Mathematics Department - Graduate Course Descriptions - Fall 2013

### Theory of Functions of a Real Variable I

Text: G. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, Wiley Interscience, 1999. ISBN 0-471-31716-0

Prerequisites: This course assumes familiarity with real analysis at the level, roughly, of W. Rudin,Principles of Mathematical Analysis.

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; signed measures, Radon-Nikodym theorem, and Lebesgue's differentiation theorem.

NOTE: There is a problem session scheduled on Wednesdays, second period, for courses 16:640:501 and 16:640:503 in Hill Center, Room 423, Busch Campus.

### Theory of Functions of a Complex Variable I

Text: Complex Analysis, Elias M. Stein and Rami Shakarchi, Princeton Lectures in Analysis II, Princeton University Press (April 7, 2003). ISBN-10: 0691113858

Prerequisites: Acquaintance with analytic arguments at the level of Rudin's Principles of Mathematical Analysis is necessary. Some knowledge of algebra and point-set topology is useful.

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 Mathematical 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.

NOTE: There is a problem session scheduled on Wednesdays, second period, for courses 16:640:501 and 16:640:503 in Hill Center, Room 423, Busch Campus.

### Functional Analysis I

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

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

Description: We shall begin by reviewing some material which has been covered by Professor Carlen this term in Math 502: the definition of Banach space, Hilbert space and locally convex topological vector space, the Baire category theorem, the Hahn-Banach theorem (including the geometric versions of Hahn-Banach) and the Banach Alaoglu theorem, but we shall present some different applications of these theorems. We shall discuss the weak topology on Banach spaces, reflexivity in Banach spaces, the Eberlein-Smulyan theorem and the connection between uniform convexity of norms and reflexivity. There will be an excursion to the topic of extreme points of convex sets in locally convex topological vector spaces and the Krein-Milman theorem.The basic general theory of bounded linear operators on Banach space will be presented: the open mapping theorem, the closed graph theorem, the nonemptiness of the spectrum, etc. We shall also discuss basic theory for some important classes of bounded linear operators, e.g., compact linear operators. The Schauder fixed point theorem will be presented and applied to the invariant subspace problem (Lomonosov's theorem). All abstract results will be illustrated with examples from analysis or operator theory. Toward the end of the course we shall present results concerning partial orderings in Banach spaces, e.g., the geometry of cones in Banach spaces and the famous Krein-Rutman theorem. The hope is to continue in Math 508 with a course on the theory and applications of positive operators (linear and nonlinear) in finite and infinite dimensions.

### Selected Topics in Analysis

Subtitle: Sobolev Maps with Values into the Circle

Text: Linked below is a (provisional) table of contents for the book in preparation with Mironescu. For the course I will pick a few selected topics.

Prerequisites: Students are required to have a good knowledge of Functional Analysis and Sobolev spaces, as presented e.g. in the book "Haim Brezis , Functional Analysis, Sobolev Spaces and PDEs, Springer (2011)".

Description: PLEASE NOTE: THIS CLASS WON'T START UNTIL OCTOBER 10, 2013.

Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics.

By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in this course is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.

I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

### Selected Topics in Analysis

Subtitle: Nonlinear Partial Differential Equations & Hamiltonian Systems;Existence Theorems

Text: None

Prerequisites: This course is a continuation of 509 that I taught in Fall 2012. The assumption is that the students know the basic facts about Sobolev spaces and injections, degree theory and critical point theory. For these two latter subjects, adjustments can be made for new students. Knowledge of the basic facts about Sobolev spaces is a prerequisite.

Description: Please see the Syllabus (PDF)

### Partial Differential Equations I

Text: Partial Differential Equations, Methods and Applications, By Robert McOwen, 2nd Ed., ISBN-10: 0821849743.

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 variables course (640:501).

Description: This is the first half of a year-long introductory graduate course on partial differential equations. This introductory course should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, differential geometry, complex analysis, and, of course, partial differential equations.

Topics: First-order equations, Principles for higher order equations, The wave equation, The Laplace equation, The heat equation, Linear functional analysis, Differential calculus methods, Linear elliptic theory. Applications.

### Selected Topics in Differential Equations

Subtitle: Monge-Ampere Equation and Regularity

Text: Notes

Prerequisites: Math 501, 502, 517, 518 (not necessary, but will help)

Description: The course aims to develop Caffarelli’s theory of sections for the Monge-Ampere Equation. We will develop Alexandrov’s theory of generalized solutions for the Monge-Ampere equation, the Alexandrov-Bakelman-Pucci estimate, the Krylov-Safonov Harnack inequality and Pogorelov’s theorem for the Monge-Ampere equation.

