Mathematics Department - Graduate Course Descriptions - Fall 2010

### 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.

### 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 I

Text: Functional Analysis by Peter Lax (Wiley-Interscience, ISBN: 0471556041)

Prerequisites: 640:501 & 640:503, or permission of instructor

Description: This is a one-semester course on the rudiments of functional analysis. We will cover about half of Peter Lax's classic text. Topics include: Linear spaces and linear maps. The Hahn-Banach theorem and its applications. Normed linear spaces. Hilbert space theory. Duality. Weak convergence. Weak and weak* topologies. Bounded linear maps. Compact operators.

### Selected Topics in Analysis

Subtitle: Contact Homology via Legendrian curves: An Overview

Text:

Prerequisites:

Description: This course is a research course intended to describe all the tools required for the computation of he Contact Homology via Legendrian curves.

The examples that will be investigated are the case of the standard contact structure of $S^3$ and the case of the first exotic contact structure on $S^3$ of J.Gonzalo and F.Varela.

The outline of the course is as follows:

1.Definition of Contact forms and structures, Reeb vector-fields.

2.The datum of a vector-field $v$ in the kernel of a contact form $\alpha$. Dynamics, remarkable quantities.

3.The spaces $L_\beta$, $C_\beta$, with $\beta=d\alpha(v,.)$

1. a.Manifold structure, Topology.
2. b.The examples of $\alpha=\alpha_0$, the standard contact structure on $S^3$ and the example of $\alpha=\alpha_1$, the first exotic contact structure of J.Gonzalo and F.Varela (vector-field $v$ of Vittorio Martino).

4.The variational problem defined by the action functional $J(x)=\int_0^1\alpha_x(\dot x)dt$ on $C_\beta$. Critical points, critical points at infinity, flow-lines originating at periodic orbits.

5.Compactness of flow-lines originating and ending at periodic orbits under some algebraic restrictions.

6.Fredholm issues.

7.Arrows and Computations.

Warning: This course falls short of a full computation of the homology in the case of the first exotic contact structure of J.Gonzalo And F.Varela because the Fredholm issues are not entirely settled: the verification of the tools designed to overcome these issues is an ongoing process.

### Partial Differential Equations I

Text: The course material will be mostly drawn from "Partial Differential Equations" by Lawrence C. Evans, Second Edition published by American Mathematical Society, 2010. *Graduate Studies in Mathematics* Volume: 19 (hardcover) ISBN-10: 0-8218-4974-3 ISBN-13: 978-0-8218-4974-3

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 familiarity 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: The Yamabe problem and a fully nonlinear version of it

Text: T. Aubin, Some nonlinear problems in Riemannian geometry, SpringerMonographs in Mathematics, Springer-Verlag, Berlin, 1998. R. Schoen and S.T. Yau, Lectures on differential geometry, International Press, Cambridge, MA, 1994.

Prerequisites: 640-517 or permission by the instructor

Description:

In this course I plan to present material on the Yamabe problem and a fully nonlinear version of it. I will present one third of the material below with detailed proofs, another one third will be presented with outline of the proofs, while for the remaining one third I will describe the results and introduce some open problems. I will discuss with the registered students to give a final decision on what material goes to which parts.

My initial plan is to give a detailed proof of the Yamabe problem and its solution through the work of Yamabe (1960), Trudinger (1968), Aubin (1976), and Schoen (1984). The positive mass theorem of Schoen and Yau was used in its solution. The proofs of the theorem will be either outlined or provided in details. For most of these material, we will use of the above text books.

I will then present a fully nonlinear version of the Yamabe problem. We will present some Liouville theorems with detailed proofs. In doing so, we will use the method of moving planes, the Bernstein type arguments, the theory of viscosity solutions and in particular Jensen approximations, the Alexsandrove-Bakelman-Pucci inequality. I also plan to present related results in a series of recent joint work with Luis Caffarelli and Louis Nirenberg concerning strong maximum principles and removable singularity of viscosity solutions (part of the material available aspreprints or papers on my webaite, while some more will be available as a preprint at the begining of September). I will also present some existence and compactness of solutions of the fully nonlinear version of the Yamabe problem (part of them with detailed proofs, part of them with outlines of the proofs, and the remaining part without proofs), together with some open problems.

