Kailash C. Misra - US grants

This conference on Kac-Moody algebras will be held December 13 - 17, 1988 at North Carolina State University. It is for experts in the field as well as researchers in related areas of mathematics and mathematical physics. The conference is partly expository in nature, with Professors Kac and Moody giving a series of lectures on the mathematical aspects of Kac-Moody algebras. In addition, David Olive will lecture on the physical application of Kac-Moody algebras, and there will be technical lectures by other participants. The theory of Kac-Moody algebras began twenty years ago with the seminal work of Kac and Moody, and has connections with diverse areas of mathematics, including number theory, combinatorics, topology, completely integral systems, and particle physics.

During the last two decades representation theory of affine Lie algebras has led to many interesting and important applications in many areas of mathematics and physics. One such connection has been established with certain two dimensional solvable lattice models in statistical mechanics. The interaction between quantum affine Lie algebras and solvable lattice models has recently led to many new discoveries both in representation theory of affine and quantum affine Lie algebras as well as solvable lattice models and integrable systems. Professor Miwa is a pioneer in this field and possesses a unique overview. He can give a well organized presentation of the major ideas and recent research results, and prescribe future directions for research in this field, during the course of the ten lectures. This project will support an NSF-CBMS Regional Research Conference in the Mathematical Sciences on Application of the Representation Theory of Quantum Affine Lie Algebras to Solvable Lattice Models, to be held June 1 - 5, 1993, on the main campus of North Carolina State University in Raleigh. Professor Tetsuji Miwa will be the principal lecturer/ To stimulate interest and activity in mathematical research, the National Science Foundation each year supports a number of NSF-CBMS Regional Research Conferences in the Mathematical Sciences. Each five-day conference features a distinguished lecturer who delivers ten lectures on a topic on important current research in one sharply focused area of the mathematical sciences. The lecturer subsequently prepares an expository monograph based upon these lectures, which is normally published by the American Mathematical Society or the Society for Industrial and Applied Mathematics, or jointly by the American Statistical Association and the Institute of Mathematical Statistics. Certain features differentiate these conferences from typical research conferences. These are: (1) Focus on a single important and timely area of research by a leading practitioner, (2) Continued effect and local stimulation through regional emphasis, (3) Panel review for quality, breadth, and timeliness, and (4) Published monographs for a wider audience.

9802449 Misra This award supports a conference on "Representations of Affine and Quantum Affine Algebras and Their Applications" at North Carolina State University. This conference aims to bring together experts working in different aspects of affine Lie algebras and quantum affine algebras as well as their applications. The participants will involve senior, mid-level and junior researchers working in this area. Special efforts will be made to assure that the list of participants is diverse. This conference is concerned with a mathematical object called a Lie algebra. Lie algebras arise from another object called a Lie group. An example of a Lie group is the rotations of a sphere where one rotation is followed by another. Lie groups and Lie algebras are important in areas involving analysis of sperical motion.

9970493

The proposer will work on problems in representation theory of quantum
affine algebras, infinite-dimensional Lie algebras and Macdonald polynomials. He will continue to develop the vertex operator approach to symmetric functions. In particular he will use this to study the realization of affine canonical basis in terms of Macdonald polynomials and will also investigate its application to crystal bases. In addition he will use the recent method of quantum wedge modules to study perfect crystal bases.

This project studies quantum affine algebras and vertex operator algebras, which are recent generalizations of Lie algebras and Lie groups. The study of Lie algebras and quantum groups is aimed at revealing more symmetry that existed in nature. This symmetry is crucial to the theory of quantum mechanics and quantum field theory, one of the most important scientific theories in this century.

The theory of Lie algebras, both finite and infinite-dimensional, have
been a major area of mathematical research with numerous applications in
many other areas of mathematics and physics, for example, combinatorics,
group theory, number theory, partial differential equations, topology,
conformal field theory and string theory, statistical mechanics and
integrable systems. In particular, the representation theory of an important class of infinite dimensional Lie algebras known as affine Lie algebras has led to the
discovery of new algebraic structures, such as vertex (operator)
algebras and quantum groups. Both of these algebraic structures have
become important areas of current mathematical research with deep
connections with many other areas in mathematics and physics. This
conference will provide an excellent setting for researchers in mathematics and physics working in the area of Lie algebras, vertex operator algebras and their applications to explore
possible new directions of research in the twenty-first century.
The focus of the conference will be on the following topics:
(i) Finite and infinite dimensional Lie algebras and quantum groups.
(ii) Vertex operator algebras and their representations.
(iii) Applications to number theory, combinatorics, conformal field
theory and statistical mechanics.

Lie algebras are a class of algebras describing continuous symmetries in
nature. They were first introduced by mathematician S. Lie in the
ninteenth century and have been studied by many prominent mathematicians and
physicists since then. During the twentieth century, the theory of Lie
algebras developed rapidly into a main research area in
mathematics with numerous important applications in physics. Vertex
operator algebras and quantum groups are relatively new class of
algebras and can be viewed as far-reaching analogues of Lie algebras.
Vertex operator algebras have been used to solve problems related to discrete symmetries and to number theory. They are also an important ingredient in a
physical theory describing phenomena such as the physical state in which
water, ice and steam coexist and in a physical theory called string
theory which some physicists are using to unify all the forces in the
universe. This conference is on Lie algebras, vertex operator algebras
and their applications and it will encourage mathematicians and
physicists to interact and, to join forces to discover
new frontiers. It will be especially beneficial to graduate
students and junior faculty members who have
just started their careers. We will encourage participation from
graduate students, junior researchers, women, minorities, and persons
with disabilities by giving them priority for financial support.

