Dear all,
We'd like to bring the following courses available Fall 2014 to your graduate students' attention, could you please forward this announcement to them?
MATH 678 - Modular Forms
Credits: 3
Requirements & Distribution: BS
Advisory Prerequisites: MATH 575, 596, and Graduate standing.
BS: This course counts toward the 60 credits of math/science required for a Bachelor of Science degree.
Repeatability: May be repeated for credit.
Instructor: Lagarias
Description:
Modular forms involve a wonderful overlap of arithmetic, algebra, analysis and geometry. This is a basic course on modular forms, expected to take an analytic viewpoint, but covering algebraic aspects. It will cover the modular group, classical modular forms, (holomorphic and non‐holomorphic) Eisenstein series, and related spectral theory for SL(2, R). This will include Hecke operators, and the connection to Dirichlet series with Euler products. There will be a discussion of the adelic viewpoint and the connection with representation theory. Applications may include theory of partitions, representations of quadratic forms, connections to elliptic curves, with and without complex multiplication. Some other possible subjects: mock theta functions, mock modular forms, weakly holomorphic Maass forms The textbooks cover more material than the course can cover.
TEXT: Automorphic Forms and Representations by Daniel Bump A First Course in Modular Forms by F. Diamond and J. Shurman, 978‐00387‐27226‐9
------------------------------
MATH 682 - Set Theory
Credits: 3
Requirements & Distribution: BS
Advisory Prerequisites: MATH 681 or equivalent.
BS: This course counts toward the 60 credits of math/science required for a Bachelor of Science degree.
Repeatability: May not be repeated for credit.
Instructor: Blass
Description:
This will be a course in combinatorial set theory, also known as infinitary combinatorics. Its central theme is that rather elementary structures on infinite sets can have surprisingly rich properties. An easily stated illustration of the sort of thing I have in mind is (a special case of)Ramsey's Theorem: If X is an infinite set and if every two elements of X are joined by a red or green thread, then there is an infinite subset Y of X such that all threads joining its elements are the same color.
In the first part of the course, I'll develop several combinatorial results that essentially involve only the smallest infinite sets, the countably infinite ones. I'll also discuss the so‐called compactness phenomenon, which relates the behavior of infinite sets and large finite sets. For example, the infinite Ramsey theorem quoted above implies various finite versions, including some that cannot be proved without a detour into the infinite.
In the second part of the course, I'll describe some of the new phenomena that occur in uncountable sets. Here is one example: If X is an uncountable, well‐ordered set and if a function f assigns to each member x of X except the first some earlier member f(x), then some single member must be f(x) for uncountably many distinct x. I'll also discuss uncountable analogs of some of the countable phenomena from the first part of the course. For example, in Ramsey's theorem quoted above, how large must X be if we want to guarantee an uncountable Y? Answer: The cardinality of the continuum is not large enough, but any larger cardinal is. If time permits, I plan to discuss properties of the "exponential" function that maps the cardinality of a set X to the cardinality of the family of all subsets of X. In addition to the easily verified (weak) monotonicity, this function has some surprisingly subtle additional properties. There will not be time in the course to treat independence results. Such results will be mentioned where appropriate but not proved. Thus, this course will be disjoint from the versions of 682 offered in some previous years. Students who have had one of these earlier versions and wish to also take this version are welcome to do so, but will have to register under a reading‐course number because 682 is not officially repeatable for credit.
The set‐theoretic prerequisites for this course are minimal. Math 582 is more than enough. I'll briefly review the necessary material, basic cardinal and ordinal arithmetic, in class. So the only real prerequisite is the "mathematical maturity" ordinarily presupposed in graduate courses.
Grading will be based on several homework assignments.
TEXT: There will be no textbook. I plan to put on reserve in the library several books whose union includes most of the course material.
------------------------------
MATH 703 - Topics in Complex Function Theory I
Credits: 3
Requirements & Distribution: BS
Advisory Prerequisites: MATH 604 and Graduate standing.
BS: This course counts toward the 60 credits of math/science required for a Bachelor of Science degree.
Repeatability: May not be repeated for credit.
Instructor: Sibony
Description:
This is an introductory course on recent developments in holomorphic dynamics in several variables. The emphasis will be on pluripotential methods (positive closed or ddc‐closed currents). For simplicity, we will first focus first on the dimension 2‐case: Rigidity results for polynomial automorphisms of $C^2$ and automorphisms of positive entropy of compact Kähler surfaces. We will then discuss the notion of entropy for meromorphic maps on compact Kähler manifolds and it's relation with dynamical degrees. Finally, we introduce the theory of super‐ potentials, which permits to develop a calculus on positive closed currents of bi‐degree (p,p). This is an essential tool for concrete equidistribution problems in holomorphic dynamics. If time permits, we will discuss some analogies with the dynamics of (singular) foliations by Riemann‐surfaces and with Nevanlinna's theory of value distribution.
TEXT: No text required
------------------------------
MATH 756 - Advanced Topics in Partial Differential Equations
Credits: 3
Requirements & Distribution: BS
Advisory Prerequisites: MATH 597 and Graduate standing.
BS: This course counts toward the 60 credits of math/science required for a Bachelor of Science degree.
Repeatability: May not be repeated for credit.
Instructor: Bieri
Description:
Prerequisites: Some knowledge in partial differential equations and differential geometry. Partial differential equations (PDE) on manifolds with rich geometrical features are studied in pure mathematics to unravel the structures of their solutions and the spaces they live in. PDE describe phenomena in the real world including physics, medicine, biology or economics. They have become essential to science, technology and to modern life. In general relativity (GR) the Einstein equations describe the laws of the Universe. GR unifies space, time and gravitation. A spacetime in GR is a Lorentzian manifold where the metric solves the Einstein equations. They can be written as a system of nonlinear, second‐order, hyperbolic PDE. The unknown is the metric. Typical physical questions are formulated as initial value problems for the Einstein equations under specific conditions. The solution will lay open the geometry of the resulting spacetime. Today, the methods of geometric analysis have proven to be most effective to investigate these structures. In this course, we introduce some of these methods which are universal and can be applied to other PDE outside GR.
This course will be taught as a mixture of lectures and seminarstyle student work. The students with guidance of the professor will explore some of the important topics in mathematical GR, and will also work through some of the latest results in research. First, we will introduce the spacetime as a solution of the Einstein equations. Then we discuss topics from linear and nonlinear wave equations on flat and on curved backgrounds. Along the way, the role of curvature in GR will be given special attention. We will study the initial value problem in GR. Finally, we will address questions in modern research on gravitational waves and their geometric‐analytic structures. These are produced during extreme events in our Universe like supernovae and when binary black holes merge. These waves are expected to be seen in experiments in the near future. This course features an outreach component. Towards the end of the semester, students will be asked to present a topic they learnt to high school students. This will be optional, and the participating students will work in groups. It is important to communicate intricate developments in mathematics and science to the public. Therefore, the students will be asked: How do you explain this topic to a broad public? What would you like to learn about it in an exhibit? And how?
TEXT: The Cauchy Problem in General Relativity by Hans Ringstroem, 2009, Optional
Thank you,
Stephanie Carroll
Graduate Program Coordinator
Mathematics Department
University of Michigan
Phone: (734) 615-3439
Fax: (734) 763-0937
Mathematics Department
University of Michigan
Phone: (734) 615-3439
Fax: (734) 763-0937