### Functions of Several Complex Variables I

Text: The course materials will be largely taken from the following:

[1] L. Hormander, {\it An introduction to complex analysis in several variables}, Third edition, North-Holland, 1990.

[2] James Morrow and K. Kodaira, {\it Complex Manifolds}, Rinehart and Winston, 1971.

[3] Xiaojun Huang, Lectures on the Local Equivalence Problems for Real Submanifolds in Complex Manifolds, Lecture Notes in Mathematics 1848 (C.I.M.E. Subseries), Springer-Verlag, 2004.

[4] Subelliptic analysis on Cauchy-Riemann manifolds, Lecture Notes on the national summer graduate school of China, 2007. (to appear)

Prerequisites: One complex variable and the basic Hilbert space theory from real analysis

Description:

A function with $n$ complex variables $z\in {\bf C}^n$ is said to be holomorphic if it can be locally expanded as power series in $z$. An even dimensional smooth manifold is called a complex manifold if the transition functions can be chosen as holomorphic functions. Roughly speaking, a Cauchy-Riemann manifold (or simply, a CR manifold) is a manifold that can be realized as the boundary of a certain complex manifold. Several Complex Variables is the subject to study the properties and structures of holomorphic functions, complex manifolds and CR manifolds. Different from one complex variable, if $n>1$ one can never find a holomorphic function over the punctured ball that blows up at its center. This is the striking phenomenon that Hartogs discovered about 100 years ago, which opened up the first page of the subject. Then Poincar\'e, E. Cartan, Oka, etc, further explored this field and laid down its foundation. Nowadays as the subject is intensively interacting with other fields, providing important examples, methods and problems, the basic materials in Several Complex Variables have become mandatory for many investigations in pure mathematics. This class tries to serve such a purpose, by presenting the following fundamental topics from Several Complex Variables.

(a)Holomorphic functions, plurisubharmonic functions, pseudoconvex domains and the Cauchy-Riemann structure on the boundary of complex manifolds

(b) H\"ormander's $L2$-estimates for the $\bar \partial$-equation and the Levi problem \noindent

(c) Cauchy-Riemann geometry, Webster's pseudo-Hermitian Geometry and subelliptic analysis on CR manifolds

(d) Complex manifolds, holomorphic vector bundles, Kahler Geomtry.

### Algebraic Geometry I

Text: No textbook.

Prerequisites: 640:503 and 640: 551 or permission of instructor

Description: This course will be an introduction to the study of algebraic varieties, that is the zero sets of polynomials in several variables. Just as linear algebra has geometric content of lines, planes and hyperplanes as well as the algebraic structure of vector spaces, subspaces and linear maps the subject of algebraic geometry has simultaneously the geometric flavor of surfaces, hypersurfaces, etc. and the algebraic structure of commutative algebra of rings of polynomial functions. The subject is the study of the interplay of these two points of view.

The emphasis of the course will be on examples of algebraic varieties and general attributes of varieties and morphisms as reflected in these examples. Examples of algebraic varieties arise in many places in physics, topology, geometry, combinatorics and number theory and the examples studied in this course will be often be drawn from other areas of mathematics. Topics include but are not limited to projective spaces, elliptic curves, line bundles, blowups. Brief introduction to the language of schemes will be given at the end of the semester if time permits.

### 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. ### Abstract Algebra I Text: Main N. Jacobson, Basic Algebra I, II (2nd edition, 1985) These books are available in paperback for under$20 from Dover Publications (2009). (ISBN: 0486471896 and 048647187X)
There are supplementary handouts for: bilinear forms over fields, simple/semisimple algebras, and group representations.

Prerequisites:

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. (Class supplement provided)

### Theory of Groups

Text: Pierre de la Harpe, Topics in Geometric Group Theory, Chicago, 2000.

Prerequisites: None, except for the most basic notions of group theory.

Description:
Geometric Group Theory

This course will be an introduction to Geometric Group Theory. There are no prerequisites, except for the most basic notions of group theory. In Geometric Group Theory, finitely generated groups are viewed as metric spaces via the path metrics on their Cayley graphs and their large-scale geometry is studied. The topics covered in this course will include:

• Growth rates of finitely generated groups, including the construction of groups of intermediate growth.
• The basic theory of amenable groups.
• Quasi-isometries and asymptotic cones of finitely generated groups

Text: Pierre de la Harpe, Topics in Geometric Group Theory, Chicago, 2000.

