Mathematics Department - Graduate Course Descriptions - Fall 2009

### Theory of Functions of a Real Variable I

Text: Modern Techniques and Their Applications, 2nd Edition, Gerald Folland

Prerequisites: Undergraduate analysis at the level of Rudin's "Principles of Mathematical Analysis," chapters 1--9, including basic point set topology, metric space, continuity, convergence and uniform convergence of functions.

Description: Basic real variable function theory, measure and integration theory prerequisite to pure and applied analysis. Topics: Riemann and Lebesgue-Stieltjes integration; measure spaces, measurable functions and measure; Lebesgue measure and integration; convergence theorems for integrals; Lusin and Egorov theorems; product measures and Fubini-Tonelli theorem; Lp-spaces. Other topics and applications (such as Lebesgue's differentiation theorem, signed measures, absolute continuity and Radon-Nikodym theorem) as time permits.

### Theory of Functions of a Complex Variable I

Text: Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics) *by Robert E. Greene and Steven G. Krantz
Publisher: American Mathematical Society; 3rd edition (March 29, 2006)
ISBN-13: 978-0821839621

Prerequisites:

Description:

The beginning of the study of one complex variable is certainly one of the loveliest mathematical subjects. It's the magnificent result of several centuries of investigation into what happens when R is replaced by C in "calculus". Among the consequences were the creation of numerous areas of modern pure and applied mathematics, and the clarification of many foundational issues in analysis and geometry. Gauss, Cauchy, Weierstrass, Riemann and others found this all intensely absorbing and wonderfully rewarding. The theorems and techniques developed in modern complex analysis are of great use in all parts of mathematics.

The course will be a rigorous introduction with examples and proofs foreshadowing modern connections of complex analysis with differential and algebraic geometry and partial differential equations. Acquaintance with analytic arguments at the level of Rudin's Principles of Modern Analysis is necessary. Some knowledge of algebra and point-set topology is useful.

The course will include some appropriate review of relevant topics, but this review will not be enough to educate the uninformed student adequately. A previous "undergraduate" course in complex analysis would also be useful though not necessary.

There are many excellent books about this subject. The official text will be Function Theory of One Complex Variable, by Greene and Krantz (American Math Society, 3rd edition, 2006). The course will cover most of Chapters 1 through 5 of the text, parts of Chapters 6 and 7, and possibly other topics. The titles of these chapters follow.
1: Fundamental Concepts; 2: Complex Lines Integrals; 3: Applications of the Cauchy Integral; 4: Meromorphic functions and Residues; 5: The Zeros of a Holomorphic Function; 6: Holomorphic Functions as Geometric Mappings.

### Functional Analysis II

Text: Reed-Simon I,II and E.B.Davies' book on Spectral theory

Prerequisites: Real Analysis, ODE, Linear Algebra

Description: The course will be focused on Spectral ans scattering theory techniques in partial differential equations and mathematical physics. It begins with review of basic notions and results from Functional analysis, including the spectral theorem for unbounded operators. Then, applications of compact operators in PDE and Spectral theory. basic constructions of Hamiltonians in Quantum Mechanics. Large time behavior: from Global existence to decay and scattering. Examples and open problems.

### Selected Topics in Analysis

Subtitle: From functional analysis to PDEs

Text: Functional analysis, Sobolev spaces and PDEs, by Haim Brezis.

Prerequisites: 640:502

Description: The course will cover approximately the following topics.
• The Hahn-Banach theorems.
• The uniform boundedness principle and the closed graph theorem.
• Weak topologies. Reflexive spaces. Separable spaces.
• L^p spaces. Reflexivity. Separability. Dual of L^p. Convolution and regularization.
• Hilbert spaces.
• Compact operators. Spectral decomposition of self-adjoint compact operators.
• The Hille-Yosida theorem.
• Sobolev spaces and the variational formulation of boundary value problems in one dimension.
• Sobolev spaces and the variational formulation of elliptic boundary value problems in N dimensions.
• Evolution problems: the heat equation and the wave equation.

### Partial Differential Equations I

Text: The course material will be mostly drawn from "Partial Differential Equations" by Lawrence C. Evans, published by AMS, 2002; and "Partial Differential Equations: Methods and Applications, Second Edition" by Robert McOwen, Prentice Hall, 2002.