### Functions of Several Complex Variables II

Text: Textbook is not required. Lecture notes will be distributed. Course material will be drawn from the following texts: [1] An introduction to complex analysis in several complex variables (Third edition), by L. H\"{o}rmander, Elsevier, 1991. [2] Partial differential equations in several complex variables, by So-Chin Chen and Mei-Chi Shaw, AMS/IP, 2001. [3] Complex analytic and algebraic geometry, by J.-P. Demailly, available online at the author’s website.

Description: The main focus of this topics course on geometric analysis in several complex variables is $L^2$-theory of the $\bar\partial$-operator and spectral analysis of complex Laplacians, and their applications to complex algebraic geometry. Topics include: $L^2$-theory of the $\bar\partial$-operator; subelliptic estimates and regularity theory in the $\bar\partial$-Neumann problem; spectral theory of complex Laplacians; holomorphic Morse inequality; invariance of plurigenera; and relevant recent developments.

### Selected Topics in Geometry

Subtitle: Topics in Metric Riemannian Geometry

Text: The main reference book is by Jeff Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds (Publications of the Scuola Normale Superiore, 2001).

Prerequisites:

Description: The main subject of this course is the Degeneration of Riemannian metrics with Ricci curvature bounded from below, and the main reference book is by Jeff Cheeger, Degeneration of Riemannian metrics under Ricci curvature bounds (Publications of the Scuola Normale Superiore, 2001).

The principal aim of the course is to present the structure theory developed by Cheeger-Colding, for metric spaces which are Gromov-Hausdorff limits of sequences of Riemannian manifolds which satisfy a uniform lower bound of Ricci curvature. The emphasis in the lectures was on the non-collapsing situation.

Note that this course may not be a normal graduate course which is usually lectured by an instructor from the beginning to the end. Instead, this course may looks like a reading/working seminar course: I will begin with lectures to introduce the background and basic results on this subject. Most participants will be assigned with a topic to present during the second half of the course.

The participants should have a basic knowledge on Riemannian geometry, basic topology and PDE.

### Introduction to Algebraic Topology I

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

1. A. Hatcher: algebraic topology, excellent collection of exercises. \$30 in paperback from Cambridge University Press, as well as online here: http://www.math.cornell.edu/~hatcher/AT/ATpage.html.

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

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

Prerequisites: Point Set Topology and Basic Algebra

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, Brouwer's fixed-point theorem, and the Jordan-Brouwer separation theorem.

### Topics in Algebraic Topology

Subtitle: Applied and Computational Topology

Text:

Prerequisites: Math 451 or Equivalent

Description: This course is an introduction to algebraic topology with an emphasis on applications and computation of homology groups, persistent homology, and induced maps on homology. Specific topics that will be covered include:

- Cell Complexes, Chain Complexes and Homology Groups

- Approximations of Continuous Maps and Induced maps on Homology

- Exact sequences

- Discrete Morse Theory

- Filtrations and Persistent Homology

- Techniques of Topological Data Analysis

- Topological Analysis of Nonlinear Dynamics

These techniques will be applied to problems associated with protein structures, fluid flow, dense granular media, materials science, and image processing. The IMA in Minneapolis is running a special emphasis year entitled “Scientific and Engineering Applications of Algebraic Topology." As much as possible the course will be structured to take advantage of the online offerings which include the workshops:

10/2-4/13-Tutorial: Introduction to Statistics and Probability for Topologists

10/7-11/13-Workshop: Topological Data Analysis

10/28-11/1/13-Workshop: Modern Applications of Homology and Cohomology

12/9-13/13-Workshop: Topological Structures in Computational Biology

### Abstract Algebra I

Text: Main text, N. Jacobson, Basic Algebra I, II (2nd edition, 1985) These books are available in paperback from Dover Publications (2009). (ISBN: 0486471896 and 048647187X)

Prerequisites: Standard course in Abstract Algebra for undergraduate students at the level of our Math 451.

Description: This is a standard course for beginning graduate students. It covers Group Theory, basic Ring & Module theory, and bilinear forms. Group Theory: Basic concepts, isomorphism theorems, normal subgroups, Sylow theorems, direct products and free products of groups. Groups acting on sets: orbits, cosets, stabilizers. Alternating/Symmetric groups. Basic Ring Theory: Fields, Principal Ideal Domains (PIDs), matrix rings, division algebras, field of fractions. Modules over a PID: Fundamental Theorem for abelian groups, application to linear algebra: rational and Jordan canonical form. Bilinear Forms: Alternating and symmetric forms, determinants. Spectral theorem for normal matrices, classification over R and C. (Class supplement provided) Modules: Artinian and Noetherian modules. Krull-Schmidt Theorem for modules of finite length. Simple modules and Schur's Lemma, semisimple modules. (from Basic Algebra II) Finite-dimensional algebras: Simple and semisimple algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.

