Naihuan Jing - US grants

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.

Recently combinatorial methods have played important role in mathematical research. The theory of symmetric polynomials and other (non-symmetric) classical polynomials is particular important due to its applications in topology, representation theory, algebraic geometry, algebraic number theory as well as mathematical physics. Schur polynomials and Schubert polynomials stand out as one of the major combinatorial objects in connections with the enumerative geometry of manifolds, Yang-Baxter equations, Hecke algebras and quantum groups. In order to make this area of investigation accessible to a wide audience we are holding an NSF/CBMS regional conference on Algebraic Combinatorics at North Carolina State University in Raleigh during June 4-8, 2001. The principal speaker is Professor Alain Lascoux, from Universite de Marne-la-Vallee, who will give a series of 10 lectures on ``Multivariate Polynomials''. As a leading figure in algebraic combinatorics, Professor Lascoux (himself and jointly with his coauthors) developed many key concepts and tools in this field. During the 10-hour lectures he will present the theories
of multi-Schur functions, lambda-rings, divided differences as well as applications to Yang-Baxter equations. The lecture notes of Prof. Lascoux will be published by CBMS within one year after the conference. The conference's main purpose is to attract newcomers and the secondary purpose is to provide a venue of discussions for experts.

The interaction between quiver representations and Lie theory has sparkled several important developments in mathematics, for instance Lusztig's introduction of canonical basis for quantum enveloping algebras in early 90's. Recently S. Fomin and A. Zelevinsky introduced a new algebraic structure called the cluster algebra in order to study positivity properties for canonical basis. This new algebraic structure has quickly
found connections and applications in quiver theory, discrete dynamical systems, thermodynamic Bethe Ansatz, generalized associahedra Grassmannians, projective configurations, Poisson geometry and Teichmueller theory, and the quantum cluster algebra is also introduced and studied actively. In order to make this new area of research accessible to a wide audience we are holding an NSF/CBMS regional conference on ClusterAlgebras and their Applications at North Carolina State University in Raleigh during June 13--17, 2006 with Professor Zelvinsky as the principal speaker. The audience will have a chance to listen to one of the founders of cluster algebras to present the fundamental materials of cluster algebras. The lectures aim to introduce graduatestudents and new comers to the field. Specifically the following topics will be covered:

1) Motivation--total positivity, canonical bases, double Bruhat cells;
2) Definitions and first examples of cluster algebras;
3) The Laurent phenomenon;
4) Rank 2 cluster algebras;
5) Y-systems and generalized associahedra;
6) Quantum cluster algebras;
7) Cluster algebras and quiver representations.

In order for an even wider audience to study the subjects the lecture notes of Professor Zelevinsky will be published by CBMS within one year after the conference. Graduate students and young researchers, in particular those from under-presented groups (women, minorities, and persons with disabilities) are strongly encouraged to attend the conference.

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.

NSF/CBMS Regional Conference in Mathematical Sciences

Quiver varieties and crystal bases of quantum affine algebras at NCSU, May 25-29, 2010


The interaction between quiver representations and Lie theory has sparkled several important developments in mathematics, On the one hand Lusztig's canonical basis or Kashiwara's crystal bases for quantum enveloping algebras have been a key propeller for the developments in Lie theory. On the other hand, Nakajima's work on quiver varieties has lead to new connection among representation theory, geometry and mathematical physics.

In order to make this new area of research accessible to a wide audience we are organizing a CBMS conference on "Quiver theory and crystal bases of quantum affine algebras" at North Carolina State University during May 25-29, 2010 with Hiraku Nakajima as the principal speaker. As the inventor of quiver varieties and a leading practioner in this field, Prof. Nakajima will give ten lectures on the subject aiming to introduce graduate students and new comers to the current research and chart possible future directions. We encourage participation from women, minorities, and persons with disabilities by giving priority for financial support.

Representation theory covers fast range of research in mathematics, and has close relationships with geometry, number theory, topology, combinatorics as well as mathematical physics. In order to survey current research and bring in new blood into the field, the International Conference series on Representation Theory was initiated in 1998 in Shanghai by a group of mathematicians from China and the US, since then American mathematicians have actively participated in each of the ICRT.

This award provides partial support for American mathematicians to attend the 5th International Conference of Representation Theory
(ICRT) during August 9--14, 2010, in Xian, China where specialists from Asia, Europe and United States will gather chart future research directions.
Xian, China has been
the cross point of cultural exchanges for more than 2000 years.
The current award contributes to new scientific exchanges between mathematicians from three continents and also promotes deeper mathematical research in representation theory.
The conference offers a great opportunity for young researchers to follow the current stage of active research in representation theory.

Representation theory provides concrete models to study various symmetries in nature and has been a central theme in mathematics and physics. Many research groups world wide do research related to representation theory from various perspectives using different tools. Thus it is especially important that a graduate student working in this area has exposure to some of the breadth of the area. In order to strengthen graduate education in representation theory and help our graduate students gain international working experiences, a joint US-China Summer School on Representation theory will be held at East China Normal University in Shanghai during July 12-26, 2010. The program will feature five short courses given by leading experts on quantum affine algebras and Kac-Moody algebras.

This proposal provides travel supports for a group of US graduate students to attend a 2-week Summer School program in East China Normal University, Shanghai, China, as well as the international workshop at the end of the summer school. In the allocation of travel funds we will give preference to women and other minorities and underrepresented groups. The summer activities will strengthen the graduate education of US students in representation theory by providing unprecedented opportunities for them to gain insights on the mathematics, and it will also provide them with a wealth of international working experiences.

In order to strengthen graduate education in representation theory and help our graduate students gain international working experiences, the proposed project is to support the travel of U.S. graduate students to attend the second US-China summer program during during June 28-July 10 2011 at South China University of Technology. The summer program will feature 3 or more short courses given by leading experts on algebraic, analytic, combinatorial, and geometric aspects of representation theory. About 50 Chinese students from more than half dozen major mathematical departments and research centers in China will attend the joint US-China summer school.

Representation theory has been a central theme in mathematics and physics for the last several decades. It is so broad that almost all major mathematical areas have connections to it. Many research groups world wide do work related to representation theory from various perspectives using different tools. Thus it is especially important that a graduate student training in this area has exposure to some of the breadth of the area.

This project will support a five-day CBMS conference on "Higher representation theory of Kac-Moody algebras" at North Carolina State University in June 2014 with Raphael Rouquier as the principal speaker. The theory of Kac-Moody Lie algebras and their quantum analogs is a timely topic with various applications to other branches of mathematics and theoretical physics. Major developments have been focused on the categorification program which realizes key algebraic structures, such as canonical bases, in terms of category of modules based on combinatorial, algebraic or geometric structures. Categorification can provide deeper understanding of the original algebraic structures and obtain new structures. In categorification of quantum enveloping algebras, Khovanov-Lauda and Rouquier independently introduced the KLR algebra.

As one of the inventors and a leading practitioner in this field, Professor Rouquier will give ten lectures on the subject which aim to introduce workshop participants, including graduate students, to current research and chart possible future directions. Prof. Rouquier will also write up a monograph of the CBMS lectures after the conference. The conference expects to gather about 40 participants. An additional five speakers are invited to assist Professor Rouquier's lectures and to talk about categorification and higher representation theory. The conference will encourage participation from women, minorities, and persons with disabilities.