Publisher: American Mathematical Society; 3rd edition (March 29, 2006)
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
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.
These topics are covered in the first semester graduate real variable course (640:501).
This introductory course should be useful for students with a variety of research interests: physics and mathematical physics, applied analysis, numerical analysis, complex analysis, differential geometry, and, of course, partial differential equations.
For an introductory course, it is more important to examine some important examples to certain depth, to introduce the formulation, concepts, most useful methods and techniques through such examples, than to concentrating on presentation and proof of results in their most general form.
This is the way the course will be conducted.
The beginning weeks of the course aim to develop enough familiarity and experience to the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation.
Fourier series/eigenfunction expansions, Fourier transforms, energy methods, and maximum principles will be introduced. More importantly, appropriate methods are introduced for the purpose of establishing qualitative, characteristic properties of solutions to each class of equations. It is these properties that we will focus on later in extending our beginning theories to more general situations, such as variable coefficient equations and nonlinear equations.
Next we will discuss some notions and results that are relevant in treating general PDEs: characteristics, non-characteristic Cauchy problems and Cauchy-Kowalevskaya theorem, wellposedness.
Towards the end of the semester, we will begin some introductory discussion on the extension of the energy methods to variable coefficient wave/heat equations, and/or the Dirichlet principle in the calculus of variations.
The purpose here is to motivate and introduce the notion of weak solutions and Sobolev spaces, which will be more fully developed in the second semester.
More concretely, we plan to do the following:
1. Quickly review strong maximum principle, Hopf Lemma, Harnack inequality, isolated singularity of solutions, comparison principles, for Harmonic functions. We will then introduce the concept of viscosity solutions, and present some recent joint and ongoing work with Caffarelli and Nirenberg on strong maximum principle for singular solutions to fully nonlinear elliptic equations of second order, and to present removable singularity results for viscosity solutions of such equations. We will also present works of Caffarelli-Gidas-Spruck on isolated singularity for semilinear elliptic equations of critical exponent, as well as some recent work with Z.C. Han and E. Teixeira on a fully nonlinear version of it. We will present Harnack inequalities, Liouville theorems, derivative estimates for conformally invariant elliptic equations, and present Bernstein type arguments in obtaining a priori estimates.
2. Introduce the method of moving planes, proving a theorem of A.D. Alexandrov asserting that embeded compact hypersufaces of constant mean curvature in Euclidean space must be round spheres. Present some variant of this theorem in joint work with Nirenberg and some open problems.
3. We will give a survey of existence theorems for the $\sigma_k$-Yamabe problem and will present some open problems in this direction.
- Differential forms on manifolds
- Vector bundles & curvature
- Riemannian metrics & geodesics
- Symplectic forms & moment maps
Topics that will be covered in the course
- First and Second Variation of Area.
- Monotonicity Formula for minimal surfaces.
- Isothermal Coordinates and Bernstein's theorem.
- Weierstrass Parametrization of Minimal Surfaces.
- Omission of points by the Gauss Map, Fujimoto's theorem.
- Heinz curvature Estimate.
- Plateau Problem.
- Stability of minimal surfaces and Schoen-Fischer-Colbrie theorem.
- Rado's theorem on branch points.
1) Francis Bonahon, "Low-dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots", Student Mathematical Library, 49.
Understanding the following articles will be our first goal:
2) Leonid O. Chekhov, Vladimir V. Fock, "Quantum Teichmuller spaces", Theoret. and Math. Phys. 120 (1999), no. 3, 1245--1259.
3) Xiaobo Liu, "The quantum Teichmuller space as a noncommutative algebraic object", J. Knot Theory Ramifications 18 (2009), no. 5, 705--726.
4) Francis Bonahon, Xiaobo Liu, "Representations of the quantum Teichmuller space and invariants of surface diffeomorphisms", Geom. Topol. 11 (2007), 889--937.
In the first part of the course, I will give a brief account of symplectic geometry and quantization, and introduce the necessary concepts in hyperbolic geometry. I will then carefully describe the (non-quantum) Teichmueller space and its associated symplectic structure. At this point we will have the necessary ingredients to introduce the quantum Teichmueller space, following the construction of Chekhov and Fock. I will explain how it can be studied using methods from representation theory, as described in the work of F. Bonahon and X. Liu.
