Mathematics Department - Graduate Course Descriptions - Fall 2012

### Theory of Functions of a Real Variable I

Text: Measure and Integral, by R. Wheeden and A. Zygmund, Taylor and Francis publisher.
The book is not required, but I will follow it closely. A supplementary text is Real Analysis by E. Stein and R. Shakarchi in Princeton Lecture Notes in Analysis III.

Prerequisites: Advanced Calculus and Elementary Topology of Euclidean Space.

Description: Basic real variable function theory needed for pure and applied analysis. Topics: bounded variation; Riemann-Stieltjes integration; Lebesgue outermeasure and Lebesgue measure of sets in Euclidean space; nonmeasurable sets; measurable functions; Lusin and Egorov theorems; convergence in measure; Lebesgue integration; convergence theorems for integrals; relations between Lebesgue integrals, Riemann integrals and Riemann-Stieltjes integrals;; product measures and Fubini, Tonelli theorems; Vitali covering lemmas; Lebesgue differentiation theorem; Hardy-Littlewood maximal function; differentiation of functions of bounded variation, 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: 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.

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.

### Selected Topics in Analysis

Subtitle: Topological Methods in Nonlinear Analysis

Text: Recommended: Chapters 2, 4, 7 of Gilbarg-Trudinger (not the entire book) are recommended for this course. They can be photo-copied from an original copy (second edition). I will make them available to the students that do not have them. Brezis - Functional Analysis, Springer 2011, M.A.Krasnosel'skii - Topological Methods in the Theory of Nonlinear Integral Equations (International Series of Monographs in Pure and Applied Mathematics Vol. 45)

Prerequisites: Knowledge of Lebesgue integration theory and of earlier courses in Real and Functional Analysis is a prerequisite for this class.

Description: PLEASE NOTE: THIS COURSE HAS REPLACED COURSE 16:640:507. IF YOU REGISTERED FOR 16:640:507, YOU HAVE BEEN MOVED OVER TO 16:640:509.
This course should prepare students to set up the framework and learn some of the early and basic techniques in order to solve some nonlinear PDEs via topological methods.
1- The first part of the course covers Lp-spaces (convolution, dual spaces), Wm,p_ spaces, Rellich-Kondrachov embeddings. It also covers the resolution of the Dirichlet problem Δu = f ; u = 0|∂Ω via Green's function representation techniques (less general, but useful). References for this first part of the course are Gilbarg-Trudinger Chapters 2,4,7 (I have the second edition, Springer 1983).
2- The second part of the course covers Fredholm operators of index ν and 0 (self-adjoint operators) and Diagonalization of compact self-adjoint operators in separable Hilbert spaces.
There are a number of references on this very classical subject in the literature, including Brezis, Functional Analysis, Springer 2011, Riesz-Nagy, Analyse Fonctionnelle (Academie des Sciences de Hongrie) and others...
3-The third part of the course is mainly based on the old and very classical book of M.A.Krasnosels'kii's Topological Methods in the Theory of Nonlinear Integral Equations (International Series of Monographs in Pure and Applied Mathematics Vol 45; I have the copy of the library...). There are again a number of other references in the literature, including notes by L.Nirenberg at Courant, J.T Schwartz (also Courant Lecture Notes) and excellent papers by A.Ambrosetti and P.H Rabinowitz... This third part should cover:
• Caratheodory conditions and continuous maps from Lp(Ω­) to Lq(Ω­).
• Finite dimensional degree theory (here, we will use a differential geometry approach. One good reference, after assuming transversality and the Morse-Sard theorem, is provided in M.Hirsch's book Differential Topology, Graduate text in Maths 33, Springer 1997)
• Leray-Schauder degree
• Weakly continuous functionals on the unit sphere or the unit ball of a separable Hilbert space (Z2 and S1-Lusternik-Schnirelman theory)
I hope to be able to cover all the topics of this course. Whatever will be covered will be covered in detail. Knowledge of Lebesgue integration theory and of earlier courses in Real and Functional Analysis is a prerequisite for this class.
Required and recommended sources:
Chapters 2, 4, 7 of Gilbarg-Trudinger (not the entire book) are recommended for this course. They can be photo-copied from an original copy (second edition). I will make them available to the students that do not have them.
Brezis' book, cited above, is recommended, so is M.A.Krasnoselsk'ii's book cited above.

