Graduate Course Descriptions |
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Publisher: American Mathematical Society; 3rd edition (March 29, 2006)
ISBN-13: 978-0821839621
The beginning of the study of one complex variable is certainly one of the loveliest mathematical subjects. It's the magnificent result of several centuries of investigation into what happens when R is replaced by C in "calculus". Among the consequences were the creation of numerous areas of modern pure and applied mathematics, and the clarification of many foundational issues in analysis and geometry. Gauss, Cauchy, Weierstrass, Riemann and others found this all intensely absorbing and wonderfully rewarding. The theorems and techniques developed in modern complex analysis are of great use in all parts of mathematics.
The course will be a rigorous introduction with examples and proofs foreshadowing modern connections of complex analysis with differential and algebraic geometry and partial differential equations. Acquaintance with analytic arguments at the level of Rudin's Principles of Modern Analysis is necessary. Some knowledge of algebra and point-set topology is useful.
The course will include some appropriate review of relevant topics, but this review will not be enough to educate the uninformed student adequately. A previous "undergraduate" course in complex analysis would also be useful though not necessary.
There are many excellent books about this subject. The official text
will be Function Theory of One Complex Variable, by Greene and
Krantz (American Math Society, 3rd edition, 2006). The
course will cover most of Chapters 1 through 5 of the text, parts of
Chapters 6 and 7, and possibly other topics. The titles of these
chapters follow.
1: Fundamental Concepts; 2: Complex Lines Integrals; 3: Applications of the Cauchy Integral;4: Meromorphic functions and Residues; 5: The Zeros of a Holomorphic Function; 6: Holomorphic Functions as Geometric Mappings.
The examples that will be investigated are the case of the standard contact structure of $S^3$ and the case of the first exotic contact structure on $S^3$ of J.Gonzalo and F.Varela.
The outline of the course is as follows:
1.Definition of Contact forms and structures, Reeb vector-fields.
2.The datum of a vector-field $v$ in the kernel of a contact form $\alpha$. Dynamics, remarkable quantities.
3.The spaces $L_\beta$, $C_\beta$, with $\beta=d\alpha(v,.)$
- a.Manifold structure, Topology.
- b.The examples of $\alpha=\alpha_0$, the standard contact structure on $S^3$ and the example of $\alpha=\alpha_1$, the first exotic contact structure of J.Gonzalo and F.Varela (vector-field $v$ of Vittorio Martino).
4.The variational problem defined by the action functional $J(x)=\int_0^1\alpha_x(\dot x)dt$ on $C_\beta$. Critical points, critical points at infinity, flow-lines originating at periodic orbits.
5.Compactness of flow-lines originating and ending at periodic orbits under some algebraic restrictions.
6.Fredholm issues.
7.Arrows and Computations.
Warning: This course falls short of a full computation of the homology in the case of the first exotic contact structure of J.Gonzalo And F.Varela because the Fredholm issues are not entirely settled: the verification of the tools designed to overcome these issues is an ongoing process.
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.
In this course I plan to present material on the Yamabe problem and a fully nonlinear version of it. I will present one third of the material below with detailed proofs, another one third will be presented with outline of the proofs, while for the remaining one third I will describe the results and introduce some open problems. I will discuss with the registered students to give a final decision on what material goes to which parts.
My initial plan is to give a detailed proof of the Yamabe problem and its solution through the work of Yamabe (1960), Trudinger (1968), Aubin (1976), and Schoen (1984). The positive mass theorem of Schoen and Yau was used in its solution. The proofs of the theorem will be either outlined or provided in details. For most of these material, we will use of the above text books.
I will then present a fully nonlinear version of the Yamabe problem. We will present some Liouville theorems with detailed proofs. In doing so, we will use the method of moving planes, the Bernstein type arguments, the theory of viscosity solutions and in particular Jensen approximations, the Alexsandrove-Bakelman-Pucci inequality. I also plan to present related results in a series of recent joint work with Luis Caffarelli and Louis Nirenberg concerning strong maximum principles and removable singularity of viscosity solutions (part of the material available aspreprints or papers on my webaite, while some more will be available as a preprint at the begining of September). I will also present some existence and compactness of solutions of the fully nonlinear version of the Yamabe problem (part of them with detailed proofs, part of them with outlines of the proofs, and the remaining part without proofs), together with some open problems.
[1] L. Hormander, {\it An introduction to complex analysis in several variables}, Third edition, North-Holland, 1990.
[2] James Morrow and K. Kodaira, {\it Complex Manifolds}, Rinehart and Winston, 1971.
