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Research Institute for Mathematical Sciences
Kyoto University

International Workshop on Tropical Algebraic Geometry and Tropical Combinatorics

August 8-21, 2004

Speakers | Program | Abstracts | RIMS | Weather in Kyoto | Welcome to Kyoto | Japan Travel Guide| Palace Side hotel

O r g a n i z e r s:

A.N. Kirillov, M. Noumi


Invited Speakers:

A. Berenstein, (University of Oregon, Eugene, USA), abstract
M. Kashiwara, (RIMS, Kyoto University), abstract
A. Kirillov (RIMS, Kyoto University), abstract
A. Knutson (University of California, Berkeley, USA), abstract
S. Manida (St.Petersburg State University, Russia), abstract
G. Mikhalkin (University of Toronto, Canada), abstract
E. Miller (University of Minnesota, USA), abstract
T. Nakashima (Sophia University, Tokyo), abstract
M. Noumi (Kobe University), abstract
M. Okado (Osaka University), abstract
D. Speyer (University of California at Berkeley, USA), abstract
T. Takagi (National Defense Academy, Kanagawa, Japan), abstract
O. Viro (Uppsala University, Sweden), abstract
Y. Yamada (Kobe University), abstract


P r o g r a m


The Workshop is to be held at RIMS, Room 115
August 9-14 -- lectures according to the schedule below
August 16-21 -- free disscussions

Tentative lectures schedule

time August 9
August 10
August 11
August 12
August 13
10:30 - 11:30 Berenstein Miller Miller Miller
11:40 - 12:40 Speyer Berenstein Mikhalkin Mikhalkin
12:50 - 14:20 Lunch Lunch Lunch Lunch
14:30 - 15:30 Kashiwara Knutson Knutson Noumi Okado
15:45 - 16:45 Kirillov Takagi Speyer Nakashima Yamada
17:00 - 18:00 Viro


A b s t r a c t s


Arkady Berenstein ( University of Oregon, Eugene, USA)
Geometric and unipotent crystals
Abstract: Geometric crystals emerged as a tool for proving certain corollaries from local Langlands conjectures. The approach (developed by A. Braverman and D. Kazhdan) for proving the corollaries requires an action of the Weyl group W (of a reductive algebraic group G) on an algebraic variety X fibered over the Cartan torus T of G. Each such a W-action has to be compatible with the natural W-action on T and agree with a certain volume form on X. A geometric G-crystal on X consists of a family e_i of the multiplicative group actions on X (labeled by the vertices of the Dynkin diagram of G), which actions have to satisfy certain Yang-Baxter type relations. Each action e_i gives rise to an involution s_i that also act on X, and the compatibility relations between e_i's guarantee the braid relations between s_i's, that is, one obtains a desirable W-action on X. The majority of geometric crystals can be constructed by means of unipotent bicrystals that are the varieties equipped with the two-sided action of U, the maximal unipotent subgroup of G. For instance, G itself or any Bruhat cell in G is a unipotent bicrystal, and, respectively, G/U and any Schubert cell in G/U is a geometric crystal, which amounts to a surprising W-action on G/U and on any Schubert cell in G/U.

Arkady Berenstein ( University of Oregon, Eugene, USA)
Totally positive geometric crystals and crystal bases
Abstract: This lecture is devoted to combinatorial aspects of geometric crystals. Our main motivation here is purely combinatorial: construct crystal bases for certain integrable G'-modules, where G' is the Langlands dual group of G. The construction proceeds in two stages. First, to each geometric crystal on a torus we associate an infinite Kashiwara crystal on a lattice. It is important to note that this Kashiwara crystal is free in the sense that the crystal operators \tilde e_i are bijections. Second, in order to obtain the actual crystal bases for G'-modules, we propose a ''truncation'' procedure for so constructed free crystals. This procedure requires the initial geometric crystal to be a unipotent bicrystal which admits a linear function and which is totally positive, where the kind of total positivity we need is a category C+ of algebraic tori with positive rational morphisms and a functor from C+ to the category of sets. To each totally positive unipotent bicrystal (i.e, a unipotent bicrystal in the category C+) that admits a linear function, we associate a crystal basis for an integrable G'-module using a special positive "truncation" of the previously obtained free Kashiwara crystal. In particular, if X=G/U, then the corresponding "truncated" crystal turns out to be the union of all irreducible crystal bases B_\lambda.

Anatol Kirillov (RIMS, Kyoto University)
An invitation to the Generalized Saturation Conjecture
Abstract: I will talk about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and "saturation" properties of the Littlewood-Richardson numbers and certain of their generalizations.

Allen Knutson (University of California, Berkeley, USA)
Multiple flags and scattering of honeycombs
Abstract: Honeycombs are a certain class of tropical plane curves, used to compute in the representation ring of GL(n). Chris Woodward showed that scattering honeycombs off one another gives a self-contained proof that they define an associative ring, and Terry Tao, Woodward, and I related this to the octahedron recurrence. I will explain this, and discuss some later developments due to Andre Henriques, Joel Kamnitzer, and David Speyer:
1. a closed form for the recurrence (Speyer),
2. the connection to Kashiwara's crystals (Henriques and Kamnitzer), and
3. a detropicalization of this octahedron associativity proof using work of Fock and Goncharov.