### Partial Differential Equations I

Text: Partial Differential Equations: Second Edition (Graduate Studies in Mathematics), By Lawrence C. Evans, American Mathematical Society; 2nd edition (March 3, 2010), 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 variable course (640:501).

Description: This is the first half of a year-long introductory graduate course on PDE. 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. 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. Next we will discuss first order nonlinear PDEs (e.g. characteristics, Hamilton-Jacobi equations), some ways to represent solutions (e.g. separation of variables, similarity solutions, Fourier and Laplace transforms, Hodograph and Legendre transform, singular perturbation and homogenizations, Cauchy-Kovalevskaya theorem). Then we discuss Soblev spaces and Soblev embedding theorems, second order elliptic equations of divergence form (existence, uniquesness, regularity), second order parabolic equations (existence, maximum principles, regularity).

### Harmonic Analysis on Euclidean Spaces

Text: None required

Prerequisites: Math 501, 502 and 503.

Description: The principal topics to be covered in this course are: 1. Interpolation Theorems, Stein's theorem, Marcinkiewicz interpolation theorems. 2. Hardy-Littlewood-Sobolev fractional Integration theorems. 3. Singular integrals, Calderon-Zygmund Theory. 4. Cotlar-Stein Lemma, Singular integrals of Non-convolution type. 5. Bounded Mean Oscillation, John-Niremberg theorem. 6. Littlewood-Paley theory 7. Fourier Transform restriction, Bochner-Riesz theory. 8. Strichartz estimates for wave and Schrodinger eqns. 9. Time permitting T1 theorems.

### 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.

### Introduction to Differential Geometry

Text: Lee, Introduction to Smooth Manifolds

Prerequisites: Point set topology

Description: Possible topics include differential forms on manifolds, vector bundles & curvature, Riemannian metrics & geodesics, and symplectic forms & moment maps. People interested in this course should e-mail me at ctw@math.rutgers.edu and let me know their background and interests.

### Algebraic Geometry I

Text: No textbook.

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

Description: Algebraic geometry is a study of solutions of polynomial equations. The course will be an overview of basic examples and techniques. The topics covered are likely to include: coordinate rings of affine and projective varieties, Bezout theorem, introduction to elliptic curves, blowups, introduction to sheaves and schemes, Cech cohomology, line bundles, Hurwitz formula, Riemann-Roch formula. If time permits, other topics may be covered, such as Chern classes and Grothendieck-Riemann-Roch formula, ADE singularities, toric varieties, Grassmannians, birational geometry. I will focus on the geometric intuition behind the algebraic constructions and will try to include as many examples of algebraic varieties as possible.

Text: Introduction to Algebraic Topology I, by Allen Hatcher. Publisher: Cambridge University Press; 1st edition (December 3, 2001) • Language: English • ISBN-10: 9780521795401 • ISBN-13: 978-0521795401 • ASIN: 0521795400 This book is available for $32 in paperback from Cambridge University Press, as well as (for free) online. Prerequisites: None Description: This course will be an introduction to the fundamental group and homology theory. The plan is to cover chapters 1,2 and parts of chapter 4 in Hatcher's book. Topics include: fundamental group, Van Kampen's Theorem, covering spaces homotopy groups and the homotopy category simplicial and singular homology, Brouwer's fixed-point theorem the Borsuk-Ulam theorem, and the Jordan-Brouwer separation theorem. ### Abstract Algebra I Text: Main text, 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. the end

Prerequisites: Standard course in Abstract Algebra for undergraduate students at the level of our Math 441.

