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IAS/Park City Mathematics Institute
Graduate Summer School 2002


The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. Candidates should have completed basic graduate courses, including introductory courses in number theory, modular forms and representation theory.  In general, these students will have completed their second year and in many cases will already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

The main activity of the Graduate Summer School will be a set of intensive short courses offered by the leaders in the field; these lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series.

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. The main activity of the Graduate Summer School is a set of intensive short courses offered by world leaders in their fields. These lectures, which do not duplicate course material available elsewhere, are designed to introduce students to exciting current research in mathematics.  The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

Course Descriptions:

  1. Basic Theory and the Theory of Eisenstein Series
  2. 8 lectures
    Lecturers: A. Borel, J. Bernstein

    Description: SL(2;Z)\SL(2; R) and more generally the space G \G where G is a semi-simple Lie Group and G a lattice, Siegel sets, the decomposition of G on L2(G \G), cuspidal subspace and discrete spectrum, locally symmetric spaces and invariant differential operators. Similar homogeneous spaces of Adèles. Definition and theory of Eisenstein series and in particular the very important proof of their meromorphic continuation via spectral theory (with emphasis on Fredholm theory methods which yield added analytic information such as growth rates which are critical in recent applications).

  3. Langlands–Shahidi Method and Converse Theorems
  4. 8 lectures
    Lecturers: F. Shahidi, J. Cogdell

    Description: The analysis of the constant term of the Eisenstein series by Langlands led him to the L-group, general definition of the L-function of an automorphic form and functoriality. Recent striking developments of this method (Kim–Shahidi) give for example the analytic continuation of the symmetric power (square, cube and fourth power) L-functions of a GL(2) cusp form. This uses Eisenstein series on exceptional groups. Converse theorems have proven to be one of the most powerful methods to show that certain representations are automorphic. In particular using Eisenstein series as above, spectacular new cases of functoriality have been established.

  5. Ramanujan Conjectures and Applications, e.g., Ramanujan Graphs.
  6. 10 lectures
    Lecturers: L. Clozel, J. S. Li, Winnie Li, A. Valette

    Description: The formulation of the general Ramanujan Conjectures—i.e. of bounding (in fact identifying the parameters of) the spectrum of the decomposition of L2(G(k)\G(A)), G being an algebraic group defined over the number field k. For GL(n) quite sharp bound are known using the methods from L-functions and these suffice for many applications. Also certain special cases are known—specifically for forms which can be related to cohomology of Shimura varieties. In the case of definite quaternion algebras the (proven) Ramanujan conjectures have applications to combinatorics and Computer Science, specifically to the construction of optimally highly connected but sparse graphs (Ramanujan Graphs). These "expanders" have a multitude of applications in theoretical computer science. Other applications of these Ramanujan Conjectures are to optimal distribution of points on homogeneous spaces and to counting integral points on such spaces. Ramanujan buildings may be a good student project.

  7. Analytic theory of GL(2) forms and L-functions
  8. 6 lectures
    Lecturer: P. Michel

    Description: The technique of ‘families of L-functions’ has been very successful for proving nontrivial upperbounds for automorphic L-functions at special points (as well as nonvanishing). These via formulae relating the special values to period integrals (Waldspurger, Kudla–Harris) allow one to establish a number of equidistribution results. For example it leads to the solution of Hilbert’s eleventh problem concerning the representation of integers in a number field by integral quadratic forms. There are many other striking applications.

  9. Arithmetic Quantum Chaos
  10. 4 lectures
    Lecturers: Z. Rudnick, A. Terras

    Description: The spectral theory of M = G \SL(2, R)/K offers an instance of quantization of a chaotic Hamiltonian (i.e., the motion by geodesics on a hyperbolic surface) which can be analytically studied in the semi-classical limit. These become questions about automorphic forms and less obviously about their L-functions. The Riemann hypothesis for these would settle most of the pressing problems but much can be achieved unconditionally. There are finite field analogues of these problems for which more can be established.

  11. Unipotent Flows on G \G and Applications
  12. 4 lectures
    Lecturer: A. Eskin

    Description: Ratner’s work on the classification of the ergodic measures for unipotent flows on G \G has striking applications to the diophantine properties of irrational quadratic forms, e.g., the Oppenheim Conjecture and to the spectral statistics of certain integrable Hamiltonians.

Participants in the Graduate Summer School also may wish to become involved in the Undergraduate Program, attend parts of the Research Program, or participate in the programs of the Education component. Graduate students are expected to participate in Institute-wide activities such as the "Cross Program Activities" and may be asked to contribute some time to volunteer projects related to running the Summer Session.

A limited number of graduate students who have not completed the basic courses may attend. These students will attend some graduate level courses and may be involved as teaching assistants in other programs or work as audio-visual assistants.


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