Mathematics Department - Graduate Course Descriptions - Fall 2014

### 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: Serge Lang, Complex Analysis. Publisher: Springer, Graduate texs in Mathematics Vol 103, 1999. ISBN: 978-1-4419-3135-1 (Print) 978-1-4757-3083-8 (Online)

Prerequisites: Familiarity with real analysis at the level, roughly, of W. Rudin Principles of Mathematical Anaalysis.

Description: The course will present a rigorous introduction to the basic ideas of Complex Analysis, focusing on the study of functions of one complex variable and explaining in detail how this theory is fundamentally different from that of functions of one or several real variables. Topics covered will be: the complex plane, complex differentiation, holomorphic functions, the Cauchy-Riemann equations and the Delta-bar operator, line integrals, Goursat's Theorem, homotopy of loops, simply connected domains, the Cauchy Integral Theorem and the Cauchy Integral Formula, homology of curves in the plane, the winding number, calculus of residues, Taylor and Laurent series, conformal mapping, the open mapping theorem and the maximum principle, the principle of analytic continuation, meromorphic functions, convergence of sequences of holomorphic functions, compact sets of holomorphic functions ("normal families"), harmonic functions of two real variables, and the Riemann Mapping Theorem.

### Selected Topics in Analysis

Subtitle: Sobolev maps with values into the circle

Text: The content of the course is based on a Monograph in preparation with P. Mironescu and I plan to provide written material to the students.

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: 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 modelling of superconductors. 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 on the physical aspects, but rather the intrinsic study of the function space W^{1,p} of maps from a 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 will 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 tools occurring in Image Processing and suggest exciting interactions with this field.

### Selected Topics in Analysis

Subtitle: Yamabe-type problems and Contact form Geometry

Text: None

Prerequisites: A basic knowledge of Sobolev spaces; also some knowledge of Algebraic Topology is better; but since I did learn most of it by myself, I can help students to acquire this knowledge and I can help them sail around most complicated arguments, until a basic core is reached. Then, there is the need to learn (what is left of) the techniques. A basic knowledge of Riemannian Geometry is also required, but here again we can "adjust".

Description: The aim of this course is to introduce its participants to classical (by now) techniques in Conformal Geometry. The current problems in Conformal Geometry include the problem of the Q-curvature. The solution of this problem has used so far various techniques, including a beautiful work by Z.Djadli and A.Malchiodi (Annals of Maths, Vol 168, 2008).
Some resonant cases are left unsettled and are currently being investigated using other variational techniques (barycenter spaces). These techniques are topological and analytical techniques that have been introduced in the framework of Yamabe-type problems. We will therefore revisit the proof of the Yamabe conjecture using two very different techniques:
a. We will discuss the topological argument giving this proof in dimension 3, 4 and 5 and the proof in the conformally flat case. There is in particular a very explicit form of this argument in the case of two "masses" that we will emphasize. This is the argument currently used to try to solve the resonant cases that are left (Mohamed Ben Ayed, Marcello Lucia, Cheikh Birahim N'Diaye, Mohameden Ould Ahmedou and of course A.Malchiodi). for the Q-curvature problem.
b. We will also study the results of R. Schoen and S.T. Yau (Comm. Math.Physics, Vol 45,1979) about the positive mass conjecture. We will focus on the outline and various steps of these latter results for n = 3; they involve several other results about regularity, embedding and convergence properties of minimal surfaces in codimension 1 and in a non-compact framework (asymptotically flat ends), which we will state and refer to in as much precise details as possible. They are the supporting results for the argument proposed by R. Schoen (Jour.Diff.Geometry (20), 1984, pp 479-495) in order to prove the Yamabe conjecture.

### 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: 640:501 or instructor permission.

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: Their is no text for the course. There are a set of notes of mine which have been split into four chapters and which will be available to the students.

Prerequisites: Math 501, 502 and 503 are essential. Some knowledge of basic functional analysis is desirable, though not necessary.

Description: This is a basic Harmonic Analysis course that deals with Harmonic Analysis on Rn. We will start with interpolation theorems, the general Marcinkiewicz theorem and Stein's theorem on complex interpolation. Next we will study singular integrals, the Calderon-Zygmund theory both in the convolution case and the non-convolution treatment by M. Cotlar. We will then proceed to BMO and the John-Nirenberg theorem. Then we will prove the Hormander multiplier theorem and the basics of Littlewood-Paley theory. Time permitting we will study Bochner-Riesz operators, Restriction theorems for the Fourier transform and Strichartz estimates for the time dependent Schrödinger equation.

