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.
Text: (1) An Invitation to C*-Algebras (Graduate
Texts in Mathematics 39), AMS, by William Arveson
(2) Completely bounded maps and operator algebras,
Cambridge University Press,
by Vern I. Paulsen
Prerequisites: Math 502 or permission of the instructor
Description: This course will provide an introduction tho the modern theory of operator algebras, developing the theory of their structure and representation. Recent results in non-commutative probability and quantum information that are based on this theory will be discussed, along with an introduction to some open problems that are the focus of current research. This course will be accessible to all students who have completed Math 502 or equivalent
Description: This course will be a continuation of the PDE course, 517, offered in the fall semester. We will take a similar approach as in 517, i.e., we will emphasize the methods than the most general forms of the results. One main theme will be how to extend the results and methods for dealing with constant coefficients prototype equations to related variable coefficients and nonlinear equations. We will develop the notions of weak solutions and Sobolev spaces and explain their applications to various initial/boundary value problems, including nonlinear ones. In the selection of topics to be covered, I will try to make a balanced choice to prepare students to use the tools of PDE in a variety of fields.
Text: (recommended)
D. Christodoulou, Mathematical Problems of General Relativity I, European Math. Soc., 2008. ISBN: 978-3-03719-005-0
H. Ringstrom, The Cauchy Problem in General Relativity, European Math. Soc., 2009. ISBN: 978-3-03719-053-1
N.M.J. Woodhouse, General Relativity, Springer, London, 2007. Available for download from the Rutgers Library Site, just click on the link:
Rutgers Library Site
Prerequisites: 517 and 532, or permission of instructor.
Description: This is an introductory course on Einstein's theory of general relativity and gravitation, emphasizing the geometric-PDEs point of view and some of the more recent advances in the field. Although welcomed and very much appreciated, no previous knowledge of physics is assumed. Some knowledge of partial differential equations and differential geometry, at the level of a first graduate course in these topics, is helpful. The following is an outline of the course:
I. The Geometry of Space-time:
Causal structure, curvature and gravitation, the energy tensor and the matter equations. Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit.
II. Einstein Vacuum Equations:
The symbol and the characteristics of EVE. The local existence theorem in wave coordinates. The Penrose singularity theorem. Black holes and cosmic censorship conjectures.
III. Conservation Laws and Noether's Theorem:
Lagrangian and Hamiltonian formulations. The Noether current in the theory of maps. Asymptotic flatness. The definition of global energy, momentum and angular momentum. The Positive Energy Theorem.
IV. Reduction under Symmetry:
Homogeneous and isotropic spacetimes. Static, spherically symmetric spacetimes: Schwarzschild, Reissner-Nordstrom, Hoffmann. Birkhoff’s theorem. Spacetimes with two Killing fields. Ernst formulation and harmonic maps. Kerr and Kerr-Newman spacetimes. Wave maps and cosmological solutions. Integrability.
Hardy's inequalities in one dimension, Vitali's covering lemma in R^n,
Hardy-Littlewood maximal function for balls or cubes, Marcinkiewicz
interpolation theorem, Hardy-Littlewood maximal function for rectangles
in R^n, Riesz fractional integrals and the Hardy-Littlewood-Sobolev theorem,
John-Nirenberg bounded mean oscillation space and exponential
integrability, Rademacher-Stepanov theorem (existence of first
differential for Lipschitz functions), local Poincare and Sobolev
inequalities for Lipschitz functions (including endpoint cases:
Gagliardo-Nirenberg and Trudinger estimates), Calderon-Zygmund
decomposition, Hilbert transform and Calderon-Zygmund singular integral
operators, non-homogeneous singular integrals, Littlewood-Paley theory, ...
Prerequisites: The prerequisite for this topic course are the following:
1. Basic notions and properties related to differentiable manifolds and
Riemannian manifolds such as: tangent bundles, differential forms,
integral of forms, Levi-Civita connection, geodesics, exponential maps,
curvature tensors.
2. Basic notions and properties in topology: open, closed, compact subsets,
convergence of sequences in metric spaces, completeness.