### Selected Topics in Algebra

Subtitle: Vertex operator algebras and the theory of partitions

Text: Main: G. Andrews, The Theory of Partitions, Cambridge University Press, 1984 (paperback).

Also, some selected material from the following will be developed and used:

I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988.

J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Birkhauser, 2004.

I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Memoirs Amer. Math. Soc., Vol. 104, Amer. Math. Soc., 1993.

Additional material, including certain research papers, will be handed out.

Prerequisites: No prerequisites for the theory of partitions. Prerequisites for relations with vertex operator theory: Basic algebra and some familiarity with Lie algebras. We will develop the relevant aspects of vertex operator algebra theory as they arise in the course. Students who don't have any experience with vertex operator algebras and are potentially interested in this course are encouraged to consult me.

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-developing theory of such identities turns out to be deeply related to the representation theory of vertex operator algebras, including currently-unfolding research in vertex operator algebra theory. In this course we will develop the theory of, and prove, natural families of partition identities, using the Rogers-Ramanujan and Rogers-Selberg recursions. We will show how this classical theory suggests, and reflects, ideas and structures in vertex operator algebra theory, structures that we will in fact use to conceptually derive such recursions and study such identities. The aspects of vertex operator algebra theory that we will develop are certain vertex-algebraic structures associated with modules for affine Lie algebras, including intertwining operators among such modules. These vertex-algebraic structures are also deeply related to other branches of mathematics, including finite group theory, the theory of modular functions, and tensor category theory, and to conformal field theory and string theory in physics. We will emphasize a range of potential research problems involving the structures developed in the course.

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.

### Representation Theory

Subtitle: Hecke Algebras

Text:

Prerequisites: Good understanding of Algebra

Description: This course will be a self contained - introduction to Hecke algebras (finite, affine, double, triple) and their degenerations (graded Hecke algebras, Cherednik algebras, etc.). We will also discuss the connection with symmetric functions and Macdonald polynomials.

### 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 is the engine behind algebraic geometry and algebraic number theory. In addition, problems from other fields such as combinatorics or optimization can sometimes be phrased as commutative algebra problems.

This course will be an introduction to the basics of commutative algebra, including localization, primary decomposition, integrality, flatness, and dimension.

We will roughly follows Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. Computational aspects and examples relevant to algebraic geometry will be emphasized, but the only prerequisite is 551/552 or equivalent.

### Special Topics in Number Theory

Subtitle: Analytic Methods in Diophantine Problems

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:

Description: This course is for graduate students interested in number theory in a broad sense. It will be accessible for beginners and for advanced students. The main objective is to show how tools from harmonic analysis can be used to solve problems of Diophantine type. In particular the circle methods will be developed to treat solutions to equations in integers. Classical circle methods (due to Ramanujan, Hardy, Littlewood, Kloosterman) as well as its modern variations will be presented. Important components of the methods are estimates for exponential sums of Weyl’s type, so a great attention will be given to the theory of exponential sums. Moreover some related questions concerning character sums will be discussed.

If time permits I am also planning to cover subjects of the equidistribution of integral points on curves (the circle), on surfaces (the sphere) and lattice points in some Euclidean domains.

### 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.

### Introduction to Mathematical Physics I

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.)

### 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:

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 techniques (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics)
• Theory of finite sets, hypergraphs, combinatorial discrepancy, Ramsey theory.
• Correlation inequalities, Martingales, Probabilistic methods.
• Algebraic methods (applications of linear algebra and higher algebra in combinatorics).
• Fourier analytic techniques.

I also plan to have a parallel reading course, in which students can pursuit a topic of interest on a deeper level

### Topics in Probability and Ergodic Theory I

Text: Richard Durett: Probability: Theory and Examples, Wadsworth & Brooks/Cole Statistics/Probability Series 1991 ISBN 0-534-13206-5

Prerequisites: A graduate mathematics course in real analysis or permission of the instructor.

Description: This course will be an introduction to probability theory at the graduate level. Measure theory will be used throughout. It will introduce the main themes of the theory, both classical and contemporary:

• Foundations-- basic theory of probability spaces, random variables, xpectations;
• Large number laws and the Ergodic Theorem;
• Central Limit Theorem, Infinitely Divisible Distributionsand Stable Distributions;
• Large Deviations;
• Coupling;
• Conditional Expectation;
• Discrete Time Martingale theory.