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)

### Selected Topics in Algebra

Subtitle: Methods in Vertex Operator Algebra Theory

Text: I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, 1988. Supplementary Texts: I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Memoirs AMS, Vol. 104, Number 494, 1993; C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math., Vol. 112, Birkhauser, Boston, 1993; J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2003.

Prerequisites: Some familiarity with the basics of vertex operator algebra theory. Students without such experience and potentially interested in this course are encouraged to consult Professor Lepowsky.

Description: We will construct lattice vertex operator algebras and generalizations, and twisted modules for such structures. This will lead to the construction of the moonshine module vertex operator algebra and of the action of the Monster sporadic finite simple group. Y.-Z. Huang's complementary method for constructing the moonshine module vertex operator algebra, using tensor category structure, will also be discussed. These ideas will be related to relevant research papers and to current research problems.

Please note: The Lie Groups/Quantum Mathematics Seminar, which will meet Fridays at 12:00, 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 Logic

Subtitle: Choice versus Determinacy: An Introduction to Classical Descriptive Set Theory

Text: No Text required

Prerequisites: A knowledge of basic set theory, including cardinals, ordinals and the axiom of choice.

Description: The Axiom of Choice (AC) implies the existence of various pathologies on the real line R such as a non-measurable set of reals, an uncountable set of reals with no perfect subset, a 2-coloring of the 2-subsets of R with no uncountable monochromatic subset, etc. In this course, we will explore the question of whether such pathologies can arise within the "definable sets" of reals; namely, the Borel sets, the analytic sets, ... , etc. While these questions cannot be fully settled using the usual ZFC axioms of set theory, we will find that they all have negative answers if we assume the Axiom of Projective Determinacy (PD): an extra set-theoretic axiom which posits the existence of winning strategies for a broad class of infinite 2-player games. The course will cover the following topics: (1) Basic Descriptive Set Theory We will study the hierarchies of the Borel sets and projective sets, and analyze the structure of the sets in each of these hierarchies. (2) Determinacy We will study infinite 2-player games played on the real line R. We shall see that the existence of winning strategies for suitably defined games implies the non-existence of set-theoretic pathologies within the projective hierarchy. (3) Applications We will study various applications to the theory of Borel equivalence relations and the theory of the Turing degrees of relative computability.

### Special Topics in Number Theory

Subtitle: THE RIEMANN ZETA FUNCTION

Text: Much of the material will be taken from the best research publications. For reading I recommend the classical book by E.C. Titchmarsh, The Theory of the Riemann Zeta Function, Clarendon Press, Oxford 1986.

Prerequisites: There are no special requirements as prerequisites. Students are assumed to have some skill in basic complex variable analysis.

Description: This course is for graduate students who would like to learn what is known up to date in the theory of the Riemann zeta function. Of course the Riemann Hypothesis will be given the most attention; however many other related questions and unconditional results will be presented in considerable details. Selected topics are: -Functional properties - Subconvexity estimates -Zero-free regions -Density of zeros off the critical line -Zeros on the critical line -Gaps between zeros -Pair correlation theory -Random matrix statistics -Heuristics beyond the Riemann Hypothesis Lectures will be on Tuesdays and Fridays, 12noon – 1:20 pm in Hill Center, room 423

### Topics in Number Theory

Subtitle: Number Theory and Modular Forms

Text: Recommended text: A Course in Arithmetic, J.-P. Serre, Springer Graduate Texts in Mathematics.

Prerequisites: Must have taken or be currently enrolled in 501, 503, and 551.

Description: The course will give an overview of problems in number theory that can be solved using modular forms and L-functions. It is intended to introduce beginning graduate students to modern areas of number theory. Topics include the prime number theorem, Dirichlet's theorem on primes in progressions, and the number of ways to represent integers as a sum of squares. Depending on the interests of the students I will cover some other material such as the circle method or an overview of Wiles' proof of Fermat's last theorem. Please check my website for updated and a more precise syllabus.