NOTE: There is a problem session scheduled on Thursdays, fourth period, for this course in Hill Center, Room 525, Busch Campus.

### Selected Topics in Algebra

Subtitle: Complex analysis method in conformal field theory

Text: There is no single text for this course. The material will be from various research monographs and papers.

Prerequisites: First year graduate courses in algebra and analysis.

Description: Many basic results and techniques in complex analysis play fundamental roles in the representation theory of infinite-dimensional Lie algebras and vertex operator algebras and in its applications in mathematics and physics. The complex analysis method should be mastered by every student interested in this representation theory and its applications. I will discuss the following topics in this course:

1. The complex analytic and geometric formulations of vertex operator algebras, modules, intertwining operators, chiral and full conformal field theories.

2. Some results in complex analysis that play important roles in the representation theory of infinite-dimensional Lie algebras and vertex operator algebras.

3. Some theorems in the representation theory whose formulations and proofs need the formulations presented in Topic 1 and /or those results discussed in Topic 2 above. This part of the material will be chosen based on the interests of the students.Besides some basic results, possible topics to be discussed include wave functions for quantum Hall systems and intertwining operators, central charges and the determinant line bundle, open-string vertex operator algebras and D-branes, modular functors and full field algebras, open-closed conformal field theories.

### 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.

### Selected Topics in Logic

Subtitle: Introduction to model theory and set theory

Text: Chang and Kielser, Model Theory Marker, Model Theory

Prerequisites: Some knowledge of basic notions of logic will be helpful. We will assume familiarity with ordinals and cardinals.

Description: We will cover basic topics from model theory and set theory. From model theory, we will cover topics such as elementary equivalence, elementary chains, categoricity and ultrapower constructions. From set theory, we will cover basic notions such as measurable cardinals and their influence on simply definable sets of reals. If time permits we will also cover quantifier elimination and prove Hilbert's Nullstellensatz. The course is aimed for beginning graduate students who are interested in philosophy, mathematical logic and algebra.

### Special Topics in Number Theory

Subtitle: Geometry of Numbers, Diophantine Approximations and Transcendence

Text: There are nice basic text books on these topics by Cassels and Baker. However I shall distribute my own lecture notes which should be self sufficient. Lectures will be on Tuesdays and Fridays, 12:00 – 1:20.

Prerequisites:

Description: This course is for graduate students of any level. It comprises of three different areas in number theory, which have similar flavor. The geometry of numbers is mostly elementary, the Diophantine approximations are partially elementary but some topics are quite intricate. The theory of transcendental numbers is the hardest and it requires a good skill in complex analysis. The following topics and results will be addressed (among many other things);

-Minkowski theorem

-Continued fractions

-Markov spectrum

-Roth theorem

-Transcendence of special numbers

-Algebraic independence

-Baker’s theorem for logarithms

-Application to the class number one problem

### Topics in Number Theory

Subtitle: Elliptic Curves and Number Theory

Text: J. Silverman, Arithmetic of Elliptic Curves, ISBN 978-0387094939

Description: This course will study elliptic curves and applications to number theory. For more details see the course webpage.

### Methods of Applied Mathematics I

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

Prerequisites: Topics the student should know, together with the courses in which they are taught at Rutgers, are as follows: Introductory Linear Algebra (640:250); Multivariable Calculus (640:251); Elementary Differential Equations (640:244 or 640:252). The course Advanced Calculus for Engineering (640:421), which covers Laplace transforms, trigonometric series, and introductory partial differential equations, is a valuable preparation for Math 527, but is not required. Students uncertain of their preparation for this course should consider taking 640:421, or consult with the instructor.

Description: This is a first-semester graduate course, intended primarily for students in mechanical and aerospace engineering, biomedical engineering, and other engineering programs. Topics include power series and the method of Frobenius for solving differential equations; Laplace transforms; nonlinear differential equations and phase plane methods; vector spaces of functions and orthonormal bases; Fourier series and Sturm-Liouville theory; Fourier transforms; and separation of variables and other elementary solution methods for linear differential equations of physics, including the heat, wave, and Laplace equations. More information can be found on the course webpage.

### 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.

### Introduction to Mathematical Physics I

Subtitle: The Mathematics of Quantum Mechanics

Text: Notes of the instructor.

