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 Hahn-Banach theorems.
- The uniform boundedness principle and the closed graph theorem.
- Weak topologies. Reflexive spaces. Separable spaces.
- L^p spaces. Reflexivity. Separability. Dual of L^p. Convolution and regularization.
- Hilbert spaces.
- Compact operators. Spectral decomposition of self-adjoint compact operators.
- The Hille-Yosida theorem.
- Sobolev spaces and the variational formulation of boundary value problems in one dimension.
- Sobolev spaces and the variational formulation of elliptic boundary value problems in N dimensions.
- Evolution problems: the heat equation and the wave equation.
The former puts more emphasis on the theory, while the latter devotes some spaceto working out applications of the theory in some interesting cases, while leaving some full discussion of the theory to references. You may obtain one or both of the texts. I will put these two and some additional books on reserve in the math library:
* Jeffrey Rauch, Partial Differential Equations, Springer, 1997. * G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1976. * F. John, Partial Differential Equations, 4th ed., Springer-Verlag, 1982.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 familarity and experience to the basic phenomena, approaches, and methods in solving initial/boundary value problems in the contexts of the classical prototype linear PDEs of constant coefficients: the Laplace equation, the D'Alembert wave equation, the heat equation and the Schroedinger equation.
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.
0. Hyperbolicity: Definitions of Hyperbolicity in the Theory of Maps, Geometry of Characteristics, Energy Estimates, Domain of Dependence Theorem, Local Well-posedness, Formation of Singularities.
1. The Geometry of Space-time: Causal structure, curvature and gravitation, the energy tensor and the matter equations of motion, the Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit.
2. The Cauchy problem for Einstein Vacuum Equations: The Symbol and Characteristics of EVE. Local Existence in Wave Coordinates.
3. Formation of Singularities: The Penrose Singularity Theorem. Black Holes. Naked Singularities. Cosmic Censorship Conjectures.
4. 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. Witten’s Proof of the Positive Energy Theorem.
5. Reduction under Symmetry: Homogeneous and Isotropic solutions. Spherically symmetric solutions. One- and Two-Killing Field Reductions of Einstein-Maxwell Equations. Ernst Potentials. Kerr and Newman solutions. Wave Maps.
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.
Morse theory is concerned with the relation between the topology of differentiable manifolds and the critical points of real-valued functions defined on those manifolds. The theory has applications to engineering, topology, differential geometry, PDE, geometric group theory, etc. There are even discrete versions of Morse theory that have applications to combinatorics.
The plan is to start with a brief discussion of differential manifolds, followed by lectures from Nicolaescu's book with appropriate detours into Milnor. From the table of contents in Nicolaescu:
-
3.1 The Cohomology of Complex Grassmannians
3.2 Lefschetz Hyperplane Theorem
3.3 Symplectic manifolds and Hamiltonian flows
3.4 Morse theory of moment maps
3.5 S1-Equivariant localization
- 4.1 Basics of complex Morse theory
1.1 Local structure of Morse functions
1.2 Existence of Morse functions
2.1 Surgery, handle attachment, and cobordisms
2.2 The topology of sublevel sets
2.3 Morse inequalities
2.4 Morse-Smale Dynamics2.5 Morse-Floer Homology 2.6 Morse-Bott functions 2.7 Min-Max theory
Course Outline:
- The classical linear groups (complex and real forms)
- Closed subgroups of GL(n, R) as real Lie groups
- Linear algebraic groups and rational representations
- Structure of complex classical groups and their Lie algebras: maximal torus, roots, adjoint representation
- Semisimple Lie algebras: structure and classification
- Highest weight theory for representations of semisimple Lie algebras
- Reductivity of classical groups
- Group Theory: Basic concepts, examples and theorems.
- Groups acting on sets: orbits, cosets, stabilizers.
- Basic Ring Theory: Fields, principal ideal domains (PIDs), matrix rings, division algebras, field of fractions.
- Classification of finitely generated abelian groups, and modules over a PID. Application to linear algebra: rational and Jordan canonical form.
- Bilinear Forms: Alternating and symmetric forms, invariants.
- Categories and functors: Introduction.
Supplementary text: 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.
This is an introductory course in toric geometry. The pace will be dictated by the background of the participants who are expected to have some knowledge of algebraic, differential or arithmetic geometry. We will focus on the basics and examples.