ABSTRACT

Principal Investigator: Helminck, Aloysius G.
Proposal Number: DMS - 0913405
Institution: North Carolina State University
Title: Conference on: Quantum Groups, Algebraic Groups and Related Topics

Quantum groups have become a comprehensive and mainstream research area in mathematics with numerous applications in mathematics and theoretical physics. It involves research from a broad range of fields, including many types of algebras, like vertex algebras, Kac-Moody Lie algebras, Hecke algebras, etc. Much of the theory of quantum groups is based on results developed for algebraic groups and Lie groups and the interface of these areas continues to provide a base of new research opportunities. The scope of the areas of quantum and algebraic groups is very broad and expertise in the various aspects has been developed all over the world. In order for the subject to continue to develop and flourish it is important that there are periodic international conferences bringing together specialists from all over the world together with young mathematicians and graduate students seeking to specialize in these rich areas. This will create opportunities for exchange of ideas and to expose new frontiers.

This award supports the travel expenses of a group of research mathematicians and graduate students from USA universities to join researchers from Asia, Australia and Europe in a timely international conference on "Quantum groups, Algebraic groups and related topics" at the Peking University, Beijing, China during July 18-23, 2009. Some of the participants will also attend a second related conference hosted at the Chern Institute of Mathematics, Nankai University, Tianjin during July 23-27, 2009 on "Quantum algebras and Physics.'' There are more than half dozen major mathematical departments and research centers in the region such as Peking University, Tsinghua University, Beijing Normal University, Institute of Mathematics, etc. A large number of graduate students from local universities and research centers will attend this conference. The senior invited international experts will outline the future directions of research in quantum groups and algebraic groups. The informal sessions will provide ample opportunity for junior researchers and graduate students to interact with other experts in the field. This conference will bring unprecedented opportunities for graduate students and young researchers to gain insights on the important area of quantum groups and algebraic groups. In particular, the USA graduate students will gain international working experience. It will also help strengthen the ongoing collaboration between USA and Chinese mathematicians.

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).

In recent years deformation of algebraic and topological structures has given rise to important structures such as quantum groups, vertex algebras, and homotopy Lie algebras which in turn have important connections with other areas of mathematics and physics. The representation theory of quantum groups and vertex algebras are better understood than the representation theory of homotopy Lie algebras. An NSF/ CBMS regional conference on "Deformation Theory of Algebras and Modules" is proposed to be held at North Carolina State University during the week of May 16 - May 20, 2011 with Professor Martin Markl of the Mathematical Institute of the Czech Academy as the principal lecturer who will deliver ten lectures on deformations of algebras and modules. Professor Markl will shed new light on these seemingly unrelated algebraic structures. This in turn will generate discussions among participating algebraists, topologists as well as mathematical physicists to explore new frontiers of research in deformation theory. It is hoped that this will lead to research collaborations among participants. In particular, it may result in a better understanding of the representations of homotopy Lie algebras.

A major objective of this conference is to foster new collaborations among algebraists and topologists working on various aspects of deformation theory. The proposed lecture series will be of immense educational value to the participating graduate students and young researchers from universities in the southeastern region of USA. Furthermore, this lecture series will help enhance the research collaborations between researchers in USA and Eastern Europe, and provide international exposure to participating US graduate students and junior researchers. It will also help attract strong graduate students from Eastern Europe to NCSU and other US universities in the southeastern region.

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

Southeastern Lie Theory Workshops will be held October 9-11, 2015 at North Carolina State University and May 23-26, 2016 at the University of Virginia. These workshops are part of the annual Southeastern Lie Theory Workshop Series held at universities in the southeastern region of USA since 2009. The workshop series website is http://pi.math.virginia.edu/lieworkshops/. The aim of the workshop series is to bring together senior and junior researchers, as well as graduate students, in order to build cohesive research groups in various aspects of Lie theory; to stimulate and enhance research collaborations in Lie theory; and to infuse a strong tradition of research and education in the region. Each workshop is centered on a research theme and will feature one or two main speakers, giving two to three expository talks that will be accessible to graduate students and postdocs in the workshop's theme area. In addition, there will be four to six invited talks by other researchers and contributed talks by junior researchers and advanced graduate students. Considerable time will be devoted to informal discussions among participants in small groups to work on specific research problems led by invited team leaders chosen from among the participants.

Algebraic, combinatorial and geometric Lie theory is a major area of mathematical research with important applications to many different areas of mathematics, physics, and computer science. Research in representation theory includes the study of quantized enveloping algebras, quantum function algebras, Kac-Moody Lie algebras, Hecke algebras, canonical bases and crystal bases, vertex operator algebras, Hall algebras, A-infinity algebras, quivers, cluster algebras, Hopf algebras, and Khovanov-Lauda-Rouquier algebras. The research theme for the workshop at NCSU in 2015 will be "Algebraic and Combinatorial Representation Theory" and the research theme for the workshop at UVA in 2016 will be "Algebraic Groups, Quantum Groups and Geometry."