Description: The goal of the course is that students obtain a good understanding of quantum mechanics and the mathematical concepts and techniques relevant to it. Topics will include most of the following:

* The Schroedinger equation

* L^2 space and Hilbert space in general

* operators in Hilbert space (unitary, projection, self-adjoint operators, bounded versus unbounded operators, Fourier transformation, unitary 1-parameter groups, the spectral theorem)

* The predictive formalism of quantum mechanics (observables, projection-valued measures, positive-operator-valued measures, the collapse rule)

* tensor product spaces; trace of operators and the use of density operators

* permutation symmetries, bosons and fermions, topological phase factors

* spin and representations of the rotation group SO(3)

* relativistic versions of the Schroedinger equation

* Bell's inequality

### Numerical Analysis I

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

### Combinatorics I

Text: van Lint-Wilson is nice but optional. There is really no text; various 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. See me if in doubt.

Description: This is the first part of a two-semester course surveying basic topics in combinatorics. Topics for the full year should include most of: • Enumeration (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics) • Matching theory, polyhedral and fractional issues • Partially ordered sets and lattices, 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: Finite Fields

Text: None

Prerequisites: None

Description: This course will cover some important classical and modern themes in the study of finite fields. These will include:

Solutions of equations

Pseudorandomness

Exponential sums and Fourier techniques

Algebraic curves over finite fields, the Weil theorems

Additive combinatorics and the sum-product phenomenon

Many applications to combinatorics, theoretical computer science and number theory

### Topics in Probability and Ergodic Theory I

Subtitle: Introduction to Probability using Measure Theory

Text: The lectures will follow class notes that will be provided electronically.

Prerequisites: Real Analysis (640:501 or an equivalent) is an essential prerequisite. Students should also have had an undergraduate course, at the level of Ross's text, A First Course in Probability, so that they have a basic familiarity with elementary, combinatorial probability and with the binomial, Poisson, exponential, and normal distributions.

Description: This course will be an introduction to the issues and techniques of probability theory, at the graduate level. The topics covered will include: (i) The measure theoretic framework of modern probability theory; probability spaces and random variables; (ii) Independence and zero-one laws; (iii) Laws of large numbers and Kolmogorov's three series theorem; (iv) Convergence in distribution and the Central Limit Theorem; (v) Conditional Expectation; (vi) An introduction to martingales in discrete-time and applications to Markov chains. Time permitting, we will try to give brief introductions, where appropriate, to coupling techniques, large deviations, and Stein's method.

### Selected Topics in Applied Mathematics

Subtitle: Variational Inequalities, Obstacle Proglems, and Free Boundary Problems

Text: Please see the PDF course outlines for information regarding textbooks - Syllabus (PDF)

Prerequisites: An undergraduate (or higher-level) course on real analysis covering elementary measure theory (Lebesgue integral), and the concepts of Hilbert spaces and Banach spaces will be useful.

CO-REQUISITES: A one-semester undergraduate course on partial differential equations (for example, based on the text by Walter Strauss) or a graduate level course on partial differential equations (for example, based on the text by Lawrence Evans) would be useful, but will not necessarily be assumed.

Description: The goal of the course is to introduce graduate students to methods for solving variational inequalities, obstacle problems, and free boundary problems and their applications. Applications will include examples arising in mathematical finance, such as the American-style option pricing problem, and examples drawn from other areas of pure and applied mathematics based on the interests and background of students. Our emphasis will be on the universal applicability of the methods introduced in the course to a wide variety of examples in all areas of pure and applied mathematics.

Topics selected will depend on the interests and backgrounds of the audience, but may include:

(1) obstacle problems, variational inequalities, and free boundary problems;

(2) application to the American-style option pricing problem;

(3) existence, uniqueness, and regularity of solutions to variational inequalities and obstacle problems;

(4) optimal regularity of solutions to obstacle problems near the free boundary;

(5) numerical solution of obstacle problems;

(6) introduction to viscosity solutions;

(7) background material on elliptic and parabolic PDEs.

Students will be polled near the start of the class regarding their interests and preferences. Please see the PDF course outline and contact the instructor for additional information.

### 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.

### Portfolio Theory and Applications

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

### High Frequency 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.

### Seminar in Mathematical Finance

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

### Topics in Mathematical Physics

Subtitle: Nonequilibrium Statistical Mechanics

Text: All the topics related to the course can be found in the Course Resource Page linked below in the Description section. I shall draw the subjects of the lectures from the books and papers. The course will be a path through them and they will provide further reading and complements.

Prerequisites:
a) Good understanding of the basics of Classical Mechanics and Probability theory

b) Familiarity with ODE's

Description:
1) Equilibrium and stationary states
2) chaos theory
3) Symbolic dynamics
4) Nonequilibrium thermodynamics: friction and time reversal
5) Onsager-Machlup theory
6) Fluctuations
7) Exactly computable examples

Please see the Course Resource Page

This page was last updated on August 01, 2017 at 11:00 am and is maintained by grad-director@math.rutgers.edu.