorem in Banach spaces, a complete proof will be given.

Description: This will be an introduction to the theory of ordinary differential equations. We will discuss existence and uniqueness theorems for the initial value problem. linearization and linear theory, stability (Lyapunov functions), omega limit sets, Poincare-Bendixson theory and invariant manifolds. If time permits, we may also discuss the Hopf bifurcation theorem and some topics from the theory of order-preserving dynamical systems.

### Partial Differential Equations II

Text: "Partial Differential Equations" by Lawrence C. Evans, published by AMS, 2002. In addition, there will be reading from several recent, and one not so recent, research papers on topics to be developed in the course.

Prerequisites: 640:517, Partial Differential Equations I.

Description: This course will build directly on the content covered in 640:517, Partial Di fferential Equations I, taught by Yanyan Li in Fall 2009. In particular, we shall pick up on the notions of weak solutions and Sobolev spaces introduced towards the end of 640:517. We shall begin with this circle of ideas, developing it from the beginning, proving results on Sobolev spaces and embeddings, and explaining their applications to nonlinear PDE. We shall continue with topics that build on, as well as complement, these ideas, including weak lower semicontinuity in variational methods for PDE's, solving nonlinear elliptic equations using sub and super solutions, the theory of viscosity solutions, the Hamilton-Jacobi equation, and the ideas of DiGiorgi, Nash, and Moser on regularity of parabolic and elliptic equations with rough coefficients. (Nash's 1958 paper is the "not so recent" paper mentioned above.) If time permits, we shall also discuss recent progress on the Monge-Ampere equation.

### Introduction to Differential Geometry

Text: "Riemannian geometry: A metric entrance" by Karsten Grove (Publisher: Univ. of Aarhus (1999) ASIN: B000J4WP10)

Prerequisites:

Description: This is an introduction course for Riemannian geometry and differential geometry. In this course, we will NOT assume a knowledge related to a manifold (tangent bundle, connection, etc), which will be introduced during the course. The following are basic contents of this course:
1. Riemannian Length and Distance
2. Geodesics
3. The First Variation of Arc Length
4. The Levi-Civita Connection
5. The Exponential Maps
6. Isometries
7. Jacobi Fields and Curvature
8. Curvature Identities
9. Second Variation of Arc Length and Convexity
10. Parallel Transports
11. Manifolds and Maps
12. Completeness
13. Global Effects of Curvature
14. Vector Bundles and Tensors
15. Connections and Differential Forms
16. Submanifolds
17. Relative Curvature
18. Space Forms
19. Riemannian Submersions
20. Lie groups ans homogeneous spaces

### Introduction to Algebraic Topology II

Text: Allen Hatcher "Algebraic Topology," available from Cambridge University Press, as well as online at hatcher's site

Prerequisites: Math 540 or knowledge of its contents.

Description:

This course will be a sequel to Math 540, but can also be viewed as a mostly independent course on cohomology and homotopy theory for students who already have had an introduction to homology.

The plan is to start with cohomology in Chapter 3 (the extent of the coverage depending on how far Math 540 gets into this chapter). We will then cover basic results on homotopy groups in Chapter 4, such as the long exact sequences for pairs of spaces and fiber bundles, and will take up a number for further topics that relate homotopy groups to homology and cohomology. We will also select from the additional topics, and will study the homotopy groups of classical groups and the cohomology of fiber bundles.

Depending on available time, the course may end with an introduction to vector bundles and characteristic classes, following a further book in progress by Hatcher on his web site, or Milnor and Stasheff's book Characteristic Classes.

### Lie Algebras

Text: Goodman-Wallach text (Springer GTM 255) supplemented by additional notes as necessary

Prerequisites:

Description: We will discuss the classification of reductive Lie algebras, algebraic Groups, and their representations, with the objective of providing an introduction to the Atlas of Lie groups. Much of the material will be new even for students from Prof. Goodman's class in Fall 2009 and from my course in Spring 2009. Attendance in those earlier course will be helpful though not a prerequisite.

### Abstract Algebra II

Text: Jacobson, "Basic Algebra", Volumes 1 and 2, second edition.

Note: These volumes are out of print. Students may be able to obtain used copies online (be sure it is the second edition) through addall.com or other websites. In the fall, photocopies will be available for purchase.