The former puts more emphasis on the theory, while the latter devotes some spaceto working out applications of the theory in some interesting cases, while leaving some full discussion of the theory to references. You may obtain one or both of the texts. I will put these two and some additional books on reserve in the math library:

* Jeffrey Rauch, Partial Differential Equations, Springer, 1997. * G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1976. * F. John, Partial Differential Equations, 4th ed., Springer-Verlag, 1982.

Prerequisites: A strong background on advanced calculus involving multivariables (esp. Green's Theorem and Divergence Theorem). We will also use some basic facts of Lp function spaces and the usual integral inequalities (mostly completeness and Holder inequalities in L2 setting).

These topics are covered in the first semester graduate real variable course (640:501).

Description: This is the first half of a year-long introductory graduate course on PDE. PDE is an enormously vast field. PDEs arise from very diverse fields: from classical to modern physics, to more applied sciences such as material sciences, mathematical biology, and signal processing, etc, and from the more pure aspects of mathematics such as complex analysis and geometric analysis.

This introductory course should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, complex analysis, differential geometry, and, of course, partial differential equations.

For an introductory course, it is more important to examine some important examples to certain depth, to introduce the formulation, concepts, most useful methods and techniques through such examples, than to concentrating on presentation and proof of results in their most general form.

This is the way the course will be conducted.

The beginning weeks of the course aim to develop enough familarity and experience to the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation.

Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing qualitative, characteristic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.

Next we will discuss some notions and results that are relevant in treating general PDEs: characteristics, non-characteristic Cauchy problems and Cauchy-Kowalevskaya theorem, wellposedness.

Towards the end of the semester, we will begin some introductory discussion on the extension of the energy methods to variable coefficient wave/heat equations, and/or the Dirichlet principle in the calculus of variations.

The purpose here is to motivate and introduce the notion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.

### Selected Topics in Differential Equations

Subtitle: Hyperbolic PDEs and the Mathematical Foundations of Relativistic Physics

Text: None will be required, but the material is largely drawn from the following books by D. Christodoulou: The Action Principle and Partial Differential Equations (1999, Princeton University Press) And Mathematical Problems of General Relativity I (2008, European Mathematical Society)

Prerequisites: Common Sense, or permission of the instructor

Description: This will be a self-contained introductory course on hyperbolic partial differential equations and the geometry of space-time. We'll cover the basic theory of hyperbolic PDE's, emphasizing equations that arise in continuum physics, i.e. Maxwell's equations of electromagnetics, Euler's equations of fluid dynamics, and Einstein's equations of General Relativity, before focusing on the latter. Although welcomed and very much appreciated, no previous knowledge of physics or of partial differential equations is assumed. Some prior knowledge of differential geometry and functional analysis is helpful. The following is an outline of the course:

0. Hyperbolicity: Definitions of Hyperbolicity in the Theory of Maps, Geometry of Characteristics, Energy Estimates, Domain of Dependence Theorem, Local Well-posedness, Formation of Singularities.

1. The Geometry of Space-time: Causal structure, curvature and gravitation, the energy tensor and the matter equations of motion, the Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit.

2. The Cauchy problem for Einstein Vacuum Equations: The Symbol and Characteristics of EVE. Local Existence in Wave Coordinates.

3. Formation of Singularities: The Penrose Singularity Theorem. Black Holes. Naked Singularities. Cosmic Censorship Conjectures.

4. Conservation Laws and Noether's Theorem: Lagrangian and Hamiltonian formulations. The Noether current in the theory of maps. Asymptotic flatness. The definition of global energy, momentum and angular momentum. Witten’s Proof of the Positive Energy Theorem.

5. Reduction under Symmetry: Homogeneous and Isotropic solutions. Spherically symmetric solutions. One- and Two-Killing Field Reductions of Einstein-Maxwell Equations. Ernst Potentials. Kerr and Newman solutions. Wave Maps.

### Potential Theory

Text: No Textbook

Prerequisites: Students should have some background knowledge on Remannian geometry and real analysis.

Description: This course will be an introduction to the complex potential theory and its applications in differential and algebraic geometry. The course will begin with a slow-paced introduction to complex manifolds and the constant scalar curvature equations on Riemann surfaces. Topics include complex Monge-Ampere equations, the Calabi conjecture, pluripotential theory and their relations to algebraic geometry and the Ricci flow.