References: ``Riemannian Geometry'' by Peter Petersen, 2nd Edition, GTM 171
Description:Chapter 1. Ricci curvature comparison
1.1. Review of basic Riemannian manifold.
1.2. Curvature equations for distance functions
1.3. Basic comparison
1.4. Ricci curvature comparison and relative volume comparison
1.5. Application of relative volume comparison
1.6. Splitting and maximal principles
1.7. Splitting theorems
1.8. Manifolds with non-negative Ricci curvature
Chapter 2. Sectional Curvature Comparison
2.1. Basic sectional curvature comparison
2.2. Sign of curvature and fundamental groups
2.3. Curvature and injectivity radius estimates
2.4. Morse theory for distance functions
2.5. Connectedness principle for positive curvature
Chapter 3. Manifolds of Positive and Non-negative Sectional Curvature
3.1. Toponogov theorem
3.2. Bounding Fundamental groups
3.3. Gromov Betti number estimates
3.4. Sphere theorems
3.5. Soul theorem
(Optional)
Chapter 4. Convergence and Collapsing
theory in Riemannian Geometry
4.1. Gromov-Hausdorff convergences
4.2. Gromov precompactness
4.3. Cheeger-Gromov convergence theorem
4.4. Collapsing with bounded curvature and diamter
Text: Donaldson-Kronheimer "Gauge theory and 4-manifolds"
Donaldson "Floer homology groups in Yang-Mills theory", and Kronheimer-Mrowka "Monopoles and 3-manifolds"
Prerequisites:
Description:Possible topics include:
I Connections, curvature, and gauge transformations
II Moduli spaces of instantons
III Semistable bundles and correspondence theorems
IV Donaldson invariants
V Instanton Floer theory and invariants for manifolds with boundary
V Seiberg-Witten equations and monopole Floer homology
Text: Linear Representations of Finite Verlag (Graduate texts in Mathematics) (v. 42)
Springer-Verlag ISBN: 978-0387901909
Prerequisites: TBA
Description: This course will be a three-part introduction to the representation theory of Lie groups, requiring no specialized knowledge beyond first year graduate algebra and analysis.
The first part of the course will deal with representation theory of finite and compact groups - following parts 1 and 2 of Jean-Pierre Serre's book [1]. The second part will explore the connection between (noncompact) Lie groups and Lie algebras following Roger Howe"s article [2]. The third part of the course will discuss unitary representations os SL (2,R) essentially following ideas that go back to V. Bargmann [3].
REFERENCES
[1] J-P Serre, Linear Representations of Finite Groups (Grad. Texts in Math.) (v. 42)
[2] R. Howe, Very Basic Lie theory, Amer. Math. Monthly 90, 600-623
[3] V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. Math 48, 568-640
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
Subtitle: Chevalley Groups and infinite dimensional generalizations
Text: No Textbook
Prerequisites: Some familiarity with finite dimensional Lie algebras will be assumed.
Description: Chevalley's approach to studying Lie groups was to define them in terms of automorphisms of the underlying Lie algebra. With some additional external data involving the universal enveloping algebra, Chevalley was able define Lie groups, including the exceptional ones, over arbitrary fields, and also over Z. The Steinberg presentation for Chevalley groups gives a way to describe these groups in terms of generators and relations. This approach to Lie theory may be generalized to infinite dimensions, namely it is possible to define Kac-Moody groups as infinite dimensional Chevalley groups. In the affine case, the Chevalley theory was developed in detail by Garland. We study Chevalley groups and the Steinberg presentation for finite dimensional Lie groups, and we build all the necessary machinery to generalize the Chevalley construction to infinite dimensions.
Text: Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, North Holland, Amsterdam
# ISBN-10: 0444868399
# ISBN-13: 978-0444868398
Available at:www.amazon.com, just click on the link:
Theory-Studies-Logic-Foundations-Mathematics
Prerequisites: A knowledge of basic set theory, including cardinals, ordinals and the axiom of choice.
Description:
This is an introductory course on proving independence results in set
theory. Here a statement S is said to be independent of set theory
iff S can neither be proved nor disproved from the classical ZFC
axioms of set theory. For example, it is well-known that the
Continuum Hypothesis CH is independent of set theory.
Initially we shall follow the lazy man's approach to obtaining independence
results; namely, we shall study the consequences of the following two
extra set-theoretic axioms.
The Diamond Axiom(♢): a combinatorial principle which
says intuitively that there exists a fortune-teller who correctly
predicts the future often enough to be useful.
Martin's Axiom(MA + ¬ CH): a
powerful strengthening of the negation of the Continuum Hypothesis.
We shall see that there are many independent statements
S such that ZFC + MA + ¬ CH implies S, while
ZFC + ♢ implies ¬S. For example, this is true of
the following statement.
The Souslin Hypothesis: Suppose that the linear order
{ X, < } is complete and dense without endpoints.
If there does not exist an uncountable family of
pairwise disjoint nonempty open intervals of X,
then { X, < } ~ { R, < }.
However, there are many statements S whose independence cannot be
established in this manner, such as:
2^{ℵn} = ℵ_{n+1} iff n is prime.