Prerequisites: Any standard course in abstract algebra for undergraduates and/or Math 551

Description:

Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. Representative topics will be:

• Galois Theory
• Finite algebraic extensions, resolutions of equations by radicals (and without radicals)
• Noetherian Rings
• Rings of polynomials, Hilbert basis theorem, Dedekind domains, Finitely generated algebras over fields, Noether normalization, Nullstellensatz
• Basic Module Theory
• Projective and injective modules, resolutions, baby homo- logical algebra, Hilbert syzygy theorem

### Selected Topics in Algebra

Subtitle: Modular invariance in conformal field theory

Text: I will use several research papers, including, Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237--307

and

Yi-Zhi Huang, Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649--706.

Prerequisites: Basic knowledge in algebra and complex analysis. It will be helpful if the student has some knowledge in Lie algebras and vertex operator algebras. But I will start from the definition of vertex operator algebra.

Description: Vertex operator algebras and their representations are basic ingredients in conformal field theory, a theory playing important roles in both condensed matter physics and string theory. For a vertex operator algebra satisfying certain reductivity and finiteness conditions, a theorem of Yongchang Zhu says that certain traces of vertex operators on the representations of the algebra form a basis of a module for the modular group SL(2, Z). In particular, for vertex operator algebras associated to affine Lie algebras, the Virasoro algebra, lattices and the moonshine module for the Monster group, we have this modular invariance property. I will discuss a proof of this theorem, various concepts and tools needed in this proof and some generalizations and applications.

### Homological Algebra

Text: An introduction to homological algebra, by C.~Weibel, Cambridge U. Press, paperback edition (1995).

Prerequisites: First-year knowledge of groups and modules.

Description: This will be an introduction to the subject of Homological Algebra. Homological Algebra is a tool used in many branches of mathematics, especially in Algebra, Topology and Algebraic Geometry.

The first part of the course will cover Chain Complexes, Projective and Injective Modules, Derived Functors, Ext and Tor. In addition, some basic notions of Category Theory will be presented: adjoint functors, abelian categories, natural transformations, limits and colimits.

The second part of the course will study Spectral Sequences, and apply this to several topics such as Homology of Groups and Lie Algebras. Which topics we cover will be determined by the interests of the students in the class.

### Model Theory

Text: Wilfrid Hodges, Model Theory, Cambridge University Press ISBN-10: 0521066360 ISBN-13: 978-0521066365

Prerequisites: A previous course in Mathematical Logic such as Math 561, or permission of the instructor.

Description:

Model theory deals with models of axiomatized theories, in particular the algebraic structures, combinatorial structures, and models of set theory, by unified methods. We will cover both the general methods of Model Theory, and some representative applications in each of these three areas.

Topics include the Lowenheim-Skolem theorems, Compactnes, indiscernible sequences, omitting types, countable and uncountable categoricity, model completeness, saturation, and stability.

Applications include the consistency of the Continuum Hypothesis, the nonexistence of measurable cardinals in the universe of constructible sets, the solution of Hilbert's 17th problem, and results on locally finite generalized quadrangles, universal graphs, and classes of permutations with forbidden patterns.

For the theory and some of the applications we follow Wilfrid Hodges' text, Model Theory. The remaining applications are found in various specialized texts and the journal literature, and may include some work in progress.

### Special Topics in Number Theory

Subtitle: Analytic Theory of Automorphic L-functions

Text: In many cases I will distribute my personal notes on the subjects during the course. There is no one book which covers all the material, so I shall refer to specific publications when needed.

Prerequisites: Knowledge of the spectral theory will be helpful

Description: This course is a continuation of the course given in the Fall 2009 on Spectral Theory of Automorphic Forms (see the description pasted below). Although the knowledge of the spectral theory will be helpful, a new student may still be able to follow and learn the new material by simply accepting basic theorems without studying their development. I will often recall these basic theorems during my course. The topics of the spring 2010 course include:
1. The theory of Hecke operators
2. Analytic properties of automorphic L-functions
3. Rankin-Selberg convolution L-functions
4. Symmetric power L-functions
5. Non-vanishing on the boundary of the critical line
6. Spectral power-moments of L-functions
7. Subconvexity bounds of L-functions on the critical line
8. Central values of L-functions
9. Applications to problems of equidistribution

These lectures will be on Tuesdays and Fridays 12:00-1:20pm in Hill 124.

