Brian Parshall - US grants

This project is concerned with the Kazhdan-Lusztig conjecture in the modular representation theory of semisimple algebraic groups. The principal investigators will make use of quasi-hereditary algebras and highest weight categories in attacking this problem. Questions involving representations of finite groups and Lie algebras will also be considered. The central problem to be considered in this project is the Kazhdan-Lusztig conjecture which states that certain invariants attached to an algebraic group can be calculated from polynomials. This is a difficult and central problem in group theory. It also has implications in other areas of mathematics.

Abstract
Parshall

This award will partially support a conference to be held at the University of Virginia, May 21-25, 2000. The conference will focus on the interaction between infinite dimensional Lie theory in mathematics and conformal field theory in physics. There has been considerable activity in this area in the past two decades that has been enriching for both sciences. The conference will be structured around three mini-courses delivered by three international leaders in their respective areas, and complemented by additional one hour talks. This is an important area, and the conference will allow many graduate students and young investigators to gain first hand knowledge from the areas' leaders.

Principal Investigators: Brian Parshall, David Evans, Jeffrey Holt, Olivier Pfister and
Harold Ward
Proposal Number: DMS- 0308708
Institution: University of Virginia

Abstract: Coding theory and quantum computing

The Department of Mathematics and the Institute for Mathematical Sciences at the University of Virginia will sponsor an interdisciplinary conference on quantum computing and coding theory at the University of Virginia on May 20-26, 2003. The conference will begin with an instructional workshop consisting of three mini-courses presented by senior experts aimed at providing an introduction to quantum computing and coding theory. The target audience for the workshop will consist of advanced graduate students and junior faculty. Quantum computing has connections with many different areas of mathematics, so that the educational impact of the program on mathematicians attending the conference should be considerable. The workshop will be followed by a short conference featuring invited speakers discussing recent developments. Other conference participants will also have the opportunity to present contributed talks on their work.

One cannot overstate the potential impact of quantum computing. Current methods of encrypting data for private transmission are secure not because they are unbreakable but
rather because today's computers and algorithms do not have the speed or efficiency to break the code. When developed, quantum computers will make current encryption methods insecure and obsolete. These computers will also provide new means of quickly searching databases of information, which would have many important applications in
biotechnology and data mining.

The funds are being requested to organize a meeting on infinite-dimensional aspects of representation theory and applications. The conference will be held at University of Virginia in May 2004.
The meeting will feature three mini courses meant to help graduate students to enter new and exciting areas of mathematics, as well as plenery talks by many leaders of the field. More than half of the requested funds will cover the student and junior faculty participation.

ABSTRACT: SOUTHEASTERN LIE THEORY WORKSHOP SERIES

Lie theory represents a major area of mathematical research. Besides its increasing importance within mathematics to geometry, topology, combinatorics, and algebra, it has important applications outside of mathematics in areas such as physics, chemistry and computer science. Over the last 30 years, the universities in the southeastern region of USA have hired a steady stream of mathematicians working on different aspects of Lie theory and its applications. This project focuses on using the current infrastructure to build a sense of regional unity and foster cohesive research collaborations in the region. Annual regional workshops (for 3-4 days) in Lie theory will be held. These workshops will bring together senior and junior researchers (including graduate students), and create an academic environment which will encourage the exchange of ideas along with the awareness of research opportunities.

This project will also setup a regional network of researchers in Lie Theory to improve communication and collaboration. This network will significantly strengthen the recruitment and retention of graduate students in mathematics, the placement of graduate students and postdoctoral fellows who have been trained in the southeast, the visibility of area mathematics departments at both the national and international levels, and the quality of graduate education and research in mathematics. Funding from this grant will support the first three annual workshops in Lie theory. These will be held at the North Carolina State University (2009), the University of Georgia (2010), and the University of Virginia (2011).