Depending on students interest and if time permits, I will also survey its possible relationship with some of the following topics: quantum invariants of knots and 3-manifolds, conformal field theory and TQFT, the study of the mapping class groups.
The plan is to cover chapters 1 and 2 of Hatcher's book. Topics include fundamental group, Van Kampen's Theorem, covering spaces, simplicial and singular homology, Brouwer's fixed-point theorem, the Borsuk-Ulam theorem, and the Jordan-Brouwer separation theorem.
Topics will include elementary properties of linear algebraic groups, their finite-dimensional representations, and their Lie algebras. The Lie algebras of the classical groups will be studied using root systems and Weyl groups relative to a maximal torus. The complete reducibility of finite-dimensional representations will be proved and the Cartan-Weyl highest weight theory of irreducible finite-dimensional representations will be developed. For the classical simple Lie algebras explicit models for the irreducible representations will be constructed. The structure and classification of semisimple Lie algebras will be also covered.
If time permitted, more advanced topics (Kac-Moody algebras, etc) will be discussed.
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.
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)
Certain papers by Y.-Z. Huang, J. Lepowsky and L. Zhang developing logarithmic tensor category theory for modules for a vertex algebra. Parts of the introductory monograph listed above, covered in the course, provide sufficient background for these papers.
J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math., Vol. 227, Birkhauser, Boston, 2003.
I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, 1988.
Please note: The Lie Groups/Quantum Mathematics Seminar, which will meet Fridays at 11:45, will sometimes be related to the subjects of the course. Students planning to take the course should also try to arrange to attend the seminar, although the seminar will not be required for the course.
The first part of the course will cover Chain Complexes, Projective and Injective Modules, Derived Functors, Ext and Tor. In addition, some basic notions of Category Theory will be presented: adjoint functors, abelian categories, natural transformations, limits and colimits.
The second part of the course will study Spectral Sequences, and apply this to several topics such as Homology of Groups and Lie Algebras. Which topics we cover will be determined by the interests of the students in the class.
This is an introductory course in Mathematical Logic aimed at graduate students in mathematics rather than prospective logicians.
In Set Theory, we will discuss basic topics such as ordinals, cardinals and the various equivalents of the Axiom of Choice; as well as more advanced topics such as the club filter and stationary sets. (The latter are needed to prove results such as the existence of 2ℵ1 nonisomorphic dense linear orders of cardinality ℵ1.)
In Model Theory, we will begin with basic results such as the Completeness and Compactness Theorems. Then we will cover some more advanced topics focusing on the structure of the countable models of a complete theory in a countable language.
The course has no formal prerequisites.
- Geometry of the hyperbolic plane
- Discontinuous groups
- Eisenstein series
- Spectral resolution of automorphic forms
- Selberg's trace formula
- Spectral theory of Kloosterman sums
- Applications to L-functions
Students uncertain of their preparation for this course should consider taking 640:421, or consult with the instructor.
More information is on the www.math.rutgers.edu/courses/527/
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.
The course will cover fundamentals of graph theory and the following topics:
- Electrical networks
- Flows, Connectivity and Matching
- Extremal Problems
- Ramsey Theory
- Random Graphs
- Graphs, Groups and Matrices
- Random Walks on Graphs
- The Tutte Polynomial
- 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
- Probability Spaces: construction; monotone class theorems; independence.
- Random variables: distribution; Kolmogorov extension theorem; expectation; types of convergence.
- Laws of large numbers.
- Convergence in distribution and the central limit theorem.
- Brownian motion: construction; basic properties.
- L'evy processes and infinitely divisible distributions: construction.
- Martingales in discrete time.
- Martingales in continuous time.
This course introduces interested students to these fundamental conceptual problems, explains the deficiencies of the currently prescribed remedies, and then develops in detail a mathematical framework which promises to restore a well-defined finiteness to the whole affair.
Then, we discuss more advanced topics, like construction of self-adjoint operators and finding their spectral properties; the relation to PDE and Quantum Mechanics will be treated in detail. Some problems, motivated by current research in nonlinear dispersive equations will be described.
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 microscopic 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, email@example.com