### Introduction to Algebraic Topology I

Text: There will be no textbook for the course. Below are some nice references:

1. Marvin J. Greenberg, J. R. Harper, Algebraic Topology: A First course. Publisher: Westview Press .

2. A. Hatcher: algebraic topology, excellent collection of exercises. \$30 in paperback from Cambridge University Press, as well as online here

3. James W. Vick, Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics), Springer.

Prerequisites:

Description: This course will be an introduction to algebraic topology and basic manifold theory. The plan is to cover the following topics: fundamental group, Van Kampen's Theorem, covering spaces, simplicial and singular homology, cohomology, Brouwer's fixed-point theorem, and the Jordan-Brouwer separation theorem.

### Topology of Manifolds

Text: An Invitation to Morse Theory - Liviu Nicolaescu, Morse Theory - Milnor

Prerequisites: Algebraic topology as taught in math 540.

Description:

Morse theory is concerned with the relation between the topology of differentiable manifolds and the critical points of real-valued functions defined on those manifolds. The theory has applications to engineering, topology, differential geometry, PDE, geometric group theory, etc. There are even discrete versions of Morse theory that have applications to combinatorics.

The plan is to start with a brief discussion of differential manifolds, followed by lectures from Nicolaescu's book with appropriate detours into Milnor. From the table of contents in Nicolaescu:

1.1 Local structure of Morse functions

1.2 Existence of Morse functions

2.1 Surgery, handle attachment, and cobordisms

2.2 The topology of sublevel sets

2.3 Morse inequalities

2.4 Morse-Smale Dynamics

2.5 Morse-Floer Homology 2.6 Morse-Bott functions 2.7 Min-Max theory

1. 3.1 The Cohomology of Complex Grassmannians 3.2 Lefschetz Hyperplane Theorem 3.3 Symplectic manifolds and Hamiltonian flows 3.4 Morse theory of moment maps 3.5 S1-Equivariant localization

2. 4.1 Basics of complex Morse theory

### Lie Groups

Text: Symmetry, Representations, and Invariants (Springer, 2009), by Roe Goodman and Nolan R. Wallach.

Prerequisites: Real analysis, linear algebra, abstract algebra, and elementary topology at the beginning graduate or honors undergraduate level. No prior knowledge of Lie algebras, Lie groups, or representation theory will be assumed.

Description: This course will be an introduction to Lie groups, Lie algebras, algebraic groups, and finite-dimensional representation theory.

Course Outline:

1. The classical linear groups (complex and real forms)
2. Closed subgroups of GL(n, R) as real Lie groups
3. Linear algebraic groups and rational representations
4. Structure of complex classical groups and their Lie algebras: maximal torus, roots, adjoint representation
5. Semisimple Lie algebras: structure and classification
6. Highest weight theory for representations of semisimple Lie algebras
7. Reductivity of classical groups
Additional topics from Lie groups and representation theory will be covered depending on the interests of the class and the time available.

### Abstract Algebra I

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

Description: This is a standard graduate level course for beginners. We will consider a lot of examples.
1. Group Theory: Basic concepts, examples and theorems.
2. Groups acting on sets: orbits, cosets, stabilizers.
3. Basic Ring Theory: Fields, principal ideal domains (PIDs), matrix rings, division algebras, field of fractions.
4. Classification of finitely generated abelian groups, and modules over a PID. Application to linear algebra: rational and Jordan canonical form.
5. Bilinear Forms: Alternating and symmetric forms, invariants.
6. Categories and functors: Introduction.

### Selected Topics in Algebra

Subtitle: Vertex Operator Algebra Theory

Text: Main text: J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2003.

Supplementary text: I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, 1988.

Prerequisites: Only basic algebra. Also, some familiarity with Lie algebras would be helpful but is not necessary. Students potentially interested in this course are encouraged to consult me.

Description: This course will develop the axiomatic theory of vertex operator algebras, including representation theory, from a contemporary point of view. Important examples, including vertex operator algebras based on lattices and on affine Lie algebras, and the vertex operator algebra underlying monstrous moonshine,'' will be discussed. Using this theory one can raise new questions and address a range of problems relating to many areas of mathematics and to conformal field theory in physics.

Please note: The Lie Groups/Quantum Mathematics Seminar, which will meet Fridays at 11:45, will sometimes be related to the subjects of the course. Students planning to take the course should also try to arrange to attend the seminar, although the seminar will not be required for the course.

### Selected Topics in Algebra

Subtitle: Topics in algebraic geometry -- intro to toric geometry.

Text: No textbook required

Prerequisites: Some familiarity with varieties and schemes.