Previous course topic (Fall 2009): Spectral Theory of Automorphic Forms

This course will be for students interested in analytic number theory as well as for those who would like to learn the basics of spectral theory in the hyperbolic plane. Large part of the subject will be devoted to general topics, but the target is to give (eventually) applications to questions in arithmetic. The material is huge, so if students will like to learn more of special topics I may continue the course in the spring semester 2010. Here are some fundamental topics:

1. Geometry of the hyperbolic plane
2. Harmonic analysis on the hyperbolic plane
3. Fuchsian groups
4. Automorphic forms
5. Eisenstein series
6. Spectral decomposition
7. Selberg’s trace formula
8. Spectral decomposition of Kloosterman sums
9. Quantum Unique Ergodicity Conjecture (with proofs)

### Topics in Number Theory

Subtitle: Holomorphic Modular Forms

Text:
Topics in classical automorphic forms,
by Henryk Iwaniec, AMS Graduate Texts. Useful secondary source:
A course in arithmetic,
by J.-P. Serre, Springer Verlag Graduate Texts.

Prerequisites: At least one course in complex analysis, and either: 1) Math 571 or Math 573, or 2) permission of instructor.

Description: The course will cover the basic theory of holomorphic modular forms on the complex upper half plane for SL(2,Z) and its congruence subgroups. Main topics to be covered include Fourier expansions, Hecke operators, Atkin-Lehner operators, Theta functions, and L-functions.

### Experimental Mathematics

Text: No textbooks just handouts

Prerequisites: There are no prerequisites, and no previous programming knowledge is assumed

Description:

Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in that direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they are doing ( or will do ) research in.

We will first learn Maple, and how to program in it. This semester we will focus on PRIMES. We will learn how to tell whether any given integer is a prime in polynomial time ( thanks to AKS ), really understand the elementary proof of the Prime Number Theorem, and, who knows?, may be one of you will prove the Goldbach conjecture?

But the actual content is not that important, it is mastering the methodology of computer-generated and computer-assisted research that is so crucial for your future.

There are no prerequisites, and no previous programming knowledge is assumed. Also, very little overlap with previous years. The final projects for this class may lead to journal publications.

### Methods of Applied Mathematics II

Text: Advanced Engineering Mathematics (2nd edition) by Michael D. Greenberg, (Prentice Hall, Upper Saddle River, NJ, 1998). ISBN 0-13-321431-1

Optional purchase:

Methods of Applied Mathematics (2nd edition) by Francis B. Hildebrand, which is available in paperback (Dover, New York, 1965) ISBN 0-486-67002-3

Prerequisites: Math 527, or else permission of the instructor

Description: This is a second-semester graduate course, appropriate for students of mechanical and aerospace engineering, biomedical, electrical, or other engineering areas, materials science, or physics. It begins with the algebra of complex numbers, complex-valued functions of complex variables, analytic functions and the Cauchy-Riemann conditions, poles and branch cuts, and conformal mappings, with applications in physics and engineering to the solution of differential equations and to fluid mechanics. Finally, we address some topics in the calculus of variations with applications.

### Numerical Analysis II

Text: There is no required textbook. Lecture notes will be posted to the course web site after each class. For those students who also wish to have another source for the material, I recommend purchasing one of the following texts: Copies will also be on reserve in the Mathematics library.

K. Atkinson, / An Introduction to Numerical Analysis,/, (second edition), Wiley, 1989.

D. Kincaid and W. Cheney: /Numerical Analysis: Mathematics of Scientific Computing/, (third edition), American Mathematical Society, 2002 (republished 2009).

A. Quarteroni, R. Sacco, and F. Saleri, /Numerical Mathematics,/, (second edition), Springer, 2004.

J. Stoer and R. Bulirsch: /Introduction to Numerical Analysis/, (third edition) Springer, 2002.

Prerequisites: Advanced Calculus, Linear Algebra, and familiarity with differential equations. .

Description: This is the second part, independent of the first, of a general survey of the basic topics in numerical analysis -- the study and analysis of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications.

This semester (642:574), we will study the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations, such as Poisson's equation and the heat equation.

In the fall semester, we considered the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential equations, and showed how all these problems are related.

Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work.

To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, the use of /Matlab/ will be encouraged. This program has many built in features which make programming easy, even for those with very little prior programming experience.

### Numerical Solutions of Partial Differential Equations

Text: Finite Elements: Theory, fast solvers, and applications in solid mechanics by Dietrich Braess, Cambridge University Press, (paperback), 2007 (3rd edition)

Prerequisites: Advanced calculus, linear algebra, and familiarity with differential equations.

Description:

The course goals consist of design, analysis and implementation of the most commonly used numerical methods for solutions to partial differential equations. We will discuss various discretization schemes such as finite difference, finite element and finite volume methods as well as iterative solvers such as multigrid methods. The course will maintain a balance between in-depth mathematical theories for algorithmic techniques and computer implementations and students will have the opportunity to study not only theoretical backgrounds in developing and understanding the numerical algorithms, but also a hands-on experience to implement the methods.