### Functions of Several Complex Variables I

Text: The course materials will be largely taken from the following, in particular [5], which I will give the pdf files during the semester:

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

[2] James Morrow and K. Kodaira, 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] Xiaojun Huang, Subelliptic analysis on Cauchy-Riemann

manifolds, Lecture Notes on the national summer graduate school of China, 2007. (to appear)

[5] Xiaojun Huang, Lecture Notes on Several Complex Variables, to appear.

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

### Differential Geometry

Text: “Riemannian geometry: A metric entrance” by Karsten Grove (Publisher: Univ. of Aarhus (1999) ASIN: B000J4WP1O)

Prerequisites:

Description: This is an introduction course for Differential and Riemannian geometry. The following are basic contents of this course:

1. Riemannian Length and Distance

2. Geodesics

3. The First Variation of Arc Length

4. The Levi-Civita Connection

5. The Exponential Maps

6. Isometries

7. Jacobi Fields and Curvature

8. Curvature Identities

9. Second Variation of Arc Length and Convexity

10. Parallel Transports

11. Manifolds and Maps

12. Vector Bundles and Tensors

13. Connections and Differential Forms

14. Submanifolds

15. Completeness

16. Global Effects of Curvature

17. Relative Curvature

18. Space Forms

19. Riemannian Submersions

20. Lie groups and homogeneous spaces

### Selected Topics in Geometry

Subtitle: Homogeneous Dynamics

Text: Bekka & Mayer, "Ergodic Theory and Topological Dynamics of Group Actions on HomogeneousSpaces", LMS Lecture Note Series 269, Cambridge, 2000.

Prerequisites: None

Description: We will study dynamics of group actions on homogeneous spaces with applications to number theory, the end goal being (time permitting) Margulis's proof of the Oppenheim Conjecture. Along the way, we will encounter the following topics (developing each from scratch, and studying them just enough to gather the tools necessary for Oppenheim, while indicating a more general theory):

• Hyperbolic Geometry, Locally Symmetric Spaces, Geodesic and Horocycle Flows

• Basic Ergodic Theory

• Unitary Representations of Semisimple Lie Groups, Howe-Moore Decay of Matrix Coefficients

• Equidistribution, Applications to Lattice Point Counting

### Algebraic Geometry I

Text: It is recommended to own a copy of Hartshorne's book, Algebraic Geometry (GTM 52). The class will eventually converge to follow this book.

Prerequisites: Some familiarity with commutative algebra is an advantage, for example Algebra II (Math 552).

Description: The course will be an introduction to algebraic geometry, with the main emphasis on algebraic varieties over an algebraically closed field (e.g. the complex numbers). Varieties are algebraic analogues of manifolds, which locally look like geometric figures cut out by polynomial equations.

Topics will include products and morphisms of varieties, projective and complete varieties, dimension, non-singular varieties, rational maps, divisors, sheafs, and line bundles. We will take a closer look at algebraic curves, especially elliptic curves and consequences of the Riemann-Roch theorem. Along the way we will also introduce the more general notion of schemes, which makes it possible to work with varieties over an arbitrary commutative ring.

### Introduction to Algebraic Topology I

Text: Algebraic Topology, 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.

• ### Lie Groups

Text: William Fulton and Joe Harris, "Representation theory. A first course".

Prerequisites: Real analysis, linear algebra, and elementary topology. No prior knowledge of Lie algebras, Lie groups, or representation theory will be assumed.

Description: This course will be an introduction to Lie groups and algebraic groups.

The classical linear groups (real and complex forms)

Closed subgroups of GL(n) as Lie groups

Linear algebraic groups and rational representations

Structure of complex classical groups: maximal torus, roots, adjoint representation, Weyl group

Highest weight theory for representations of semisimple Lie algebras

Complete reducibility of representations of semisimple Lie algebras and classical groups

### 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: Introduction to 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., J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2004.

Prerequisites: Only basic algebra. Also, some familiarity with Lie algebras would be helpful but is not necessary.

Description: This course will provide an introduction to vertex operator algebra theory and some of its many applications in mathematics and theoretical physics. Using the first text, we will begin by developing foundational examples of vertex operator constructions and their important role in representation theory. Understanding the new algebraic principles behind the calculus'' of vertex operators leads to the concepts of vertex operator algebra and of module for such a structure. Using the second text, we will also develop these concepts axiomatically, providing much greater insight into the theory, into important examples, and into applications.

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

### Selected Topics in Logic

Subtitle: An invitation to inner model theory.

Text:

Prerequisites: Basic set theory, familiarity with forcing will help.