### 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 is on the www.math.rutgers.edu/courses/527/

### 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: There are many textbooks on statistical mechanics. Each of them has much useful material. I strongly recommend that you look through some of them and find one which just suits you. Some Recommendation are: (1) H.B. Callan, Thermodynamics, John Wiley Sons, New York, 1960. Chapter 1, (2) J.W. Gibbs, Elementary Principles in Statistical Mechanics. Dover Publications, Introduction, (3) B. Simon, The Statistical Mechanics of Lattice Gases, Volume I, Princeton University Press, 1993, p. 3-34, (4) T.C. Dorlas, Statistical Mechanics, Fundamentals and Model Solutions, Institute of Physics Publishing, 1999, p. 44-45, 63-66, (5) C. Garrod, Statistical Mechanics and Thermodynamics. Oxford University Press, 1995, part of Chapter 2, (6) L.E. Reichl, A Modern Course in Statistical Physics, University of Texas Press, Austin, 1980, (7) D. Ruelle, Statistical Mecahanics: Rigorous Results, World Scientific, (8) S. Brush, The Kind of Motion We Call Heat, North Holland, p. 1-14. Please also look at the publication list on my web page.

Prerequisites:

Description: The course will cover traditional areas of statistical mechanics with a mathematical flavor. It will describe exact results where available and heuristic physical arguments where applicable. A rough outline is given below: (I.) Overview: microscopic vs. macroscopic descriptions; microscopic dynamics and thermodynamics (II.) Energy surface; microcanonical ensemble; ideal gases; Boltzmann’s entropy, typicality (III.) Alternate equilibrium ensembles; canonical, grand-canonical, pressure, etc. Partition functions and thermodynamics (IV.) Thermodynamic limit; existence; equivalence of ensembles; Gibbs measures. (V.) Cooperative phenomena: phase diagrams and phase transitions; probabilities, correlations and partition functions. Law of large numbers, fluctuations, large deviations. (VI.) Ising model, exact solutions. Griffith’s, FKG and other inequalities; Peierle’s argument; Lee-Yang theorems. (VII.) High temperature; low temperature expansions; Pirogov-Sinai theory (VIII.) Fugacity and density expansions (IX.) Mean field theory and long range potentials (X.) Approximate theories: integral equations, Percus-Yevick, hypernetted chain. Debye-H¨uckel theory. (XI.) Critical phenomena: universality, renormalization group. (XII.) Percolation and stochastic L¨oewner evolution. If you have any questions about the course please email me: lebowitz@math.rutgers.edu. We can then set up a time to meet.

### Numerical Analysis I

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

### Combinatorics I

Text: (1.) The Probabilistic Method, by Alon and Spencer (2.) Enumerative Combinatorics I and II, by Stanley (3.) Combinatorial Problems and Exercises, by Lovasz (4.) A Course in Combinatorics, by van Lint and Wilson

Prerequisites: Linear algebra, some discrete probability and mathematical maturity.

Description: We will study basic topics in Combinatorics such as enumeration, symmetry, polyhedral combinatorics, partial orders, set systems, Ramsey theory, discrepancy, additive combinatorics and quasirandomness. There will be emphasis on general techniques, including probabilistic methods, linear-algebra methods, analytic methods, topological methods and geometric methods. There will be problem sets every 2-3 weeks. There are no exams.

### Selected Topics in Discrete Mathematics

Subtitle: Probabilistic Methods in Combinatorics

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

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

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

### Topics in Probability and Ergodic Theory II

Text: I will not follow any one text, and I will provide class notes. I will draw upon the following texts mostly: (1) Bernt Oksendahl, Stochastic Differential Equations: An Introduction with Applications, Springer, latest edition (2) I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer, second edition. (3) L.C.G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales, Volumes I and II, Cambridge (4) D. Revuz and M.Yor, Continuous Martingales and Brownian Motion, Springer, third edition. The first two books are available in paperback and are excellent for a first introduction to the subject; the Oksendahl text is more elementary and less rigorous; Karatzas and Shreve fill in many of the theoretical details.