Matlab (or scilab) will be used for the computational component of the course and a number of source codes will be provided to minimize the coding efforts from students.

### Graph Theory

Text: Bollobas: Modern graph theory

Prerequisites: Permission of instructor required for students not enrolled in mathematics PhD program.

Description:

The course will cover fundamentals of graph theory and the following topics:

• Electrical networks
• Flows, Connectivity and Matching
• Extremal Problems
• Coloring
• Ramsey Theory
• Random Graphs
• Graphs, Groups and Matrices
• Random Walks on Graphs
• The Tutte Polynomial

### Combinatorics II

Text: There is no one text; various more or less relevant books will be on reserve in the library.

Prerequisites: There are no formal prerequisites, but the course assumes a level of mathematical maturity consistent with having had good courses in linear algebra (such as 640:350) and real analysis (such as 640:411) at the undergraduate level. It will help to have seen at least a little prior combinatorics, and (very) rudimentary probability will also occasionally be useful. (Of course survival of the first semester certifies preparation for the second.)

Description: This is the second part of a one-year long introductory course combinatorics Topics in this second semester are listed below:

• Probabilistic methods and random structures
• Spectral graph theory
• Discrete harmonic analysis and applications
• Applications in additive combinatorics and theoretical computer science

### Selected Topics in Discrete Mathematics

Subtitle: Probabilistic Methods in Combinatorics

Text: Alon-Spencer, The Probabilistic Method (optional, but useful).

Prerequisites: I will try to make the course self-contained except for basic combinatorics and very basic probability. See me if in doubt.

Description: We will discuss applications of probabilistic ideas to problems in combinatorics and related areas (e.g. geometry, graph theory, complexity theory). We will also at least touch on topics, such as percolation and mixing rates for Markov chains, which are interesting from both combinatorics/TCS and purely probabilistic viewpoints.

### Topics in Probability and Ergodic Theory II

Subtitle: Arithmetic structure in the integers and the primes

Text:

Prerequisites: The course assumes a level of mathematical maturity consistent with having had a good course in linear algebra (such as 640:350) and real analysis (such as 640:411) at the undergraduate level. A prior introduction to combinatorics, and rudimentary probability are all occasionally helpful.

Description: This course is suitable for anyone with an interest in number theory, harmonic analysis or combinatorics. We shall explore various approaches to establishing the existence of arithmetic structure in arbitrary subsets of the integers and the primes. These approaches include purely combinatorial methods, analytic techniques as well as a thorough discuss from a dynamical systems point of view.

This course serves as an introduction to a very active field of research that has attracted much attention recently -- a number of the results I will be discussing are less than five years old.

Syllabus:

1. Finite field models:

a) introduction to discrete Fourier analysis

b) counting 3-term progressions (Meshulam)

c)* counting 3-term progressions without Fourier analysis (Lev)

d) functions with small spectral norm (Green-Sanders)

e) uniformity norms (Gowers)

f) Freiman's theorem (Ruzsa)

g) inverse theorem for U3 (Gowers, Green-Tao)

h)* counting 4-term progressions (Green-Tao)

2. Approximate subgroups:

a) Bohr sets

b) counting 3-term progressions (Bourgain)

c)* square-difference free sets (Sarkozy)

d)* sets with the minimal number of 3-term progressions (Croot)

e) inverse theorem for U3 (Gowers, Green-Tao)

f)+ counting general linear configurations (Gowers-W.)

3. Ergodic theoretic approaches:

a) dynamical systems and factors

b) correspondence principle (Furstenberg)

c) another proof of Szemeredi's theorem (Furstenberg)

d)* U^k seminorms (Host-Kra)

e) structure theorem for the seminorms (Host-Kra)

f)+ polynomial extensions of Szemeredi's theorem (Bergelson-Leibman)

4. Long progressions in the primes:

a) transference principle (Green-Tao, Gowers)

b)* some analytic number theory (Goldston-Pintz-Yildirim)

c) putting it all together (Green-Tao)

Topics marked with * would be suitable for student presentations.

Topics marked with + will be included if time permits.

### Mathematical Finance II

This course is part of the Mathematical Finance Master's Degree Program.

### Computational Finance

This course is part of the Mathematical Finance Master's Degree Program.

### Selected Topics in Mathematical Finance

This course is part of the Mathematical Finance Master's Degree Program.

### Selected Topics in Mathematical Finance

This course is part of the Mathematical Finance Master's Degree Program.

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