The first part of the course will cover the following topics: Convex rational polytopes, fans and rational polyhedral cones. Toric varieties and morphisms. Toric birational geometry; smooth toric varieties and divisors Homogeneous coordinate ring construction. Singular toric varieties and resolution of singularities
The second part of the course will be tailored to the interests of the students in the class. Possible topics include:
Commutative algebra is broadly concerned with solutions of structured sets of polynomial and analytic equations, and the study of pathways to methods and algorithms that facilitate the efficient processing in large scale computations with such data.
This course will be an introduction to commutative algebra, with applications to algebraic geometry, combinatorics and computational algebra.
Topics:
1) (If needed by audience) Noetherian rings: Rings of polynomials, Hilbert basis theorem, Dedekind domains, Finitely generated algebras over fields, Noether normalization, Nullstellensatz.
2) The first part of the course will treat basic notions and results---chain conditions, prime ideals, flatness, Krull dimension, Hilbert functions.
3) Required material from Homological Algebra--such as the derived functors of Hom and tensor products--will be given in class, not assumed.
4) The other half of the course will study in more detail rings of polynomials and its geometry, and Groebner bases. It will open the door to computational methods in algebra (a few will be studied). Some other applications will deal with counting solutions of certain linear diophantine equations.
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.
Recommended
M. Ram Murty and Jody Esmonde, Problems in Algebraic Number Theory, Springer-Verlag Graduate Texts in Mathematics, volume 190.
Other reference books:
Lang, Algebraic Number Theory, Springer GTM, volume 110
Ireland and Rosen, A Classical Introduction to Modern Number Theory, GTM, volume 84.1. Elementary Number Theory
2. Euclidean Rings
3. Algebraic Numbers and Integers
4. Integral Bases
5. Dedekind Domains
6. The Ideal Class Group
7. Quadratic Reciprocity
8. The Structure of Units
9. Higher Reciprocity Laws
10. Zeta Functions
11. Prime Number Theorem
- 1. The theory of Hecke operators
- 2. Analytic properties of automorphic L-functions
- 3. Rankin-Selberg convolution L-functions
- 4. Symmetric power L-functions
- 5. Non-vanishing on the boundary of the critical line
- 6. Spectral power-moments of L-functions
- 7. Subconvexity bounds of L-functions on the critical line
- 8. Central values of L-functions
- 9. Applications to problems of equidistribution
These lectures will be on Tuesdays and Fridays 12:00-1:20pm in Hill 124.
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.
- Enumeration (basics, generating functions, recurrence relations, inclusion-exclusion, asymptotics)
- Matching theory, polyhedral issues
- Partially ordered sets and lattices, MÄobius functions
- Theory of ¯nite sets, hypergraphs, combinatorial discrepancy, Ramsey theory, correlation inequalities
- Probabilistic methods
- Algebraic methods
Some highlights: Chromatic number of random graphs, (Sharp-) Threshold phenomenon, Limiting distributions, Randomized algorithms, Pseudo-random graphs.
The field of (Molecular) Systems Biology mostly concerns itself with individual cells, or small collections of cells, seen as networks involving DNA, RNA, proteins, metabolites, and small molecules. An example is the study of signal transduction pathways in cells, and their disruption in cancer. It is widely recognized by leading biologists that the typical "reductionist" approach is not powerful enough to describe, analyze, and interpret the complex behaviors of such networks. Quantitative (i.e, mathematical) formalisms, concepts, tools, and models are required, and there is a major role to be played by mathematicians in applying and adapting known theory to model and understand specific systems.
This course will provide an introduction to mathematical techniques as well as to the relevant biology. No background in biology will be expected from students. To make the course accessible to a wide audience, only minimal math prerequisites will be needed in order to follow most of the material.
There are a very large number of possible topics to choose from, and the syllabus will evolve based on student's interest and input. Possible topics include the dynamics of cell signaling networks, including memories, switches, adaptation, and oscillators, as well as biological phenomena of chemotaxis, pattern formation, and neural transmission. If there is interest, we can also discuss synthetic biology.
More information may be found at this website:
http://www.math.rutgers.edu/~sontag/613.html
and/or please contact the instructor by email: sontag@math.rutgers.edu