The PIs will build on their recent work on bounds on cohomology groups for semi-simple algebraic groups. For large primes, this both directly and indirectly involves quantum groups, in methods pioneered by the PIs. There are immediate consequences, from a classical "generic cohomology" theory, obtained by the PIs years ago in collaboration with other authors, to asymptotic estimates for bounds for finite groups of Lie type, with more modern methods sometimes allowing these to be improved to actual bounds. This program has been carried out for degree 1 cohomology, and to the point of generic cohomology for all higher degrees. These results lead to important new questions involving the rates of growth of the cohomology spaces. Again the issues in the structures are intertwined, and the study of algebraic groups is a decided advantage for analyzing the quantum case, with both structures contributing to estimates for the growth of sizes of Kazhdan-Lusztig polynomials. For algebraic groups, there are many open questions, especially that of a polynomial rate of growth. Such growth rate issues occur broadly in mathematics, especially in algorithmic issues, and are prominent in theoretical computer science. The PIs will also continue to study Koszul properties for the finite dimensional algebras which come up in the representations of semi-simple groups and quantum groups. Koszul structures arise, or may be conjectured, from geometric considerations (perverse sheaves and their filtrations), but the authors have been pushing entirely algebraic methods into areas where geometry may not directly apply. Applications to the graded and filtered structures of standard (Weyl) modules have already been found, with additional results expected, as well as applications to filtrations of resolutions and cohomology groups of these modules. Often this work uses a conjecture due to Lusztig, which has been proved true for large primes. These studies exhibit deeper consequences of the conjecture (and could even provide insight for establishing it in more cases). Finally, the PIs will continue their work in small characteristic and the calculation of support varieties.

This proposal concerns the representation and cohomology theory of important algebraic structures, including semisimple algebraic groups and their finite and infinitesimal subgroups, quantum groups, and Kazhdan-Lusztig polynomials. These structures are interrelated, so can be profitably studied together. A central aspect includes representations of important classes of finite groups. Over the past century, similar theories for continuous groups played a large role in quantum theory and the theory of elementary particles. Their finite analogs have already proved valuable in the design of communications and data storage devices. Though this finite theory remains very incomplete, it will surely be even more important in the future. This project also points to the future in its manifold involvement of graduate and undergraduate students.

The Southeastern Lie Theory Workshops funded under this grant will take place at College of Charleston on October 13-15, 2012 and at Louisiana State University in late spring of 2013. Algebraic, analytic and geometric Lie theory is a major area of mathematical research with important application to many different areas of mathematics, physics, computer science, etc. There is a critical mass of mathematicians working on different aspects of Lie Theory and their applications in the southeastern region of U.S. Three years ago some of the PIs established a consortium called "Southeastern Lie Theory Network" to enhance regional research collaboration and provide a stronger educational environment for graduate students and junior researchers. Toward this end, they initiated an annual workshop series. The first three workshops were held at member institutions: North Carolina State University (NCSU) (2009), University of Georgia (UGA) (2010) and University of Virginia (UVA) (2011) with a follow up workshop to be held at North Carolina State University during April 21-22, 2012. Based on the success of these workshops the PIs plan to continue this series with a workshop at College of Charleston (CoC) in 2012 and one at Louisiana State University (LSU) in 2013. The research themes for these two workshops will be: "Vertex Algebras, Conformal Field Theory, and Integrable Systems" (2012, CoC) and "Noncommutative geometry and representation theory" (2013, LSU).

This funding will provide the needed support to organize these two workshops. Each workshop will feature one or two main speakers, giving 2-3 expository talks in the theme area, accessible to graduate students and postdocs. In addition, there will be 3-4 invited talks by other researchers. There will be ample time for informal discussion among participants. Considerable time will be devoted for informal discussion among participants in small groups (following the "AIM Model") to work on specific research problems led by invited team leaders chosen from among the participants. The consortium and workshop series has and will continue to stimulate and enhance research collaboration in Lie theory in the southeastern region of the U.S. In the long run, the proposed activity will foster a strong regional tradition of research and education, in turn helping mathematics departments attract more minorities and underrepresented groups to mathematical sciences in general. The website for this conference is:
http://coxbl.people.cofc.edu/Southeastern%20Lie%20Theory%20conference