Prerequisites: An introductory course to probability using measure theory and including conditional expectation and martingales in discrete time.

Description: This course will be an introduction to stochastic integration with respect to martingales and applications to the study of Brownian motion and stochastic differential equations. 1) Brownian Motion, Poisson processes and Levy processes; 2) Stopping times and filtrations; 3) Martingales in continuous time, Doob-Meyer decomposition and quadratic variation; 4) Stochastic Integrals and Ito's rule; 5) Applications to Brownian motion; 6) Stochastic Differential Equations and relations to parabolic equations; 7) The martingale problem, strong and weak solutions; 8) Application to diffusions.

### Mathematical Foundations for Industrial and Systems Engineering

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

Prerequisites: None

Description: This course is offered specifically for graduate students in Industrial Engineering. Proof Structure for the Development of Concepts Based on the Real Numbers Axioms for the Real Numbers Logical Principles The Continuity Axiom The supremum concept and useful implications Convergence of sequences and series Development of the Calculus of Functions of One Variable Continuous functions and basic properties Differentiable functions and basic properties (the Mean value Theorem and Taylor's Theorem) The Riemann Integral and its basic properties The Fundamental Theorem of Calculus and implications Uniform convergence of sequences of functions

### 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: Non-equilibrium statistical mechanics

Text: The lectures will be based on published articles and books freely available on the web.

Prerequisites: Thermodynamics, Real analysis and basic linear algebra; ordinary differential equations at a basic level will help.

Description: The course will cover the following topics: Equilibrium ensembles and the ergodic hypothesis Examples of ergodic systems and criticism of the need of the assumption Chaotic and ordered motions: the paradigmatic examples of quasi periodi motions nad of the geodesic motion on negative curvature surfaces The SRB distributions as extensions of the Gibbs' ensemmles Universality of non equilibrium fluctuations Simulations: thermostats and irreversibility Does entropy extend to stationary nonequilibrium? Equivalence of thermostats as an extension of the equivalence between ensembles of equilibrium statistical mechanics

### Topics in Mathematical Physics

Subtitle: Intro. to Dispersive Equations of Mathematical Physics, via Functional Analysis and Spectral Theory

Text: Functional Analysis by Reed & Simon, part I. Self-Adjointness, Reed & Simon II. The rest from notes and papers.

Prerequisites: Real analysis and basic linear algebra; ordinary differential equations at a basic level will help.

Description: We begin with basic notions of Analysis: Hilbert spaces and linear operators. Then compact operators and their applications in PDE and Math-Phys. Then, we go through the Spectral Theorem for general self-adjoint operators, and some immediate applications.Then, we turn to spectral theory: Notions of spectrum, eigenfunctions of Schroedinger type operators, Properties of continuous spectrum, decay estimates and scattering.

### Topics in Mathematical Physics

Subtitle: Mathematics and Mechanics of Materials

Text: Electrodynamics of Continuous Media by Landau, Lifshitz and Pitaevskii

Prerequisites: Advanced Calculus or Real Analysis, and Linear Algebra at the senior or graduate level, which may be replaced by a course on Applied Mathematics at the graduate level from the School of Engineering.

Description: The one-semester topic course introduces how differential equations are derived or motivated from fundamental physical laws and how empirical physical laws can be eventually interpreted by solutions to partial differential equations in mechanics, physics and materials science. Particular emphases are on the field theories for continuum media. Students in mathematics may benefit from the physical motivations of some classical equations while engineering students may benefit from the level of rigor and typical solution methods to differential equations. Precise topics will be adjusted according to the specific interest of students who take this course, but likely include the following: nonlinear elasticity, two-well problem, electrostatics and magnetostatics of continuum media, modeling of multifunctional materials.

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