Mathematics Department - Graduate Course Descriptions - Spring 2014

### Theory of Functions of a Real Variable II

Text: Real Analysis by G.B. Folland

Prerequisites: 501

Description: The course will start with covering Lemmas, which will be applied to study the theory of Differentiation of Integrals. Then, we move onto the Fundamental theorem of calculus.

We next study various inequalities on Lp spaces, interpolation of operators, the Marcinkiewicz interpolation theorem.

The last part of the course deals with the rudiments of Functional Analysis with a special focus on Hilbert spaces and applications to Lp spaces.

### Theory of Functions of a Complex Variable II

Text: Green and Krantz: Function Theory of One Complex Variable, AMS. Forster, O.: Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991

Prerequisites: Math 503

Description: This will be a continuation of Math 503. The following two parts will be covered:

(1) Classical Complex Analysis and (2) Riemann surfaces.

Part 1. Clasical Complex Analysis. We will discuss topics not covered in Math 503, such as the Riemann and Caratheodory conformal mapping theorems, Picard theorems, elliptic functions.

Part 2. Introduction to Riemann surfaces. Hyperbolic geometry and uniformization theorem. Riemann-Roch theorem.

### Functional Analysis II

Text: "Functional Analysis," by Peter D. Lax, Wiley Interscience, 2002, ISBN 0-471-55604-1.

Prerequisites: Math 501, Math 502, Math 503, and Math 507 or the equivalents. If in doubt, consult with the instructor.

Description: We shall continue the discussion of the basic theory of linear operators on Banach spaces and other topological linear spaces, as well as the properties of concrete examples of function spaces. We will cover:

(1) Fixed point theorems, especially the Schauder Fixed Point Theorem (which is an infinite-dimensional version of the Brower Fixed Point Theorem) and the Cellina, Kakutani, and Fan-Glicksberg theorems. This part will be based on notes by the instructor.)

(2) the spectral theory of bounded and unbounded self-adjoint operators on Hilbert spaces (starting with the study of commutative C-star algebras and the Gelfand Representation Theorem), symmetric operators and Friedrichs' extension theorem, unitary operators, unitary one-parameter groups, and Stone's Theorem,

(3) Sobolev spaces, Sobolev inequalities, boundary trace theorems, and applications to elliptic boundary value problems (this part will be based on lecture notes such as the ones in www.iadm.uni-stuttgart.de/LstAnaMPhy/Weidl/fa-ws04/Suslina_Sobolevraeume.pdf),

(4) distributions.

In each case, the theory will be illustrated with examples, such as eigenfunction expansions related to various differential operators, and in particular Fourier series and Fourier transforms, and applications to partial differential equations.

### Partial Differential Equations II

Text: Recommended: Degenerate diffusions by Panagiota Daskalopoulos and Carlos E. Kenig; Smoothing and decay estimates for nonlinear diffusion equations by Vazquez.

Prerequisites: A first semester graduate PDE course (such as our 640:517) or equivalent

Description: We will be dealing with the simple nonlinear equation u_t = \Delta u^m, where m > 0.

For m > 1 this is called the porous medium equation and for m < 1 it is called the fast diffusion equation. There are a number of physical applications of this equation. We will discuss the questions of existence and uniqueness of a solution and the main properties of the solutions. We will study the initial value problem (Cauchy problem) and the Dirichlet problem. Our approach to these problems is through the use of local regularity estimates and Harnack type inequalities, which yield equicontinuity and hence compactness for families of solutions. We also want to discuss the notion of weak solutions.

The goal is to emphasize different techniques used to tackle problems regarding the porous medium and the fast diffusion equations since they may have a wider scope of applicability to studying other nonlinear equations.

### Selected Topics in Differential Equations

Subtitle: Some aspects of Navier-Stokes equations

Text: References:

1. Temam, Roger Navier-Stokes equations. Theory and numerical analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001.

2. Galdi, G. P. An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011.

3. Various papers.

Prerequisites: 517 or Permission of Instructor

Description: In this course we will introduce some results on Navier-Stokes equations. The material being presented will be as follows.

