Mathematics Department - Graduate Course Descriptions - Spring 2011

### Theory of Functions of a Real Variable II

Text: We will continue to use the 501 text, Real Analysis: Modern Techniques and Their Applications, by G. Folland (Wiley-Interscience; 2nd edition, 1999. ISBN-10: 0471317160). We will supplement the main text by material from a variety of other sources.

Prerequisites: 640:501 or permission of Instructor

Description:

This course will be a continuation of 640:501.Our hope is to cover material from Chapters 6-9 and Chapter 11 of Folland's book.Thus we will discuss aspects of the theory of L^p spaces, Radon measures, the Riesz representation theorem for Radon measures, Fourier series, the Fourier transform, Sobolev spaces, some distribution theory and some discussion of Hausdorff measure and Hausdorff dimension. There will be occasional excursions into functional analysis: the weak topology on Banach spaces, reflexive Banach spaces and aspects of linear operator theory. As we shall try to illustrate, this material has applications to wide areas of mathematics, for example, ordinary and partial differential equations, differential-delay equations,and integral equations. Some applications are unexpected, e.g., a connection between Hausdorff dimension and the spectral radius of certain bounded linear operators.

### Theory of Functions of a Complex Variable II

Text: Green and Krantz: Function Theory of One Complex Variable, AMS.

Forster, O.: Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991

Prerequisites: Math 503

Description: This will be a continuation of Math 503. The following two parts will be covered: (1) Classical Complex Analysis and (2) Riemann surfaces.

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

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

### Selected Topics in Analysis

Text: 1. Regularity theory for the mean curvature flow by Klaus Ecker, 2. Hamilton's Ricci flow by Chow, Lu, Ni.

Prerequisites: some basic knowledge on partial differential equations and differential geometry.

Description:

In this course we will first recall some basic properties of the heat equation, such as the maximum principle, the Harnack estimate, etc. We will next see how many of these techniques can be used to study different parabolic geometric flows, such as the Ricci flow and the mean curvature flow.

In the course, we will give an introduction to the Ricci flow, with a brief review of Riemannian geometry. We will start the study of the Ricci flow with considering special solutions to the Ricci flow. We will talk about the short time existence, maximal principle, Li-Yau-Hamilton type differential Harnack inequality and derivative estimates. We will give a comprehensive treatment of the Ricci flow on surfaces. We will also study Hamilton's original paper on three-manifolds with positive Ricci curvature. Finally, Perelman's entropy functional will be discussed at the end.

For the mean curvature flow we shall cover various important estimates and their geometric implications. For example, Huisken's theorem on shrinking convex hypersurfaces to a point will be discussed and some of the regularity theory developed by White, Brakke.

### Partial Differential Equations II

Text: "Partial Differential Equations" by Lawrence C. Evans, published by AMS, 2002. In addition, there will be reading from several recent, and one not so recent, research papers on topics to be developed in the course.

Prerequisites: 640:517, Partial Differential Equations I.

Description: This course will build directly on the content covered in 640:517, Partial Di fferential Equations I, taught by Zheng-Chao Han in Fall 2010. In particular, we shall pick up on the notions of weak solutions and Sobolev spaces introduced towards the end of 640:517. We shall begin with this circle of ideas, developing it from the beginning, proving results on Sobolev spaces and embeddings, and explaining their applications to nonlinear PDE. We shall continue with topics that build on, as well as complement, these ideas, including weak lower semicontinuity in variational methods for PDE's, solving nonlinear elliptic equations using sub and super solutions, the theory of viscosity solutions, the Hamilton-Jacobi equation, and the ideas of DiGiorgi, Nash, and Moser on regularity of parabolic and elliptic equations with rough coefficients. (Nash's 1958 paper is the "not so recent" paper mentioned above.) If time permits, we shall also discuss recent progress on the Monge-Ampere equation.

