Research Training Group in Representation Theory, Geometry and Combinatorics

About the RTG

• EMSW21 postdocs. Three-year postdoctoral positions funded jointly by NSF and the university, with a reduced teaching load of one course per semester, support for summer research in each of the first two years, and additional funds for travel and other research-related expenses. We currently have three EMSW21 postdocs: David Hill, Joshua Sussan, and Michael Rose. Future postdoc openings will depend on the possibility of renewing our current grant, which expires in 2010.
  
• Graduate student fellowships. Fellowship support is avaliable during the period 2004-2010 for a core group of 8 to 10 graduate students whose research interests belong to the areas covered by the RTG. RTG students normally receive six semesters of research fellowship support, and are expected to teach two semesters. Eligibility is limited to US citizens or permanent residents.
  
• Seminars. The RTG runs a joint research seminar, meeting weekly for two hours, featuring research talks by students, faculty and outside visitors. The usual talk format iss an introductory overview in the first hour, followed by a short break and more technical material in the second hour.
  We also run instructional seminars on topics not readily available in the standard courses. In a typical instructional seminar, talks are given by the participating students with guidance from the seminar organizer. Some instructional seminars concentrate on a single topic for the whole semester, others may devote a few weeks to each of several related topics.
  
• Summer workshops. Each summer from 2005 through 2009, the RTG has given week-long workshops featuring lecture series by distinguished invited speakers plus individual talks by workshop participants. The workshops are in the style of a mathematical summer school, intended for a graduate student to postdoctoral level audience. Participation is open to attendees from all institutions. Funding is available to support travel by workshop participants.
  

Faculty members associated with the RTG

The faculty members affiliated with the group are listed below with brief descriptions of their research interests.

Richard Borcherds: I used to work on vertex algebras, infinite dimensional Lie algebras, and automorphic forms. I am currently trying to figure out what a quantum field theory really is.

Edward Frenkel's research centers on representation theory of infinite-dimensional Lie algebras and quantum groups, integrable systems such as the KdV hierarchy, and the geometric Langlands correspondence.

Alexander Givental works in Gromov-Witten theory and its relationships with other subjects such as symplectic topology, singularity theory, mirror symmetry, integrable hierarchies, representations or combinatorics.

Mark Haiman works on combinatorial problems connected with symmetric functions, representations, and algebraic geometry. Some of his topics of current interest are Macdonald polynomials, LLT polynomials, Hecke algebra characters, and quantum groups.

Martin Olsson works on problems in algebraic and arithmetic geometry. Much of his current work is on stacks and their applications to the study of moduli spaces, group actions, and arithmetic.

Nicolai Reshetikhin: In recent years many questions in representation theory, combinatorics and geometry appeared as problems at the interface of these subjects with mathematical physics. Some of them are: representation theory of infinite dimensional Lie algebras and quantum groups, combinatorics of weight multiplicities, invariants of knots and 3-manifolds, geometry of moduli spaces of flat G-bundles over surfaces, etc. This is roughly the direction of my research.

Vera Serganova: I work in representation theory. Right now I am mostly interested in geometric methods such as D-modules, localization and associated varieties. Also working on Lie superalgebras and quantum groups.

Bernd Sturmfels works on polyhedral combinatorics and algebraic geometry. He is particularly interested in computational aspects and applications (e.g. to statistics, optimization and biology).

Lauren Williams is interested in algebraic, enumerative, and topological combinatorics, and their connections with algebraic geometry, representation theory, and physics. In particular, she is interested in total positivity, tropical geometry, cluster algebras, and statistical mechanics.

Two former members of our group are Allen Knutson, now at Cornell University, and Constantin Teleman, now a member of the Research Training Group in Geometry, Topology and Operator Algebras run by Peter Teichner and Robion Kirby.

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