1. Survey and exposition on the existence of solutions to the nonhomogeneous stationary Navier-Stokes equationsin dimension two. This will includes works from some pioneering work of Leray and new developments in the last few years.

2. Exposition on some works on Leray's problem of steady Navier-Stokes flow past a body in the plane.

3. Regularity of solutions of stationary Navier-Stokes equations in dimension less than 5, partial regularity of stationary Navier-Stokes equations in dimension bigger than 4, existence of regular solution of the stationary Navier-Stokes equations in dimension 5.

4. Existence of weak solutions in three space dimension (Jean Leray) and the existence and smoothness of Navier-Stokes solutions in two space dimension (Ladyzhenskaya).

5. Global existence of solution for Navier-Stokes if the initial data is small in various scale invariant spaces.

6. Partial regularity of weak solutions in three space dimension (Caffarelli-Kohn-Nirenberg, and a simplified proof by Fanghua Lin).

7. L^\infty([0,T], L^3(R^3)) solutions of the Navier-Stokes and backward uniqueness (Escauriaza-Seregin-Sverak).

### Differential Geometry

Subtitle: Riemannian geometry

Text: Do Carmo's "Riemannian geometry" and some of my own notes.

Prerequisites: Multivariable Calculus

Description: This course covers basic Riemannian geometry: geometry of Riemannian manifolds, connections, curvature, completeness, geodesics, Jacobi fields, Bonnet-Myer's theorem and various comparison theorems.

The course is aim to prepare the students for research in Riemannian geometry, metric geometry and global geometric partial differential equations arising from physics such as canonical metrics of Einstein type as well as the Ricci flow and mean curvature flow.

### Selected Topics in Geometry

Subtitle: Floer homology

Text: None

Prerequisites: Differential Forms on Manifolds

Description: "Symplectic manifolds; foundations of pseudoholomorphic curves; quantum and Floer homology, depending on the interests of the audience. If you are interested please contact the instructor at ctw@math.rutgers.edu to join the discussion about possible topics."

### Lie Algebras

Text: Representation Theory, by W. Fulton and J. Harris

Prerequisites: First semester of Abstract Algebra for graduate students, permission of instructor required for students not enrolled in the mathematics Ph.D. program.

Description: This course will be an introduction to Lie algebras in the context of linear algebraic groups, with emphasis on the classical complex Lie algebras (the general and special linear Lie algebra, orthogonal Lie algebra, and symplectic Lie algebra).

Topics will include elementary properties of linear algebraic groups, their finite-dimensional representations, and their Lie algebras. The Lie algebras of the classical groups will be studied using root systems and Weyl groups relative to a maximal torus. The complete reducibility of finite-dimensional representations will be proved and the Cartan-Weyl highest weight theory of irreducible finite-dimensional representations will be developed. For the classical simple Lie algebras explicit models for the irreducible representations will be constructed. The structure and classification of semisimple Lie algebras will be also covered.

If time permitted, more advanced topics (Kac-Moody algebras, etc) will be discussed.

### Abstract Algebra II

Text: Jacobson, Basic Algebra, 2nd edition, Volumes I and II

Prerequisites: 551

Description: The main topics of the course will be introductions, as time permits, to

Field theory,

Commutative rings,

The classical groups, and

Modules and the representation theory of finite groups.

### Selected Topics in Algebra

Subtitle: The theory of partitions and vertex operator algebras

Text: Selected parts of The Theory of Partitions'' by G. Andrews, Cambridge University Press, 1984 (paperback). Also, some selected material from Vertex Operator Algebras and the Monster'' by I. Frenkel, J. Lepowsky and A. Meurman, Academic Press, 1988. Additional material, including some research papers, will be distributed.

Prerequisites: No prerequisites for the theory of partitions. Prerequisites for relations with vertex operator theory: Besides basic algebra, some familiarity with Lie algebras will be helpful.