### Harmonic Analysis on Euclidean Spaces

Text:

Prerequisites:

Description:

Hardy inequalities, Vitali's covering lemma, Hardy-Littlewood maximal function for balls, Marcinkiewicz interpolation theorem, Hardy-Littlewood maximal function for rectangles, Riesz fractional integrals and the (global) Hardy-Littlewood-Sobolev theorem, John-Nirenberg bounded mean oscillation space, Rademacher-Stepanov theorem (first differential for Lipschitz functions), local Poincare and Sobolev inequalities for Lipschitz functions, Calderon-Zygmund decomposition, Calderon-Zygmund singular integral operators, non-homogeneous singular integrals, Littlewood-Paley theory, ...

### Functions of Several Complex Variables II

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, Subelliptic analysis in Cauchy-Riemann Geometry and Complex Geometry, 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 topics from Several Complex Variables.

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

(b) Complex manifolds, holomorphic vector bundles, Sheaf-Cohomology theory, Kahler manifolds

(c) D-bar equations on complex manifolds, Kodaira vanishing theorem and Kodaira embedding theorem, Kodaira-Spencer's deformation theory and Kodaira' classification theory of complex surfaces

### Selected Topics in Geometry

Subtitle: An introduction to Alexandrov geometry

Text:

Prerequisites:

Description:

This is an introduction course on Alexandrov geometry. An Alexandrov space with curvature bounded below by k is a length metric space such that any geodesic triangle can be compared with one of the same size on a space form of constant curvature k. Riemannian manifolds with sectional bounded from below are Alexandrov spaces and a general Alexandrov is singular both in geometry and topology. Alexandrov geometry studies geometric and topological structure of Alexandrov spaces, and provides tools to generalize many results that are known on Riemannian manifolds with curvature bounded from below.

The first part we will cover basic properties of an Alexandrov space (definition, tangent cone, gradient-exponential map, first variation of distance function, dimension, etc), and prove some global results.

The second part will introduce basic theory of semi-concave functions on Alexandrov space. This will provide some analytic tools that are useful in doing certain analysis on Alexandrov spaces.

The references are :

1. A course in metric geometry, D. Burago, Y. Burago and S. Ivanov
2. Semi concave functions on Alexandrov spaces, (by A. Petrunin) Surveys in J. Diff. Geom. XI (2007), 1-26.
3. Plaut, C., Metric spaces of curvature >k. Handbook of geometric topology, 819898, North-Holland, Amsterdam, 2002.

### Algebraic Geometry II

Text: Hartshorne, Algebraic Geometry (Springer GTM 52).

Prerequisites: Math 535. Familiarity with commutative algebra is an advantage, but is not required.

Description:

This course continues the study of algebraic geometry from the fall by replacing algebraic varieties with the more general theory of schemes, which makes it possible to assign geometric meaning to an arbitrary commutative ring. One major advantage of schemes is the availability of a well-behaved fiber product. Combined with Grothendieck's philosophy that properties of schemes should be expressed as properties of morphisms between schemes, fiber products make the theory very flexible. The goal of the course is to cover the basic definitions and properties of schemes and morphisms, and to introduce and study the cohomology of sheaves, which provides a powerful tool for settling geometric questions. For example, one can use cohomological methods to give a simple proof of the classical Riemann-Roch theorem for curves.

### Introduction to Algebraic Topology II

Text: "Algebraic Topology" by Hatcher. It's also available for free online at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Prerequisites: 540 or permission of the instructor

Description:

Topics: Cohomology, cup product, Poincare duality, universal coefficients, Kunneth Formula, cohomology of SO(n), direct and inverse limits, transfer homomorphism, homotopy groups, Hurewicz theorem, fiber bundles, obstruction theory, cohomology of fiber bundles, Whitehead torsion, finiteness obstruction.

We will continue MA 540 with chapter 3 of Hatcher's book and continue with selected topics from chapter 4. After that, we will move on to Whitehead's simple-homotopy classification of finite CW complexes and Wall's characterization of CW complexes that are homotopy equivalent to finite CW complexes and Swan's characterization of groups that act freely on CW complexes homotopy equivalent to a sphere.

### Abstract Algebra II

Text: Jacobson, "Basic Algebra", Volumes 1 and 2, second edition. These volumes are currently available from Dover (www.doverpublications.com).