Description:

This is an introductory course in toric geometry. The pace will be dictated by the background of the participants who are expected to have some knowledge of algebraic, differential or arithmetic geometry. We will focus on the basics and examples.

The first part of the course will cover the following topics: Convex rational polytopes, fans and rational polyhedral cones. Toric varieties and morphisms. Toric birational geometry; smooth toric varieties and divisors Homogeneous coordinate ring construction. Singular toric varieties and resolution of singularities

The second part of the course will be tailored to the interests of the students in the class. Possible topics include:

• Monomial ideals.
• Cohomology rings of toric varieties and Betti numbers.
• Counting lattice points in lattice polytopes.
• Toric modular forms.

• ### Commutative Algebra

Text: David Eisenbud, Commutative Algebra with a view to Algebraic Geometry, Springer.

Prerequisites: Any graduate course in abstract algebra, or permission of the instructor

Description:

Commutative algebra is broadly concerned with solutions of structured sets of polynomial and analytic equations, and the study of pathways to methods and algorithms that facilitate the efficient processing in large scale computations with such data.

This course will be an introduction to commutative algebra, with applications to algebraic geometry, combinatorics and computational algebra.

Topics:

1) (If needed by audience) Noetherian rings: Rings of polynomials, Hilbert basis theorem, Dedekind domains, Finitely generated algebras over fields, Noether normalization, Nullstellensatz.

2) The first part of the course will treat basic notions and results---chain conditions, prime ideals, flatness, Krull dimension, Hilbert functions.

3) Required material from Homological Algebra--such as the derived functors of Hom and tensor products--will be given in class, not assumed.

4) The other half of the course will study in more detail rings of polynomials and its geometry, and Groebner bases. It will open the door to computational methods in algebra (a few will be studied). Some other applications will deal with counting solutions of certain linear diophantine equations.

### Mathematical Logic

Text: No textbook

Prerequisites: No prerequisites

Description:

This is an introductory course in Mathematical Logic aimed at graduate students in mathematics rather than prospective logicians.

In Set Theory, we will discuss basic topics such as ordinals, cardinals and the various equivalents of the Axiom of Choice; as well as more advanced topics such as the club filter and stationary sets. (The latter are needed to prove results such as the existence of 21 nonisomorphic dense linear orders of cardinality ℵ1.)

In Model Theory, we will begin with basic results such as the Completeness and Compactness Theorems. Then we will cover some more advanced topics focusing on the structure of the countable models of a complete theory in a countable language.

The course has no formal prerequisites.

### Number Theory I

Text:

Recommended

M. Ram Murty and Jody Esmonde, Problems in Algebraic Number Theory, Springer-Verlag Graduate Texts in Mathematics, volume 190.

Other reference books:

Lang, Algebraic Number Theory, Springer GTM, volume 110

Ireland and Rosen, A Classical Introduction to Modern Number Theory, GTM, volume 84.

Prerequisites: *Permission of instructor required for students not enrolled in the mathematics Ph.D. program.

Description: Topics: This will be an introductory graduate course, designed to cover the main prerequisites for our further graduate course offerings, such as Iwaniec's usual graduate courses. I will start with algebraic topics, but present them with an analytic perspective. I will then turn to analytic techniques proper, such as the proof of the prime number theorem. If time permits, we will discuss the basics of modular and maass forms, and perhaps elliptic curves. My overall aim is to cover many topics and describe the role of the key ideas involved.

1. Elementary Number Theory
2. Euclidean Rings
3. Algebraic Numbers and Integers
4. Integral Bases
5. Dedekind Domains
6. The Ideal Class Group
8. The Structure of Units
9. Higher Reciprocity Laws
10. Zeta Functions
11. Prime Number Theorem

### 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. 1. The theory of Hecke operators
2. 2. Analytic properties of automorphic L-functions
3. 3. Rankin-Selberg convolution L-functions
4. 4. Symmetric power L-functions
5. 5. Non-vanishing on the boundary of the critical line
6. 6. Spectral power-moments of L-functions
7. 7. Subconvexity bounds of L-functions on the critical line
8. 8. Central values of L-functions
9. 9. Applications to problems of equidistribution

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

### Methods of Applied Mathematics I

Text: M.Greenberg, Advanced Engineering Mathematics(second edition); Prentice, 1998 (ISBN# 0-13-321431-1))

Prerequisites: Topics the students should know, together with the courses in which they are taught at Rutgers, are: Introductory Linear Algebra (640:250); Multivariable Calculus (640:251); Elementary Differential Equations (640:244 or 640:252); Advanced Calculus for Engineering(Laplace transforms, sine and cosine series, introductory pde)(640:421).