Description: The Rogers-Ramanujan partition identities and generalizations due to Gordon, Andrews and others have long been of great interest in combinatorial analysis. The classical and still-unfolding theory of such identities turns out to be deeply related to the representation theory of vertex operator algebras. In this course we will develop the theory of natural families of partition identities, using such methods as the Rogers-Ramanujan and Rogers-Selberg recursions. We will also cover recent approaches to such identities, approaches that are still being developed in current research. We will introduce and motivate relevant aspects of the theory of affine Lie algebras and vertex operator algebra theory as they arise in the course, but the core of the course will be elementary.'' A range of research problems in both the theory of identities and vertex operator algebra theory will be highlighted.

Please note: The Lie Groups/Quantum Mathematics Seminar, which will meet Fridays at noon, 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 certainly not be required for the course.

### Commutative Algebra

Text: David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, GTM 150, 1995

Prerequisites: 640:551/640:552 or equivalent

Description: Commutative algebra provides the basic toolbox in a number of fields including number theory, invariant theory, and algebraic geometry. This course will be an introduction to the subject which mostly focuses on applications to algebraic geometry. The course will emphasize the viewpoint that a commutative ring corresponds to a geometric space. Furthermore, the geometric properties of this space, like its dimension and singularities, are reflected by the algebraic properties of the ring. The course will introduce basic notions such as localization, primary decomposition, integrality, flatness, and dimension. We will also discuss Groebner bases which make it possible to automatically compute solutions to many problems in algebraic geometry on a computer.

### Selected Topics in Logic

Subtitle: Forcing and applications

Text: We will not follow any specific book, but two standard texts are Jech's Set Theory and Kunen's Set Theory. I highly recommend Schoenfield's paper Unramified forcing".

Prerequisites: None

Description: Continuum Hypothesis, which was the first problem on Hilbert's list, has been a source of many deep ideas and techniques. It was eventually shown by Godel and Cohen that it can neither be proved nor disproved from the usual axioms of set theory. Godel discovered his constructable universe L and showed that it satisfies CH while Cohen discovered forcing, a technique for constructing models of set theory from existing ones, and showed that L has an extension in which CH fails. Cohen was awarded Fields medal for his work. Today forcing is the most fundamental tool used to establish independence results and often even theorems of ZFC. In this course we will systematically develop the theory of forcing and then use it to construct model of ZFC with various interesting properties. Some specific list of topics include the following. 1. Martin's Axiom 2. Levy Collapse 3. Product Forcing 4. Solovay's Model in which all sets of reals are Lebesgue Measurable 5. Interplay with large cardinals, Prikry Forcing

### Special Topics in Number Theory

Subtitle: Spectral Theory of Automorphic Forms

Text: Spectral Methods of Automorphic Forms, AMS Graduate Studies in Math V.53 by Henryk Iwaniec

Prerequisites: Complex Analysis

Description: This course is for graduate students of any level who are interested in analytic number theory. The following topics will be covered:

- Harmonic analysis on the hyperbolic plane

- Fuchsian groups

- Automorphic forms

- Spectral theory

- Fourier coefficients and Hecke eigenvalues

- Kloosterman sums

- Trace formulas

- Applications to number theory

Students are assumed to be knowledgeable of basic facts from complex analysis and elementary number theory. The lectures will be Tuesdays and Fridays, 12:00 – 1:20pm.

### Experimental Mathematics

Text: None (Links to websites - TBA)

Prerequisites: None

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 do Numerical Analysis (Numerical solutions of ODEs (Runge-Kutta) and PDEs(using both finite difference schemes and the finite element method)) via Symbolic Computation. We will also experiment with Approximation Theory, and see how symbolic computation can help it.

In addition to the actual, very important, content, mastering painlessly a very important part of applied mathematics (whose knowledge would look very good on the CV), students will master the methodology of computer-generated and computer-assisted research that is so crucial for their future.

There are no prerequisites, and no previous programming knowledge is assumed. Also, no overlap with previous years. The final projects will, with high probability, lead to publishable articles, and with strictly positive probability, to seminal ones.

### Methods of Applied Mathematics II

Text: Advanced Engineering Mathematics by Michael Greenberg

Prerequisites: Methods in Applied Mathematics I

Description: A second semester graduate course primarily intended for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. There will be three parts:

1. Complex variable theory, including the differential and integral calculus of functions of a complex variable, conformal mapping, Taylor series, Laurent series and the residue theorem. Introduction to the calculus of variations.