In order to prove the independence of such statements, we shall need to
learn set-theoretic forcing. However, at this point in the course, those
students who have successfully mastered the use of Martin's Axiom
will realise that they already know how to force.
Text:Text:There will be no required text for this course. Diverse texts will
be on reserve and notes will be distributed for certain topics .
Other reference books:
M. Ram Murty and Jody Esmonde, Problems in Algebraic Number Theory,
Springer-Verlag Graduate Texts in Mathematics, volume 190.
Lang, Algebraic Number Theory, Springer GTM, volume 110
Ireland and Rosen, A Classical Introduction to Modern Number Theory,
GTM,
volume 84.
Course Format: There will be periodic problem
assignments and term projects involving algebraic number theory which
will investigate applications and extensions of the material.
Prerequisites:Prerequisites: We will assume a working knowledge of first year
graduate level algebra. *Permission of instructor required for
students not enrolled in the mathematics Ph.D. program.
Description: This will be a introductory course in
Algebraic Number Theory. The subject matter of the course should be
useful to students in areas of algebra and discrete mathematics, which
often have a number theoretic component to problems, as well as
students in number theory and algebraic geometry. The basic
invariants of field extensions of finite degree over the rational
field (so-called number fields) will be introduced --- ring of
integers, class number, units group, zeta functions, adele rings and
group of ideles. The relation of these abstract invariants to the
problem of solving polynomial equations in integers will be developed.
Special examples of number fields such as quadratic and cyclotomic
fields which have a rich structure will be used to illuminate the
theoretical aspects. Algorithmic computation of these invariants will
be analyzed, and open questions detailed.
Topics:
1. Number fields, lattices and rings of integers
2. Dedekind domains and their ideals and modules
3. Ideal class groups and Class number
4. Zeta functions of number fields
5. Quadratic fields and binary forms
6. Cyclotomic fields and Gauss sums
7. Diophantine problems and algebraic number theory
8. Algorithms in number theory
9. Adeles and Ideles of number fields
Text: The On-Line Encyclopedia of Integer Sequences (OEIS) and other on-line resources, and handouts.
Prerequisites: There are no prerequisites, and no previous programming knowledge is assumed. Also, very little overlap with previous years.
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 Sequences, both in general-studying important families of sequences and how to detect them- and in particular, studying particularly interesting members of Neil Sloane's zoo
In addition to the actual, very important content, students will master the methodology of computer-generated and computer-assisted research that is so crucial for their future. We will also have one guest lecture by Mr. Edinah Gnang, who will give a tutorial on the open-source beautiful (and powerful)
computer algebra system SAGE, and one by the high priest himself.
There are no prerequisites, and no previous programming knowledge is assumed. Also, very little overlap with previous years. The final projects will
definitely lead to new sequences that should be entered into the OEIS, with due credit to the contributing students, thereby guaranteeing them
immortality. Some of the final projects may lead to journal publications.
Subtitle: Tensor categories in representation theory
Text: 1. Y.-Z. Huang, Introduction to representation theory and
tensor categories, Lecture notes, 2011. Here is the pdf file:
Lecture Notes, 2011
2. Y.-Z. Huang, J. Lepowsky and Lin Zhang,
Logarithmic tensor category theory, I-VIII, to appear; Part I to
Part VI are already in the archive. The numbers are:
arXiv:1012.4193, arXiv:1012.4196, arXiv:1012.4197, arXiv:1012.4198,
arXiv:1012.4199, arXiv:1012.4202
Part VII and Part VIII will also be posted soon so that they will be in the archive
in the spring semester.
Prerequisites: Algebra and complex analysis at the level of first year graduate
course. The second part of the course also requires the students to be familiar
with the material presented in Lepowsky's course in Fall, 2011.
Description: Tensor categories are generalizations of monoids and groups.
They appear naturally in representation theory and in quantum physics.
Many mathematical structures and physical phenomena can be studied
using the theory of tensor categories. One particularly interesting class of tensor categories is the class from conformal field theories. These tensor categories have applications in algebra, topology, geometry, string theory, condensed matter physics and quantum computation.
The first part of the course will be an introduction to tensor categories using representations of groups, associative algebras and Lie algebras as motivating examples. The second part of the course discusses tensor category structures on suitable module categories for suitable vertex operator algebras, assuming that the students are familiar with the material presented in Lepowsky's course in Fall, 2011.
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.
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.
Text: Since detailed lecture notes for this course will be available on the instructor's web site, there is no assigned textbook. However, since many students find a textbook useful, the following two, both available in paperback, are recommended.
Finite Elements: Theory, fast solvers, and applications in solid mechanics by Dietrich Braess, Cambridge University Press, (paperback), 2007 (3rd edition).