Students who are not prepared for this course should consider taking 640:421.

Description: A first semester graduate course intended primarily for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. Power series and the method of Frobenius for solving differential equations; nonlinear differential equations and phase plane methods; vector spaces of functions, Hilbert spaces, and orthonormal bases; Fourier series and Sturm-Liouville theory; Fourier and Laplace transforms; separation of variables and other elementary solution methods for the linear differential equations of physics: the heat, wave, and Laplace equations.

### Linear Algebra and Applications

Text: Gilbert Strang, "Linear Algebra and its Applications", 4th edition, ISBN #0030105676, Brooks/Cole Publishing, 2007

Prerequisites: Familiarity with matrices, vectors, and mathematical reasoning at the level of advanced undergraduate applied mathematics courses.

Description: Note: This course is intended for graduate students in science, engineering and statistics.

This is an introductory course on vector spaces, linear transformations, determinants, and canonical forms for matrices (Row Echelon form and Jordan canonical form). Matrix factorization methods (LU and QR factorizations, Singular Value Decomposition) will be emphasized and applied to solve linear systems, find eigenvalues, and diagonalize quadratic forms. These methods will be developed in class and through homework assignments using MATLAB. Applications of linear algebra will include Least Squares Approximations, Discrete Fourier Transform, Differential Equations, Image Compression, and Data-base searching.

Grading: Written mid-term exam, homework, MATLAB projects, and a written final exam.

### Statistical Mechanics I: Equilibrium

Text:

Prerequisites:

Description:

### Numerical Analysis I

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

### Combinatorics I

Text: There is no one text; we will make use of several books that will be on reserve in the library.

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 is the first part of a two-semester course surveying basic topics in combinatorics. Topics in the first semester will be some subset of:

• Enumeration (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics)
• Matching theory, polyhedral issues
• Partially ordered sets and lattices, MÄobius functions
• Theory of ¯nite sets, hypergraphs, combinatorial discrepancy, Ramsey theory, correlation inequalities
• Probabilistic methods
• Algebraic methods

### Selected Topics in Discrete Mathematics

Subtitle: Random and Pseudo-Random Structures

Text:

Prerequisites: Basic courses in probability and graph theory or 642:582 or 642:581 or permission of instructor 642:581

Description: We are going to study basic random objects such as random graphs, random matrices, random walks etc and their applications. I am going to introduce essential techniques in these areas, including martingales, concentration, correlation, fourier analyis, convexity etc.

Some highlights: Chromatic number of random graphs, (Sharp-) Threshold phenomenon, Limiting distributions, Randomized algorithms, Pseudo-random graphs.

### Mathematical Foundations of Systems Biology

Text: Professor's notes will be made available online. No textbook will be used, but the website contains notes written by the instructor (to be updated) as well as additional material.

Prerequisites: Linear algebra, differential equations, and basic probability, though some topics that require more advanced prerequisites will be also covered. Please e-mail instructor if you have any questions.

Description:

The field of (Molecular) Systems Biology mostly concerns itself with individual cells, or small collections of cells, seen as networks involving DNA, RNA, proteins, metabolites, and small molecules. An example is the study of signal transduction pathways in cells, and their disruption in cancer. It is widely recognized by leading biologists that the typical "reductionist" approach is not powerful enough to describe, analyze, and interpret the complex behaviors of such networks. Quantitative (i.e, mathematical) formalisms, concepts, tools, and models are required, and there is a major role to be played by mathematicians in applying and adapting known theory to model and understand specific systems.

This course will provide an introduction to mathematical techniques as well as to the relevant biology. No background in biology will be expected from students. To make the course accessible to a wide audience, only minimal math prerequisites will be needed in order to follow most of the material.

There are a very large number of possible topics to choose from, and the syllabus will evolve based on student's interest and input. Possible topics include the dynamics of cell signaling networks, including memories, switches, adaptation, and oscillators, as well as biological phenomena of chemotaxis, pattern formation, and neural transmission. If there is interest, we can also discuss synthetic biology.

### Mathematical Finance I

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

### Credit Risk Modeling

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.

### Topics in Mathematical Physics

Subtitle: TBA

Text:

Prerequisites:

Description: Description to come.

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