2. Calculus of variation, including the motivation of variational principles from physical laws, derivation of Euler-Lagrange equations, stability criterion, linearization, and brief introduction of the foundation of finite element method.

3. Perturbation methods, including applications to ode systems, examples of boundary layer, multiple-scale problems, and eigenvalue problems.

Emphasis on applications and calculations which graduate students in engineering may encounter in their courses.

### Introduction to Mathematical Physics II

Subtitle: Classical Physics

Text: (No Textbook) Kiessling's lecture notes

Prerequisites: Working knowledge of basic ODEs and the linear wave equation. Some exposure to analysis at the level of the "baby Rudin." Basic knowledge of Euclidean geometry. A curiosity for the physical secrets of the universe.

Description: The course introduces the student to a modern mathematical treatment of the classical physical theories of the universe: space(-)time, matter, gravity and electromagnetism.

Topics:

1. The Newtonian universe (Galileian space and time, point particles, Newton's law of motion, Newton's law of gravitational force, Coulomb's law of electrical force, Lorentz' law of magnetic force; symmetries and conservation laws; other formulations of mechanics: Hamilton, Hamilton-Jacobi, and Lagrange; derivation of the celestial two-body (=Kepler) problem from the N-body problem for Newtonian atoms)

2. The Einsteinian universe (Minkowski's spacetime, Maxwell's electromagnetic field equations, electromagnetic waves, relativistic energy and momentum; and in brief outlinealso: Lorentzian manifolds, Einstein's gravitational field equations, geodesics, black holes, gravitational waves)

3. Limits of validity of the classical theories (the joint Cauchy problem for fields and point particles, the problem of self-interactions; energy and momentum laws and the dawn of quantum theory.)

### Rigorous Results in Statistical Mechanics II: Nonequilibrium

Text: Choice of specific topics will be based on student interest. For background material see the following books:
1. Mathematical Biology I and II, J.D. Murray (Springer)
2. Large Scale Dynamics of Interacting Particles, H. Spohn (Springer)
3. A Kinetic View of Statistical Physics, P. Krapivsky, S. Redner and E. Ben-Naim (Cambridge)
4. Dynamics of Self-Organized and Self-Assembled Structures, R. Desai and R. Krapral (Cambridge)
5. Evolutionary Games and population Dynamics, J. Hofbauer and K. Sigmund (Cambridge)
The course will be informal and interactive.

Description: Course will start with a broad overview of the physics and mathematics of equilibrium and nonequilibrium statistical mechanics: This will focus on the elucidation and derivation of collective behavior of macroscopic systems made up of very many individual components from the microscopic dynamics of the individual components. I will then consider application of statistical mechanics to real world problems of current interest. An example of such an application is pattern formation. This occurs in both equilibrium and nonequilibrium systems. The former generally represent low temperature phases in materials and can be studied via ensembles. The latter involve dynamical microscopic considerations and are generally described on the macroscopic level by reaction-biological systems. They range in scale from microns for cells forming an organism to hundreds of meters for flocking birds.

### Numerical Analysis II

Text: (Recommended) K. Atkinson “An Introduction to Numerical Analysis" 0-471-62489-6;
A. Quarteroni, R. Sacco, and F. Saleri, /Numerical Mathematics, (second edition), Springer, 2004, 0-387-98959-5.

Prerequisites: Numerical Analysis I - 16:642:573 is desirable, but not required

Description: This is the second part 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. 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 see how all these problems are related. In the spring 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. 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, however, 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.
This course is also part of the Mathematical Finance Master's Degree Program

Text:

Prerequisites:

Description:

### Graph Theory

Text: Bela Bollabas, Modern Graph Theory

Prerequisites: Open to any Mathematics Ph.D. student, and to others by permission of instructor.

Description: This course will serve as an advanced introduction to graph theory. in graph theory. We'll work mainly from the text by Bela Bollobas on Modern Graph Theory, with some excursions outside the book. Some of the topics we will cover include: Matchings, cuts, flows, connectivity, planar graphs, graph colorings, random graphs, extremal graph theory, Ramsey theory, linear algebra methods, and expander graphs.