Prerequisites: Any standard course in abstract algebra for undergraduates and/or Math 551

Description:

Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. The course will cover the following topics (and perhaps some others).

1) Basic module theory and introductory homological algebra - most of Chapter 3 and part of Chapter 6 of Basic Algebra II: hom and tensor, projective and injective modules, resolutions, completely reducible modules, the Wedderburn-Artin theorem

2) Commutative ideal theory and Noetherian rings - part of Chapter 7 of Basic Algebra II: rings of polynomials, localization, primary decomposition theorem, Dedekind domains, Noether normalization

3) Galois Theory - Chapter 4 of Basic Algebra I and part of Chapter 8 of Basic Algebra II: algebraic and transcendental extensions, separable and normal extensions, the Galois group, solvability of equations by radicals

### Selected Topics in Algebra

Subtitle: Infinite dimensional Lie groups and applications

Text: None

Prerequisites: some familiarity with finite dimensional Lie groups and lie algebras is preferable but not required.

Description:

Kac-Moody groups are natural generalizations to infinite dimensions of finite dimensional simple Lie groups. The subclass of real forms of hyperbolic Kac-Moody groups have recently been shown to appear in the study of algebraic symmetries of general relativity and a theory known as supergravity, which incorporates both general relativity and supersymmetry.

The discrete symmetries arising from forms of these groups over the integers, play a particularly important role, related to quantization of charges in particle physics.

In this course, we study the mathematics suggested by symmetries of supergravity, focusing on the occurrence of hyperbolic Kac-Moody groups and algebras and their properties. We draw many of our observations from analogy with finite dimensional Lie groups and their relation to classical spacetime symmetries.

The action of the Kac-Moody group G on a simplicial complex, known as the Tits building, and the related structure theory for G and its subgroups is an important tool that we will make use of. We also consider the infinite dimensional symmetric space G/K, where K is the fixed point subgroup of the Cartan involution. This space is an infinite dimensional analog of the Poincare upper half plane and has a prominent role in the coset models of supergravity theories.

We also explore the open problems connecting hyperbolic Kac-Moody groups and algebras with physical theories such as supergravity and M-theory, which is a speculative theory which proposes to unify all superstring theories.

The basic structure theory of infinite dimensional Lie groups and Lie algebras will be covered. Some familiarity with finite dimensional Lie groups and Lie algebras is preferable though not required.

### Representation Theory

Text: W. Fulton and J. Harris, Representation Theory. A First Course

Prerequisites: The required background is some basic algebra (main concepts of linear algebra and the theory of groups, rings and modules) and analysis. No knowledge of representation theory is assumed; the course will provide an introduction to its basic concepts and techniques. An emphasis will be made on a detailed study of specific examples such as the symmetric group, the general linear group, other classical groups and their Lie algebras.

Description:

Representation theory is one of the cornerstones of modern mathematics. It provides a mathematical formalism for studying symmetry, and has a very wide range of applications to other mathematical disciplines and other branches of science (physics, chemistry, economics, etc.). The course focuses on finite-dimensional representations of finite groups, semisimple complex Lie groups and Lie algebras.

Despite being one of the best developed parts of mathematics, the representation field is still full of natural open problems and is a subject of an active current research.

### Topics in Number Theory

Subtitle: The Arithmetic of Lie groups

Text: recommended texts for math 574 to my course website: http://www.math.rutgers.edu/~sdmiller/math574spring2011/

Prerequisites: Permission of instructor required for students not enrolled in the mathematics Ph.D. program.

Description:

This course will cover the basics of automorphic forms on Lie groups. I will mainly stick to Chevalley groups for simplicity, and begin the course by summarizing the simplest example of the subgroup SL(2,Z) of SL(2,R), along with its modular forms. This has been the topic of a number of recent graduate courses, and while none of these are formally a prerequisite for the material in this class, they certainly motivate the material of this course. The goal is to give students background to work with these objects on more general groups, which have become crucial to many recent advances in analytic number theory. Additionally, arithmetic subgroups of Lie groups are important in group theory, dynamics, and topology.