Description: Inner model theory is the branch of set theory that deals with consistency issues surrounding ZFC. One of its main goals is to construct standard models for various natural extensions of ZFC. Here by standard we mean what is usually meant when one says that the set of natural models is the standard model of Peano Arithmetic. What is the standard model of ZFC? We will give an answer to this and several other similar questions. We will investigate two different hierarchies, the determinacy hierarchy and the large cardinal hierarchy and build a bridge between them while developing the basic inner model theoretic tools to inter-translate facts from determinacy to large cardinals and a vie versa. The highlight of the course will be Woodin's proof of AD in L(R) (from infinitely many Woodin cardinals and a measurable above them all).

### Special Topics in Number Theory

Subtitle: Automorphic Forms and Representations

Text: (Recommended): Bump, “Automorphic Forms and Representatoins”, Borel, “Automorphic forms on SL(2,R)”

Prerequisites: 501, 502, 503, 551, or equivalent

Description: This course will cover the basics of automorphic forms for SL(2,R), with an emphasis on setting up the theory to apply to general Chevalley groups. The topics include:

• Differential operators on the symmetric spaces

• Hecke operators

• Fourier decompositions

• The representation theory of SL(2,R)

• Automorphic L-functions

### Topics in Number Theory

Subtitle: Algorithmic Number Theory

Text: Notes and online references will be provided

Prerequisites: Mathematical maturity

Description: This course will be an introduction to basic algorithmic number theory.

Topics include:

• Primality testing

• Solving polynomial equations

• Lattices and Diophantine approximation

• Integer factorization

• Polynomial factorization

• Elliptic curve algorithms

• Number field algorithms

• The complexity of algebraic computation

### 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 thecourse 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.

### Introduction to Mathematical Physics I

Subtitle: Quantum theory and Analysis

Text: Galindo-Pascual I on Quantum Mechanics, and any other book you prefer on QM. Functional Analytsis I, by Reed& Simon.

Prerequisites: Real Analysis, ODE, Linear Algebra. Will help: Classical mechanics, some PDE

Description: This is an introductory course in mathematical physics. It is focused on teaching the basics of Quantum Mechnics together with the deep relations to Mathematical Analysis, in particular Functional Analysis. Will cover Schroedinger equation, basic properties of solutions, tunneling, exactly solvable potentials, self-adjoint operators, spectrum, spin, perturbation methods for bound states. If time permits: Scattering, resonances.

### 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: There is no required text. Van Lint-Wilson, A Course in Combinatorics, and Stasys Jukna, Extremal Combinatorics are useful references. Various other books will be on library reserve.

Prerequisites: Open to Ph.D. Students in Mathematics. Other students require permission of the instructor. Subject prerequisites are not extensive (we will occasionally need some Linear algebra at the level of 640:350, some Real Analysis at the level of 640:411, and some Probability at the level of 640:477), but the ability to understand and to write subtle mathematical arguments, is essential.

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: Applications of combinatorics in other fields

Text: None

Prerequisites: None

Description: I will focus on different applications of combinatorics in number theory, combinatorics, Fourier analysis, geometry and probability theory. The choice of material is very subjective: these are my personal favorites.

### 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

### Selected Topics in Applied Mathematics

Subtitle: Obstacle problems

Text: 1. P. Feehan, Lectures on variational inequalities, obstacle, and free boundary problems in mathematical finance, Rutgers University, Fall 2011 and 2013, Columbia University, Spring 2013.

2. J-F. Rodrigues, Obstacle problems in mathematical physics, North-Holland, New York, 1987.

Prerequisites: 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 (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 of obstacle problems arise in many areas of pure and applied mathematics and engineering, including mathematical finance – in particular, the American-style option pricing problem. 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) the American-style option pricing problem and other examples from applied mathematics;<

(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 by finite difference and finite element methods; and

(6) introduction to viscosity solutions for non-linear partial differential equations.

Students will be polled at the start of the class regarding their interests.

### Selected Topics in Applied Mathematics

Text:

Prerequisites:

Description:

### Mathematical Finance I

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.

### 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 and irreversibility

Text: The program is based on papers and books freely available on internet; mainly on:

Prerequisites:

Description: Birth of kinetic theory

Heat theorem and Ergodic hypothesis

Stationary Nonequilibrium

Finite thermostatsM

SRB distributions

Symbolic dynamics and chaos

Examples of hyperbolic symbolic dynamics

The SRB distribution: its physical meaning

Counting phase space cells out of equilibrium

SRB potentials

Chaos and Markov processes

Large deviations

Intermittency and developed turbulence

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