This course is a continuation of 640:501 from Fall 2009. The goal is to give an introduction to core topics in real and functional analysis that every professional mathematician should know.
The choice of topics will be somewhat influenced by what is covered in the 501 course in fall 09, but will focus on concepts and techniques that have broad applications to different areas of mathematics. More concretely, we will discuss different modes of convergences and their applications (including weak convergence); manifestations of completeness from different perspectives ( including some Banach space theory and basic theorems involving bounded linear operations); compactness and applications; and some elementary aspects of spectral theory of (compact) linear operators. All the general ideas will be illustrated in some concrete contexts of applications, which include Fourier series and transforms, ODE's, integral and partial differential equations , and probability.
Although many of the topics to be covered are not on the syllabus of the written qualifying exam, they provide ample space for students to witness the applications of the ideas and tools learned in 501 in a variety of contexts, and to practice problem solving skills.
This will be a continuation of Math 503. We will emphasis on the relationship between classical complex analysis and other related fields (algebraic geometry, geometry, and analysis) through Riemann surfaces.
The theory of Riemann surface is a pillar in 20th century mathematics. It appears in such seemingly diverse areas as integrable systems, number theory, algebraic geometry, and string theory. We would like to concentrate on the interaction between the complex analytic, classical geometric, and algebraic geometry points of view.
The following two parts will be covered: (1) Classical Complex Analysis and (2) Riemann surfaces.
Part 1. Analytic continuation, the monodromy theorem, normal families and Riemann mapping theorem, Picard theorems, harmonic functions and elliptic functions.
Part 2. Introduction to Riemann surfaces and algebraic curves. Hyperbolic geometry and uniformization theorem. Riemann-Roch theorem, Abel and Jacobi theorems.
The reference for the first part:
 Green and Krantz: Function Theory of One Complex Variable, AMS. Ahlfors, Lars V. Complex analysis. McGraw-Hill Book Co.
The reference for the second part:
 Farkas, H & Kra, I.: Riemann Surfaces (2nd ed.), Springer-Verlag
 Forster, O.: Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991
 Narasimhan, R.: Compact Riemann surfaces. Birkhäuser Verlag, Basel, 1992.
 Griffiths, P.: Introduction to Algebraic Curves, American Mathematical Society,
- Riemannian Length and Distance
- The First Variation of Arc Length
- The Levi-Civita Connection
- The Exponential Maps
- Jacobi Fields and Curvature
- Curvature Identities
- Second Variation of Arc Length and Convexity
- Parallel Transports
- Manifolds and Maps
- Global Effects of Curvature
- Vector Bundles and Tensors
- Connections and Differential Forms
- Relative Curvature
- Space Forms
- Riemannian Submersions
- Lie groups ans homogeneous spaces
This course will be a sequel to Math 540, but can also be viewed as a mostly independent course on cohomology and homotopy theory for students who already have had an introduction to homology.
The plan is to start with cohomology in Chapter 3 (the extent of the coverage depending on how far Math 540 gets into this chapter). We will then cover basic results on homotopy groups in Chapter 4, such as the long exact sequences for pairs of spaces and fiber bundles, and will take up a number for further topics that relate homotopy groups to homology and cohomology. We will also select from the additional topics, and will study the homotopy groups of classical groups and the cohomology of fiber bundles.
Depending on available time, the course may end with an introduction to vector bundles and characteristic classes, following a further book in progress by Hatcher on his web site, or Milnor and Stasheff's book Characteristic Classes.
Note: These volumes are out of print. Students may be able to obtain used copies online (be sure it is the second edition) through addall.com or other websites. In the fall, photocopies will be available for purchase.
Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. Representative topics will be:
- Galois Theory Finite algebraic extensions, resolutions of equations by radicals (and without radicals)
- Noetherian Rings Rings of polynomials, Hilbert basis theorem, Dedekind domains, Finitely generated algebras over fields, Noether normalization, Nullstellensatz
- Basic Module Theory Projective and injective modules, resolutions, baby homo- logical algebra, Hilbert syzygy theorem
Yi-Zhi Huang, Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649--706.
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.
Model theory deals with models of axiomatized theories, in particular the algebraic structures, combinatorial structures, and models of set theory, by unified methods. We will cover both the general methods of Model Theory, and some representative applications in each of these three areas.