I will assume no particular knowledge of Lie groups, and instead build up from scratch. My plan is to cover the following topics, stressing the SL(2) example as motivation.

1. Overview of SL(2,Z) as a discrete subgroup of SL(2,R), and its automorphic forms.
2. The notion of algebraic group, and some key features, such as Borel subgroups.
3. Root systems and Weyl groups.
4. Cartan-Killing classifications of Lie groups and symmetric spaces.
5. Chevalley groups, Steinberg and Serre relations
6. Finite dimensional representations, and theory of highest weight.
7. Arithmetic subgroups, examples and constructions. Rigidity Theorems.
8. Reduction theory.
10. Hecke operators
11. Differential operators, such as the Laplacian
12. Automorphic representations
13. Ramanujan conjecture
14. Selberg Laplace eigenvalue conjecture
15. Langlands' Functoriality conjectures

### Experimental Mathematics

Text: Classical Articles,
• The monotone circuit complexity of Boolean functions by Noga Alon and R.B. Boppana
• Bounds for sorting by prefix reversal By Bill Gates and Christos Papadimitriou.
• More to TBA

• Prerequisites: There are no prerequisites, and no previous programming knowledge is assumed. Also, very little overlap with previous years. The final projects for this class may lead to journal publications.

Description:

Experimental Mathematics used to be considered an oxymoron, but the future of mathematics is in that direction. In addition to learning the philosophy and methodology of this budding field, students will become computer-algebra wizards, and that should be very helpful in whatever mathematical specialty they are doing (or will do) research in.

We will first learn Maple, and how to program in it. This semester we will focus on ALGORITHMS, both general (complexity theory, trying to prove that P != NP or at least get closer to it than people did before), but first learning what is a computation and what is a Turing Machine and other computational models (simulating them all in Maple). This would be one part. The other part will focus on a very specific kind of algorithms, sorting algorithms, in particular those useful in bioinformatics (and for sorting pancakes by flipping).

But the actual content is not that important, it is mastering the methodology of computer-generated and computer-assisted research that is so crucial for your future.

### Methods of Applied Mathematics II

Text: Advanced Engineering Mathematics (2nd edition) by Michael D. Greenberg, (Prentice Hall, Upper Saddle River, NJ, 1998). ISBN 0-13-321431-1

Optional purchase:

Methods of Applied Mathematics (2nd edition) by Francis B. Hildebrand, which is available in paperback (Dover, New York, 1965) ISBN 0-486-67002-3

Prerequisites: Math 527, or else permission of the instructor

Description: This is a second-semester graduate course, appropriate for students of mechanical and aerospace engineering, biomedical, electrical, or other engineering areas, materials science, or physics. It begins with the algebra of complex numbers, complex-valued functions of complex variables, analytic functions and the Cauchy-Riemann conditions, poles and branch cuts, and conformal mappings, with applications in physics and engineering to the solution of differential equations and to fluid mechanics. Finally, we address some topics in the calculus of variations with applications.

### Introduction to Mathematical Physics II

Subtitle: The Mathematics of Quantum Mechanics

Text: notes of the instructor

Description: The goal of the course is that students obtain a good understanding of quantum mechanics and the mathematical concepts and techniques relevant to it. Topics include:
• The Schroedinger equation
• Hilbert space and operators (unitary, projection, self-adjoint operators, the spectral theorem, Fourier transformation)
• The predictive formalism (observables, projection-valued measures, positive-operator-valued measures, Bohmian mechanics)
• tensor product spaces; trace of operators and the use of density operators; permutation symmetries; spin and representations of the rotation group SO(3)
• relativistic versions of the Schroedinger equation.
It is not necessary to have attended 561 Introduction to Mathematical Physics I.

### Numerical Analysis II

Text: There is no required textbook. Lecture notes will be posted to the course web site after each class. For those students who also wish to have another source for the material, I recommend purchasing one of the following texts: Copies will also be on reserve in the Mathematics library.