While I will not assume that students have prior knowledge of graph theory, I will assume a basic knowledge of discrete probability and linear algebra (at an undergraduate level).

### Combinatorics II

Text: There is no one text; relevant books will be on reserve.

Prerequisites: 642.582 or permission of the instructor

Description: This is the second part of a two-semester course surveying basic topics in combinatorics. In the second term we will discuss most, but probably not all of:

Partially ordered sets, lattices and Mobius functions

Theory of finite sets, hypergraphs, combinatorial discrepancy

Ramsey theory

Correlation inequalities

Probabilistic methods

Algebraic and Fourier methods

Entropy methods

### Selected Topics in Discrete Mathematics

Subtitle: Topics in Discrete Geometry

Text: No required text, however I will be using the following books/surveys as a reference: Incidence Theorems and their Applications (Zeev Dvir), Lectures on Discrete Geometry (Jiri Matousek), Lectures on Discrete and Polyhedral Geometry (Igor Pak), Using the Borsuk-Ulam Theorem (Jiri Matousek)

Prerequisites: Basic discrete math, mathematical maturity

Description: This course will cover a diverse collection of results and techniques in discrete geometry and incidence geometry spanning a wide range of topics, including some of the classical gems as well as some of the more recent results.

The following is a partial and tentative list of topics:

* The polynomial method – applications to the joints conjecture, Erdos distance problem, Kakeya conjecture

* Metric embeddings - Johnson-lindenstrauss Lemma, Bourgain's embedding etc.

* Incidence theorems (Szemeredi trotter and extensions) and applications to sum/product theorems

* Sylvester-Gallai type theorems and applications

* Topological methods - applications of the Borsuk-Ulam theorem

### 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.

### Fixed Income Securities and Derivative 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.

### Seminar in Mathematical Finance

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

### Topics in Mathematical Physics

Subtitle: The Foundation of Statistical Mechanics (continued in description)

Text: None

Prerequisites: There are no formal prerequisites, but prior exposure to equilibrium statistical mechanics and to quantum theory would be helpful, as would some knowledge of probability theory. The relevant mathematical and physical concepts will be developed as we proceed.

Description: Full Subtitle: The Foundations of Statistical Mechanics: The Arrow of Time, Irreversibility and the Origin of Randomness in Classical and Quantum Physics

This course will be concerned with foundational and conceptual issues in statistical mechanics. We shall address the following questions:

• How can the time-symmetric laws of microscopic physics lead to, or be consistent with, the irreversible (time-asymmetric) laws of macroscopic physics?

• In particular, how should entropy be understood in microscopic terms, and how can the second law of thermodynamics, Boltzmann’s equation and Boltzmann’s H-theorem be compatible with, and derivable from, the laws of microscopic physics?

• Why do macroscopic systems in nonequilibrium evolve in such a way that they converge in due course to equilibrium states characterized by the thermodynamic ensembles?

• How relevant are the concepts of ergodic theory, such as ergodicity and mixing, to the demonstration of convergence to thermodynamic equilibrium and to the justification of the use of the thermodynamic ensembles as a description of equilibrium?

• How is our understanding of the foundations of statistical mechanics affected by the transition from classical mechanics to quantum mechanics?

• What are the implications of the concepts of equilibrium and nonequilibrium for the origin of randomness and the (im)possibility of certain knowledge?

### Topics in Mathematical Physics

Subtitle: Applications to PDE

Text: Functional Analysis I,II by Reed&Simon, Schroedinger Operators, by Cycon, Froese, Kirsch, Simon, the 2008 edition from Springer. Selected papers from the recent literature.

Prerequisites: Basic Knowledge of real analysis is required.

Description: The aim of the course is to introduce modern methods of Analysis in Partial differential equations of Mathematical Physics, and general spectral theory. I begin with review of basic topics from analysis, like Hilbert spaces, compact operators and applications, Spectral Theorem. Then we study the spectral properties of Self-adjoint operators of classical type, and of general interest. Then, the subject of decay estimates for wave and Schroedinger type equations will be developed. Finally, if time permits, some consequences for nonlinear Dispersive equations will be described. There will be no exam.

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