Topics include the Lowenheim-Skolem theorems, Compactnes, indiscernible sequences, omitting types, countable and uncountable categoricity, model completeness, saturation, and stability.
Applications include the consistency of the Continuum Hypothesis, the nonexistence of measurable cardinals in the universe of constructible sets, the solution of Hilbert's 17th problem, and results on locally finite generalized quadrangles, universal graphs, and classes of permutations with forbidden patterns.
For the theory and some of the applications we follow Wilfrid Hodges' text, Model Theory. The remaining applications are found in various specialized texts and the journal literature, and may include some work in progress.
- The theory of Hecke operators
- Analytic properties of automorphic L-functions
- Rankin-Selberg convolution L-functions
- Symmetric power L-functions
- Non-vanishing on the boundary of the critical line
- Spectral power-moments of L-functions
- Subconvexity bounds of L-functions on the critical line
- Central values of L-functions
- Applications to problems of equidistribution
These lectures will be on Tuesdays and Fridays 12:00-1:20pm in Hill 124.
Previous course topic (Fall 2009): Spectral Theory of Automorphic Forms
This course will be for students interested in analytic number theory as well as for those who would like to learn the basics of spectral theory in the hyperbolic plane. Large part of the subject will be devoted to general topics, but the target is to give (eventually) applications to questions in arithmetic. The material is huge, so if students will like to learn more of special topics I may continue the course in the spring semester 2010. Here are some fundamental topics:
- Geometry of the hyperbolic plane
- Harmonic analysis on the hyperbolic plane
- Fuchsian groups
- Automorphic forms
- Eisenstein series
- Spectral decomposition
- Selberg’s trace formula
- Spectral decomposition of Kloosterman sums
- Quantum Unique Ergodicity Conjecture (with proofs)
- Topics in classical automorphic forms,
- A course in arithmetic,
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 PRIMES. We will learn how to tell whether any given integer is a prime in polynomial time ( thanks to AKS ), really understand the elementary proof of the Prime Number Theorem, and, who knows?, may be one of you will prove the Goldbach conjecture?
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.
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.
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
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.
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.
The course goals consist of design, analysis and implementation of the most commonly used numerical methods for solutions to partial differential equations. We will discuss various discretization schemes such as finite difference, finite element and finite volume methods as well as iterative solvers such as multigrid methods. The course will maintain a balance between in-depth mathematical theories for algorithmic techniques and computer implementations and students will have the opportunity to study not only theoretical backgrounds in developing and understanding the numerical algorithms, but also a hands-on experience to implement the methods.
Matlab (or scilab) will be used for the computational component of the course and a number of source codes will be provided to minimize the coding efforts from students.
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
- Probabilistic methods and random structures
- Spectral graph theory
- Discrete harmonic analysis and applications
- Applications in additive combinatorics and theoretical computer science
This course serves as an introduction to a very active field of research that has attracted much attention recently -- a number of the results I will be discussing are less than five years old.
1. Finite field models:
a) introduction to discrete Fourier analysis
b) counting 3-term progressions (Meshulam)
c)* counting 3-term progressions without Fourier analysis (Lev)
d) functions with small spectral norm (Green-Sanders)
e) uniformity norms (Gowers)
f) Freiman's theorem (Ruzsa)
g) inverse theorem for U3 (Gowers, Green-Tao)
h)* counting 4-term progressions (Green-Tao)
2. Approximate subgroups:
a) Bohr sets
b) counting 3-term progressions (Bourgain)
c)* square-difference free sets (Sarkozy)
d)* sets with the minimal number of 3-term progressions (Croot)
e) inverse theorem for U3 (Gowers, Green-Tao)
f)+ counting general linear configurations (Gowers-W.)
3. Ergodic theoretic approaches:
a) dynamical systems and factors
b) correspondence principle (Furstenberg)
c) another proof of Szemeredi's theorem (Furstenberg)
d)* U^k seminorms (Host-Kra)
e) structure theorem for the seminorms (Host-Kra)
f)+ polynomial extensions of Szemeredi's theorem (Bergelson-Leibman)
4. Long progressions in the primes:
a) transference principle (Green-Tao, Gowers)
b)* some analytic number theory (Goldston-Pintz-Yildirim)
c) putting it all together (Green-Tao)
Topics marked with * would be suitable for student presentations.
Topics marked with + will be included if time permits.