K. Atkinson, / An Introduction to Numerical Analysis,/, (second edition), Wiley, 1989.

D. Kincaid and W. Cheney: /Numerical Analysis: Mathematics of Scientific Computing/, (third edition), American Mathematical Society, 2002 (republished 2009).

A. Quarteroni, R. Sacco, and F. Saleri, /Numerical Mathematics,/, (second edition), Springer, 2004.

J. Stoer and R. Bulirsch: /Introduction to Numerical Analysis/, (third edition) Springer, 2002.

Prerequisites: Advanced Calculus, Linear Algebra, and familiarity with differential equations. .

Description: This is the second part, independent of the first, of a general survey of the basic topics in numerical analysis -- the study and analysis of numerical algorithms for approximating the solution of a variety of generic problems which occur in applications.

This semester (642:574), we will study the numerical solution of linear systems of equations, the approximation of matrix eigenvalues and eigenvectors, the numerical solution of nonlinear systems of equations, numerical techniques for unconstrained function minimization, finite difference and finite element methods for two-point boundary value problems, and finite difference methods for some model problems in partial differential equations, such as Poisson's equation and the heat equation.

In the fall semester, we considered the approximation of functions by polynomials and piecewise polynomials, numerical integration, and the numerical solution of initial value problems for ordinary differential equations, and showed how all these problems are related.

Despite the many solution techniques presented in elementary calculus and differential equations courses, mathematical models used in applications often do not have the simple forms required for using these methods. Hence, a quantitative understanding of the models requires the use of numerical approximation schemes. This course provides the mathematical background for understanding how such schemes are derived and when they are likely to work.

To illustrate the theory, in addition to the usual pencil and paper problems, some short computer programs will be assigned. To minimize the effort involved, the use of /Matlab/ will be encouraged. This program has many built in features which make programming easy, even for those with very little prior programming experience.

### Numerical Solutions of Partial Differential Equations

Text: Dietrich Braess, Finite Elements: Theory, fast solvers, and applications in solid mechanics, 3rd ed., Cambridge University, 2007. Note: Most of lectures will be based on hand outs prepared by the instructor. Students may have one of the aforementioned textbooks depending on their preference.

Prerequisites: At least one of Numerical Analysis I (16:642:573) or Numerical Analysis II (16:642:574).

Description:

In this course, we study finite difference, finite element, and finite volume methods for the numerical solution of elliptic, parabolic, and hyperbolic partial differential equations. The course will concentrate on the key ideas underlying the derivation of numerical schemes and a study of their stability and accuracy. Students will have the opportunity to gain computational experience with numerical methods with a minimal of programming by the use of Matlab's PDE Toolbox software. Since there are many sophisticated computer packages available for solving partial differential equations, one might think that a thorough understanding of the numerical methods employed is no longer necessary. A striking example of why naive use of such codes can lead to disastrous results is the sinking of the Sleipner A offshore oil platform in Norway in 1991, resulting in an economic loss of about \$700 million. The post accident investigation traced the problem to inaccurate finite element approximation of the linear elastic model of the structure (using the popular finite element program NASTRAN). The shear stresses were underestimated by 47%, leading to insufficient design. More careful finite element analysis, made after the accident, predicted that failure would occur with this design at a depth of 62m, which matches well with the actual occu

### Combinatorics II

Text: None. Various references will be put on library reserve

Prerequisites: 642:582 or permission of instructor

Description: This course is a continuation of 642:582. The topics will be chosen from:
• Combinatorial correlation inequalities and applications
• Finite partially ordered sets and lattices
• Mobius inversion in partially ordered sets
• Extremal theorems for hypergraphs: Sperner's Theorem, Kruskal-Katona Theorem, Erdos-Ko-Rado theorem, Isoperimetric theorems, Sauer-Shelah lemma
• Ramsey theory, the van der Waerden and Hales-Jewett Theorems.
• Probabilistic methods
• Linear algebraic methods
• Topological methods
• Matroids
Topics may be adjusted depending upon what has been covered in 582, student background and interest, and time constraints.

### Arithmetic Combinatorics

Text: Additive combinatorics (Tao and Vu, Cambridge Univ. Press).