Partial Differential Equations with Numerical Methods by Stig Larsson and Vidar Thomée, Texts In Applied Mathematics, (paperback) Volume 45, Springer, 2009.
Prerequisites: Students should have some knowledge of partial differential equations (at least on an undergraduate level). Although a previous course in numerical analysis is useful, it is not required. For students who have taken 642:573 Numerical Analysis I and are deciding whether to continue with 642:574 Numerical Analysis II or take 642:575, the following advice is relevant. Math 642:575 is a more challenging course than 642:574 (and covers different topics). Generally, students with grades below B+ in 642:573 should take 642:574, or seek the permission of the instructor before enrolling in
642:575.
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 various software packages.
Text: Hassan K. Khalil, Nonlinear Systems, Third Edition, Prentice Hall, 2002
Prerequisites: Working knowledge of undergraduate analysis and linear
algebra from engineering and other non-math department students;
640:501-502 from math students. (Assignments will be more challenging
for math students.)
Description: I plan to cover basic aspects of nonlinear dynamics and
control systems, using Khalil's book as a guide but material from my
own textbook at certain points. The following is a possible outline
(with textbook sections): Introduction (1.1-1.2 & 3.1-3.2),
Second-order systems (2.1-2.5 & 2.7), Stability of equilibrium points
(4.1-4.5, 4.7, 8.2 & 9.1), Passivity (6.1-6.5 & 7.1), Input-state and
Input-output stability (4.8, 9.2, 4.9, 5.1-5.4 & 6.4), Special
nonlinear (forms 13.2-13.3), Stabilization (12.1-12.2, 13.4.1, 14.3,
14.4), Robust stabilization (14.1.1-14.1.2, 14.2 & 14.3), Tracking
(13.4.2, 14.1.3), Observers (14.5), Regulation via integral control (
12.3-12.4, 14.1.4, 14.5.3).
Prerequisites: Linear algebra and advanced calculus. There are no physics prerequisites, but prior exposure to standard quantum theory would be helpful. Some knowledge of probability theory would also be good.
Description: Quantum theory is the most successful physical theory yet devised. While based on simple enough mathematics, it is nonetheless rather difficult to understand as physics. It was controversial shortly after its founding, with two of its founders, Einstein and Schr\"odinger, among its harshest critics. It remains controversial today.
This course will be an introduction to Bohmian mechanics, a formulation of nonrelativistic quantum mechanics that is clear both as mathematics and as physics. In fact Bohmian mechanics is the deterministic theory of particles in motion which naturally emerges from orthodox quantum theory when we demand conceptual clarity and physical precision.
Possible topics:
Review of orthodox quantum theory
Conceptual difficulties of orthodox quantum theory
Bohmian mechanics
Global existence and uniqueness for the Bohmian dynamics
The empirical equivalence between Bohmian mechanics and orthodox quantum theory: the origin of randomness in a determistic dynamics; the origin of operators as observables in quantum theory.
Bohmian mechanics for configuration spaces having nontrivial
topology; Bohmian mechanics for systems of identical particles
Bohmian scattering theory
The classical limit of Bohmian mechanics
Nonlocality
Relativistic extensions of Bohmian mechanics
Bohmian quantum field theory
Quantum cosmology and the meaning of the wave function
Subtitle: Collective Phenomena in Equilibrium and Nonequilibrium Systems
Text:
Prerequisites: For information about prerequisites contact Joel Lebowitz, lebowitz@math.rutgers.edu
Description: The 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.
I will then consider application of statistical mechanics to real world problems of current interest. An example of such an application is pattern formation.
This occurs in both equilibrium and nonequilibrium systems. The former generally represent low temperature phases in materials and can be studied
via equilibrium ensembles. The latter involve dynamical microscopic considerations and are generally described on the macroscopic level by reaction diffusion type equations. The resulting patterns are visible everywhere in biological systems. They range in scale from microns for cells forming an organism to hundreds of meters for flocking birds.
Choice of specific topics will be based on student interest.
For background material see the following books:
Mathematical Biology I and II, J.D. Murray (Springer)
Large Scale Dynamics of Interacting Particles, H. Spohn (Springer)
A Kinetic View of Statistical Physics, P. Krapivsky, S. Redner and E. Ben-Naim (Cambridge)
Dynamics of Self-Organized and Self-Assembled Structures, R. Desai and R. Krapral (Cambridge)
Evolutionary Games and Population Dynamics, J. Hofbauer and K. Sigmund (Cambridge)
The course will be informal and interactive. For information about prerequisites please contact me: Joel Lebowitz, lebowitz@math.rutgers.edu.