[3] Xiaojun Huang, Lectures on the Local Equivalence Problems for Real Submanifolds in Complex Manifolds, Lecture Notes in Mathematics 1848 (C.I.M.E. Subseries), Springer-Verlag, 2004.
[4] Subelliptic analysis on Cauchy-Riemann manifolds, Lecture Notes on the national summer graduate school of China, 2007. (to appear)
A function with $n$ complex variables $z\in {\bf C}^n$ is said to be holomorphic if it can be locally expanded as power series in $z$. An even dimensional smooth manifold is called a complex manifold if the transition functions can be chosen as holomorphic functions. Roughly speaking, a Cauchy-Riemann manifold (or simply, a CR manifold) is a manifold that can be realized as the boundary of a certain complex manifold. Several Complex Variables is the subject to study the properties and structures of holomorphic functions, complex manifolds and CR manifolds. Different from one complex variable, if $n>1$ one can never find a holomorphic function over the punctured ball that blows up at its center. This is the striking phenomenon that Hartogs discovered about 100 years ago, which opened up the first page of the subject. Then Poincar\'e, E. Cartan, Oka, etc, further explored this field and laid down its foundation. Nowadays as the subject is intensively interacting with other fields, providing important examples, methods and problems, the basic materials in Several Complex Variables have become mandatory for many investigations in pure mathematics. This class tries to serve such a purpose, by presenting the following fundamental topics from Several Complex Variables.
(a)Holomorphic functions, plurisubharmonic functions, pseudoconvex domains and the Cauchy-Riemann structure on the boundary of complex manifolds
(b) H\"ormander's $L2$-estimates for the $\bar \partial$-equation and the Levi problem \noindent
(c) Cauchy-Riemann geometry, Webster's pseudo-Hermitian Geometry and subelliptic analysis on CR manifolds
(d) Complex manifolds, holomorphic vector bundles, Kahler Geomtry.
The emphasis of the course will be on examples of algebraic varieties and general attributes of varieties and morphisms as reflected in these examples. Examples of algebraic varieties arise in many places in physics, topology, geometry, combinatorics and number theory and the examples studied in this course will be often be drawn from other areas of mathematics. Topics include but are not limited to projective spaces, elliptic curves, line bundles, blowups. Brief introduction to the language of schemes will be given at the end of the semester if time permits.
1. Marvin J. Greenberg, J. R. Harper, Algebraic Topology: A First course. Publisher: Westview Press .
2. A. Hatcher: algebraic topology, excellent collection of exercises. $30 in paperback from Cambridge University Press, as well as online here
3. James W. Vick, Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics), Springer.
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)
This course will be an introduction to Geometric Group Theory. There are no prerequisites, except for the most basic notions of group theory. In Geometric Group Theory, finitely generated groups are viewed as metric spaces via the path metrics on their Cayley graphs and their large-scale geometry is studied. The topics covered in this course will include:
- Growth rates of finitely generated groups, including the construction of groups of intermediate growth.
- The basic theory of amenable groups.
- Quasi-isometries and asymptotic cones of finitely generated groups
Text: Pierre de la Harpe, Topics in Geometric Group Theory, Chicago, 2000.
Also, some selected material from the following will be developed and used:
I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988.
J. Lepowsky and H. Li, Introduction to Vertex Operator Algebras and Their Representations, Birkhauser, 2004.
I. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Memoirs Amer. Math. Soc., Vol. 104, Amer. Math. Soc., 1993.
Additional material, including certain research papers, will be handed out.
The Rogers-Ramanujan partition identities and generalizations due to Gordon, Andrews and others have long been of great interest in combinatorial analysis. The classical and still-developing theory of such identities turns out to be deeply related to the representation theory of vertex operator algebras, including currently-unfolding research in vertex operator algebra theory. In this course we will develop the theory of, and prove, natural families of partition identities, using the Rogers-Ramanujan and Rogers-Selberg recursions. We will show how this classical theory suggests, and reflects, ideas and structures in vertex operator algebra theory, structures that we will in fact use to conceptually derive such recursions and study such identities. The aspects of vertex operator algebra theory that we will develop are certain vertex-algebraic structures associated with modules for affine Lie algebras, including intertwining operators among such modules. These vertex-algebraic structures are also deeply related to other branches of mathematics, including finite group theory, the theory of modular functions, and tensor category theory, and to conformal field theory and string theory in physics. We will emphasize a range of potential research problems involving the structures developed in the course.
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.
Commutative algebra is the engine behind algebraic geometry and algebraic number theory. In addition, problems from other fields such as combinatorics or optimization can sometimes be phrased as commutative algebra problems.