Prerequisites:

Description:

We focus on problems from number theory with combinatorial nature and methods to attack them.

Topics include:

• probabilistic method
• Fourier analytic method
• geometry of number
• arithmetic progression in dense sets (Roth, Szemeredi, Green-Tao theorems)
• Sum-product theorems and expansion
• Inverse theorems and applications in random matrix theory

### Selected Topics in Applied Mathematics

Subtitle: Dynamics of biomolecular networks

Text:

Prerequisites: none for mathematics graduate students; others by permission of instructor (please email instructor)

Description: (Note: this course is not the same as 613, introduction to mathematical and systems biology. It will be a more specialized course, with a focus on a narrow range of topics.) We will cover Alon's book "An Introduction to Systems Biology: Design Principles of Biological Circuits" (filling-in more mathematical details and extensions, were appropriate). Student participation will be required: sections of the book will be assigned for presentation. Grading will be based on presentations.

### Selected Topics in Applied Mathematics

Subtitle: Computational Topology and Dynamics

Text:

Prerequisites: Permission of the instructor

Description:

It is reasonably safe to assume that much of the work of this century in dynamics will involve the study of multiparameter multiscale systems, will be done using computational methods, and will be based on the analysis of large data sets. Quantitative biology provides an archetype for these types of challenges. As the following observations make clear this calls for new mathematical techniques:

1. The work of the last century has made it abundantly clear that nonlinear dynamical systems can produce extremely complicated dynamics such as chaotic invariant sets. Furthermore, the structure of the invariant sets themselves can change in fundamental ways on fractal sets of positive measure in parameter space.
2. Models of multiscale problems are often based on heuristics as opposed to being derived from first principles and thus the fine structure of the invariant sets produced by the models may be of little interest or relevance for the applications in mind.
3. Parameters for these models are often poorly defined, unknown and/or difficult to measure and thus statements about the structure of the dynamics at particular parameter values may be again be of limited relevance.

In the 70s C. Conley developed a purely topological framework for the study of dynamics. This course will describe this work but from an algorithmic perspective. In particular, it will be shown that the essential ideas of Conley theory can be recast in a combinatorial and algorithmic framework which leads to efficient novel computational methods for nonlinear systems. Furthermore, the use of algebraic topology and new computational topological tools allows one to draw mathematically rigorous conclusions even in the context of finite resolution (measurements) in both phase space and parameter space. In summary there are three essential topics that will be covered. The actual weight and emphasis will be determined by the background and interests of the students.

1. Decomposing Global Dynamics
2. Conley Index
3. Computational Homology

### Mathematical Finance II

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

### Computational Finance

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

### Selected Topics in Mathematical Finance

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

### Topics in Mathematical Physics

Subtitle: Functional Analysis in Action

Text: Reed-Simon I,II and E.B.Davies' book on Spectral theory

Prerequisites: Real Analysis, ODE, Linear Algebra

Description: This course is best described as " Functional Analysis in Action".

The course will be focused on Spectral and scattering theory techniques in partial differential equations and mathematical physics. It begins with details of basic notions and results from Functional analysis, including the spectral theorem for unbounded operators. Then, applications of compact operators in PDE and Spectral theory. Basic constructions of Hamiltonians in Quantum Mechanics. Large time behavior: from Global existence to decay and scattering.

Examples and open problems.

Weekly HW and class participation will be used for grades, recommendations.

Class hours will be modified to adapt to the interested students.

### Topics in Mathematical Physics

Text: No Textbook

Description:

Course will start with a broad overview of the physics and mathematics of equilibrium and nonequilibrium statistical mechanics: This will focus on the elucidation and derivation of collective behavior of macroscopic systems made up of very many individual components from the microscopic dynamics of the individual components.

We will then focus on application of statistical mechanics to “real word” problems of current interest using books, as well as research articles and reviews, as background and source of problems. Choice of specific topics will be based on student interest.

For background material see the following books: Mathematical Biology I: An Introduction, J.D. Murray Large Scale Dynamics of Interacting Particles, H. Spohn