Equivariant generalized Schubert calculus and its applications
Department of Mathematics and Statistics, University of Ottawa,
April 28 - May 1, 2016
The last decades have witnessed an intrusion of the methods of algebraic topology into the theory of algebraic groups, torsors and homogeneous spaces. These new methods have led to breakthroughs on a number of classical problems in algebra, which were beyond the reach by earlier purely algebraic techniques.
One striking example of this ongoing trend was the invention of equivariant K-theory by Thomason and then of an algebraic classifying space and equivariant Chow groups by Brion, Edidin-Graham, and Totaro. Merging these results with a notion of an algebraic oriented cohomology introduced by Levine-Morel and Panin-Smirnov have lead to the creation of a family of algebraic equivariant oriented cohomology theories (e.g., equivariant algebraic cobordism) which are actively studied nowadays in view of its rich connections to geometry
Another celebrated example is an application of the T-fixed point approach to study equivariant singular cohomology and K-theory of flag varieties by Kostant-Kumar. Various generalizations of this method to arbitrary equivariant oriented cohomology theories have been recently obtained by Calmes, Cooper, Ganter, Hoffnung, Malagon-Lopez, Ram, Savage, Zainoulline, Zhao, Zhong and others.
The last example we would like to stress is an invention of quantum cohomology in topology and its subsequent algebraic development by Kontsevich-Manin, Fulton, Pandharipande. Numerous applications of quantum cohomology and K-theory to combinatorics and enumerative geometry of homogeneous spaces have been obtained recently by Buch, Lenart, Perrin and others.
The workshop will be mostly focusing on
- matching various (equivariant, quantum) cohomology rings of homogeneous varieties and elements of classical interest in them (e.g. Bott-Samelson resolutions, push-pull operators, duality results, push-forward formulas, T-fixed points) with some algebraic and combinatorial objects.
- applications to the generalized Schubert calculus, e.g. smoothness of Schubert varieties, Littlewood-Richardson coefficients, positivity conjectures.
The workshop will run for three full days and a morning session during the fourth day (Thursday morning - Sunday noon) and will feature a unique combination of 2 introductory mini-courses, 7 one-hour talks given by leading experts and 5 short talks given by graduate students/postdocs. Each mini-course will have three lectures and directed towards graduate students and young researchers:
- Introduction to algebraic equivariant oriented cohomology theories
by Baptiste Calmès, Université d'Artois, France
The general goal of these lectures is to reach a workable knowledge of equivariant oriented cohomology theories, with a focus on computations related to projective homogeneous varieties under the action of a semi-simple linear algebraic group.
I will review classical topological methods and constructions in equivariant cohomology or K-theory. Then, I'll explain the general algebro-geometric framework of equivariant oriented cohomology theories. Finally, I'll introduce the algebra and combinatorics of the computation of such a cohomology theory on split projective homogeneous varieties, in terms of formal group laws and root data.
- Toward quantum cohomology and Schubert calculus
by Anders S. Buch, Rutgers University
Lecture 1: I plan to cover the ordinary (small) quantum cohomology
ring of Grassmannians and Bertram's structure theorems, regarded as
tools to solve classical enumerative problems of counting curves
meeting Schubert varieties. I will also speak about the Gromov-Witten
invariants of flag varieties G/P in various cohomology theories, which
are defined using the Kontsevich moduli space of stable maps.
Lecture 2: I will explain a kernel-span bijection that shows that the
(3 point, genus zero) Gromov-Witten invariants of Grassmannians are
special cases of the classical Schubert structure constants of
two-step flag varieties. I will also explain a more general
construction for cominuscule flag varieties, and the `quantum equals
classical' theorem for equivariant and K-theoretic Gromov-Witten
invariants, which unifies results of Chaput, Kresch, Manivel,
Mihalcea, Perrin, Tamvakis, and the speaker.
Lecture 3: In this lecture I hope to cover Givental's definition of
equivariant quantum K-theory of flag varieties G/P, Mihalcea's
Chevalley formula for the equivariant quantum cohomology of such flag
varieties, and a new Chevalley formula for the equivariant quantum
K-theory of cominuscule flag varieties proved with Chaput, Mihalcea,
- Patrick Brosnan, University of Maryland
- Samuel Evens, University of Notre Dame
- Stefan Gille, University of Alberta
- William Graham, University of Georgia
- Takeshi Ikeda,Okayama University of Science, Japan
- Saeid Molladavoudi, University of Ottawa
- Thomas Hudson, KAIST
- Oliver Pechenik, University of Illinois at Urbana-Champaign
- Arun Ram, University of Melbourne, Australia
- Vijay Ravikumar, Chennai Mathematical Institute, India
- Changjian Su, Columbia University
- Julianna Tymoczko, Smith College
- Gufang Zhao, University of Massachusetts at Amherst
List of participants.