This course will be an introduction to the basics of commutative algebra, including localization, primary decomposition, integrality, flatness, and dimension.
We will roughly follows Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. Computational aspects and examples relevant to algebraic geometry will be emphasized, but the only prerequisite is 551/552 or equivalent.
If time permits I am also planning to cover subjects of the equidistribution of integral points on curves (the circle), on surfaces (the sphere) and lattice points in some Euclidean domains.
Students who are not prepared for this course should consider taking 640:421.
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.
Topics:
1. The Newtonian universe (Galileian space and time, point particles, Newton's law of motion, Newton's law of gravitational force, Coulomb's law of electrical force, Lorentz' law of magnetic force; symmetries and conservation laws; other formulations of mechanics: Hamilton, Hamilton-Jacobi, and Lagrange; derivation of the celestial two-body (=Kepler) problem from the N-body problem for Newtonian atoms)
2. The Einsteinian universe (Minkowski's spacetime, Maxwell's electromagnetic field equations, electromagnetic waves, relativistic energy and momentum; and in brief outlinealso: Lorentzian manifolds, Einstein's gravitational field equations, geodesics, black holes, gravitational waves)
3. Limits of validity of the classical theories (the joint Cauchy problem for fields and point particles, the problem of self-interactions; energy and momentum laws and the dawn of quantum theory.)
- Enumeration techniques (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics)
- Theory of finite sets, hypergraphs, combinatorial discrepancy, Ramsey theory.
- Correlation inequalities, Martingales, Probabilistic methods.
- Algebraic methods (applications of linear algebra and higher algebra in combinatorics).
- Fourier analytic techniques.
I also plan to have a parallel reading course, in which students can pursuit a topic of interest on a deeper level
- Foundations-- basic theory of probability spaces, random variables, xpectations;
- Large number laws and the Ergodic Theorem;
- Central Limit Theorem, Infinitely Divisible Distributionsand Stable Distributions;
- Large Deviations;
- Coupling;
- Conditional Expectation;
- Discrete Time Martingale theory.
Probability models from different applied and pure areas will be discussed as examples.
Proof Structure for the Development of Concepts Based on the Real Numbers
- Axioms for the Real Numbers
- Logical Principles
- The supremum concept and useful implications
- Convergence of sequences and series
- 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
Part 2: The difference between theory and practice in quantitative finance is significant. Elements that seem superfluous in the classroom prove challenging in real life -- and the devil is always in the details. This is especially true in high frequency algorithmic trading, where there is a critical interdependence between the alpha models, strategies, and software implementation. Taught by a practitioner who has built two high frequency trading groups, this course roughly follows the stages of such an endeavor, identifying pitfalls and focusing on real calculations. Major topics include the treatment of real data, computation of returns, robust alpha modeling, aspects of strategy design, infrastructure, simulation, and performance analysis. The objectives of the course are twofold: (1) to convey the scope and components of a high frequency algorithmic trading effort, and (2) to provide the student with the tools to rigorously test and implement their own trading ideas. Participants must have a strong programming background and will be required to complete a number of projects.
| and Co-requisites ECE 503 or equivalent strong C++ programming skills (essential), Stat 563 (regression), Stat 565 (time series - co-requisite), and Math 621-622. Please visit the prerequisites page for descriptions of Rutgers undergraduate course prerequisites. A solid understanding of undergraduate probability at the level of the textbook by Sheldon Ross, A First Course in Probability, is especially important. Given this background, the course should be accessible to Mathematical Finance master's degree students and graduate students in Computer Science, Economics, Finance, Engineering, Mathematics, Physics, Operations Research, and Statistics. | "Empirical Market Microstructure" by Hasbrouck (recommended) and "Optimal Trading Strategies" by Kissell and Glantz (optional)The course will be based on the study of simple microscopic models which exhibit complex collective behavior. Of particular interest are emergent phenomena like phase transitions and pattern formation. Some of those phenomena already occur in equilibrium systems, where they can be described via properties of Gibbs measures (ensembles) which do not involve explicitely the dynamics. Examples of equilibrium cooperative phenomena are the boiling and freezing of liquids, like water.
The most intersting cases, on which we will focus, are systems far from equilibrium. These include biological, ecological and social systems. Microscopic models exhibiting behavior resembling that found in the natural world cannot be described without the dynamics, which are generally of a stochastic nature. Examples of such microscopic dynamics are: the voter model, the contact process, and reaction-diffusion models. The connection between the microscopic models and the macroscopic descriptions via deterministic equations, such as the reaction-diffusion equation, will be elucidated.
Students interested in taking this course may contact me at:\\ lebowitz@math.rutgers.edu