Probability models from different applied and pure areas will be discussed as examples.

### Mathematical Foundations for Industrial and Systems Engineering

Text: Bartle and Sherbert, Introduction to Real Analysis, 3rd Edition, Wiley & sons, 1992.

Prerequisites:

Description: This course is offered specifically for graduate students in Industrial Engineering.

Proof Structure for the Development of Concepts Based on the Real Numbers

1. Axioms for the Real Numbers
2. Logical Principles
The Continuity Axiom
1. The supremum concept and useful implications
2. Convergence of sequences and series
Development of the Calculus of Functions of One Variable
1. Continuous functions and basic properties
2. Differentiable functions and basic properties (the Mean value Theorem and Taylor's Theorem)
3. The Riemann Integral and its basic properties
4. The Fundamental Theorem of Calculus and implications
5. Uniform convergence of sequences of functions

### Selected Topics in Applied Mathematics

Subtitle: High-Frequency Finance and Stochastic Control

Text:

Prerequisites: ECE 503, Math 621 Corequisites: Stat 563, Stat 565, or equivalent

Description: Part 1: An introduction to stochastic control and estimation with emphasis on applications to problems from mathematical finance. We will study the theory in both discrete and continuous time, and cover both fully observed and partially observed stochastic control. We will treat Kalman filtering and linear quadratic control, the classical PDE approach to dynamic programming and the Hamilton-Jacobi-Bellman equation, with applications to Merton portfolio allocation, investment-consumption problems, and the martingale approach with applications to portfolio optimization.

Part 2: The difference between theory and practice in quantitative finance is significant. Elements that seem superfluous in the classroom prove challenging in real life -- and the devil is always in the details. This is especially true in high frequency algorithmic trading, where there is a critical interdependence between the alpha models, strategies, and software implementation. Taught by a practitioner who has built two high frequency trading groups, this course roughly follows the stages of such an endeavor, identifying pitfalls and focusing on real calculations. Major topics include the treatment of real data, computation of returns, robust alpha modeling, aspects of strategy design, infrastructure, simulation, and performance analysis. The objectives of the course are twofold: (1) to convey the scope and components of a high frequency algorithmic trading effort, and (2) to provide the student with the tools to rigorously test and implement their own trading ideas. Participants must have a strong programming background and will be required to complete a number of projects.

| and Co-requisites ECE 503 or equivalent strong C++ programming skills (essential), Stat 563 (regression), Stat 565 (time series - co-requisite), and Math 621-622. Please visit the prerequisites page for descriptions of Rutgers undergraduate course prerequisites. A solid understanding of undergraduate probability at the level of the textbook by Sheldon Ross, A First Course in Probability, is especially important. Given this background, the course should be accessible to Mathematical Finance master's degree students and graduate students in Computer Science, Economics, Finance, Engineering, Mathematics, Physics, Operations Research, and Statistics. | "Empirical Market Microstructure" by Hasbrouck (recommended) and "Optimal Trading Strategies" by Kissell and Glantz (optional)

### Selected Topics in Applied Mathematics

Text:

Prerequisites: Math 623

Description:

### 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.

### Topics in Mathematical Physics

Subtitle: Statistical Mechanics of Cooperative Phenomena

Text:

Prerequisites:

Description:

The course will be based on the study of simple microscopic models which exhibit complex collective behavior. Of particular interest are emergent phenomena like phase transitions and pattern formation. Some of those phenomena already occur in equilibrium systems, where they can be described via properties of Gibbs measures (ensembles) which do not involve explicitely the dynamics. Examples of equilibrium cooperative phenomena are the boiling and freezing of liquids, like water.

The most intersting cases, on which we will focus, are systems far from equilibrium. These include biological, ecological and social systems. Microscopic models exhibiting behavior resembling that found in the natural world cannot be described without the dynamics, which are generally of a stochastic nature. Examples of such microscopic dynamics are: the voter model, the contact process, and reaction-diffusion models. The connection between the microscopic models and the macroscopic descriptions via deterministic equations, such as the reaction-diffusion equation, will be elucidated.

Students interested in taking this course may contact me at:\\ lebowitz@math.rutgers.edu

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