The workshop ``Algebraic Monoids, Group Embeddings and Algebraic Combinatorics" will take place at the Fields Institute, Toronto, Ontario on July 3--6, 2012. It consists of two major components. First of all, there will be 3 minicourses on introductory topics aimed at graduate students staggered throughout the four days. These tutorials will introduce the necessary background for the remaining research talks, which form the second component of the workshop. The organizers of the workshop are Mahir Can (Tulane University), Zhenheng Li (University of South Carolina), Benjamin Steinberg (City College of New York) and Qiang Wang (Carleton University, Canada).
The theory of algebraic monoids, originated independently by M. Putcha and L. Renner in 1980, is a natural synthesis of algebraic group theory (Chevalley, Borel, Tits) and torus embeddings (Mumford, Kempf, et al). It is a significant part of the theory of spherical embeddings (Brion, Luna, Vust) and horospherical varieties (Popov, Vinberg). Symmetric varieties (De Concini, Procesi) are closely related to algebraic monoids.
Algebraic combinatorics studies discrete structures arising from an algebraic context. It is a broad and important discipline of mathematics with applications in quantum chemistry, statistical biology, statistical physics, theoretical computer science and so forth. In recent years there has been a tremendous progress on the combinatorial aspects of the above mentioned embedding theories, relating their underlying algebraic structures to the classical notions in algebraic combinatorics.
The workshop will provide a unique opportunity to bring together some of the principal investigators from different countries, junior researchers, and graduate students. It will outline future directions of research on algebraic monoids, group embeddings, and algebraic combinatorics. This workshop will prepare graduate students for research in related areas and suggest thesis projects. More information about the workshop can be found at:
www.fields.utoronto.ca/programs/scientific/12-13/monoids/index.html
This grant will provide funding for U.S. researchers to participate and benefit from this timely international workshop, which will stimulate research on the interplay between algebraic monoids, group embeddings, and algebraic combinatorics. It will help establish collaborations among the participants, including students and faculty from non-Ph.D.-granting institutions, as well as female students and professors. It will potentially help establish connections between U.S. and international academic institutions.
Summary of the Outcomes This international workshop, funded by the National Science Foundation and the Fields Institute gave the mathematical community a timely opportunity to enhance the visibility of the highly interdisciplinary fieldof algebraic group embeddings and algebraic monoids. It not only provided a unique opportunity to bring together some of the principal investigators, junior researchers and graduate students in algebraic monoids, group embeddings, representation theory, and algebraic combinatorics, but also increased the synthesis of these areas. An important aspect of the organized event was not only its scientific quality but also its opennes to all participants from different backgrounds and genders. The organizers strove for and achieved the diversity and equality in the participation. The central goal of the workshop was to seek a better understanding of the connections between algebraic monoids and the geometry of embeddings. Reductive monoids are group embeddings of a particular kind. They can be understood as reductive monoids, and also as spherical varieties. Their structure is sufficiently rich to provide hints and examples for more general problems about spherical varieties. This workshop has catalyzed the interactions between combinatorics and algebraic monoids, which resulted in advancing the understanding of the representation theory of an arbitrary (reductive) group, at the same time enriching the related combinatorics. Among the topics that the participants discussed (which were not limited to only these) were the structure and representation theory of reductive algebraic monoids, monoid schemes and applications of monoids, monoids related to Lie theory, equivariant embeddings of algebraic groups, constructions and properties of monoids from algebraic combinatorics, endomorphism monoids induced from vector bundles, as well as the Hodge–Newton decompositions of reductive monoids. During and shortly after our workshop, new collaborations were initiated, and some important problems of the aformentioned topics were solved. Guided by the senior and more experienced researchers, student participants of the workshop have gained experience, and they have had the opportunity of interacting with the most influential researchers in these fields. Since the event, many of these students obtained their Ph.D degree, and have gotten employment. The proceedings of the workshop has been published by Springer as the Volume 71 of the Communication Series of Fields Institute. The volume has made its way to university libraries and it is available for purchasing at major bookstores.