Grigory Mikhalkin (University of Toronto, Canada)
Enumerative tropical geometry in R^n
Abstract: Enumerative questions in (classical) complex or real algebraic geometry ask to compute the number of curves with given algebro-geometric properties after imposing some geometric constraints. (E.g. we fix the degree and the genus and ask that the curves pass via prescribed points or cycles or have some prescribed tangencies, etc.). In complex geometry these numbers are invariants (e.g. in some cases they are equal to the Gromov-Witten invariants or, more general, to the gravitational descendants). They only depend on the homological data of the constraints as long as the points and cycles are in general position. In real geometry these numbers depend on the choice of configuration but their variation is restricted. It turns out that Tropical Algebraic Geometry supplies tools for answering many of such enumerative questions both over C and over R. The corresponding enumerative questions in Tropical geometry are much simpler to solve, thanks to the piecewise-linear nature of this geometry, yet the resulting numbers are the same. In the talk(s) we establish a correspondence between some tropical and classical problems.

Ezra Miller (University of Minnesota, USA)
Gr"obner geometry of quiver polynomials (I and II)
Abstract: The starting point for this pair of talks is the representation space of an equioriented type A quiver, or equivalently, the space Hom(V) of sequences
V_0 --> V_1 --> ... --> V_n
of linear maps, for a fixed list V of vector spaces V_0,...,V_n (over the complex numbers, say). This representation space is the product of the vector spaces Hom(V_{i-1},V_i) for i = 1,...,n, and it carries a natural action of GL(V) = GL(V_0) x ... x GL(V_n). The orbit closures for this action on Hom(V) determine equivariant cohomology classes that can be expressed canonically as certain symmetric functions introduced by A. Buch and W. Fulton, called `quiver polynomials'. The motivation for studying these objects comes from their connections to:
- algebraic topology, where the quiver polynomials are interpreted as universal formulas for the classes of degeneracy loci for sequences of vector bundle morphisms;
- commutative algebra, where the ideals of polynomials vanishing on the orbit closures are generated by minors in products of generic matrices, and the quiver polynomials are the multigraded analogues of the degrees of these ideals;
- algebraic geometry, where (on top of the above connections) the orbit closures are related to Schubert subvarieties of flag varieties; and
- combinatorics, where the quiver polynomials are expressed via diagrams associated to symmetric functions, particularly Schur functions, Schubert polynomials, and Stanley symmetric functions.
In addition to describing these connections, the goal will be to demonstrate how the GL(V)-orbit closures in Hom(V) give rise *geometrically* to various positive combinatorial formulas for quiver polynomials. The focus in this regard will be on certain degenerations arising from Gr"obner basis theory. Among the resulting combinatorial formulas is one that had been conjectured by Buch and Fulton. The talks are based on joint work with Allen Knutson and Mark Shimozono.

Toshiki Nakashima (Sophia University, Tokyo)
Geometric crystals and affine crystals
Abstract: The notions "geometric crystals and unipotent crystals" are introduced by Berenstein and Kazhdan for reductive algebraic groups. First, we extend it to general Kac-Moody setting and define the geometric and unipotent crystal structures on Schubert cells/varieties associated with Kac-Moody groups. Perfect crystals are initiated by Kashiwara et.al., which are finite crystals associated with quantum affine algebras and play important roles in studying vertex models. We consider certain special positive structures for geometric crystals on affine Schubert cells and by using tropicalization/ultra-discretization procedure we shall show some relations of such geometric crystals with limits of perfect crystals of type affine A, B(C), D.

Masatoshi Noumi (Kobe University)
Tropical Robinson-Schensted-Knuth correspondence
Abstract: I will discuss how the Robinson-Schensted algorithm for Young tableaux can be formulated in terms of piecewise linear and totally positive birational transformations. Also it will be shown that there is a remarkable relationship between the combinatorics of Young tableaux and the discrete Toda equation. This talk is based on "Introduction to Tropical Combinatorics" by Anatol Kirillov and a joint work with Yasuhiko Yamada (Adanced Studies in Pure Mathematics, 40(2004), 371-442).

Masato Okado (Osaka University)
Geometric crystals corresponding to Kirillov-Reshetikhin modules
Abstract: Among finite dimensional modules of quantum affine algebras, there is a distinguished family called Kirillov-Reshetikhin (KR) modules, which are parametrized by two integers k, l ($1 \le k \le n, l \ge 1$). Here n+1 is the number of vertices of the corresponding Dynkin diagram. It is conjectured that any KR module has a crystal base $B^{k,l}$. Recent studies on known cases show that to a family of crystals $\{ B^{k,l} \}_{l\ge1}$ there corresponds a geometric crystal, notion introduced by Berenstein and Kazhdan. In the talk we give these results and some applications to combinatorics of KR crystals.

David Speyer (University of California at Berkeley, USA)
"The first steps in Tropical Algebraic Geometry"

David Speyer (University of California at Berkeley, USA)
Horn's Problem, Honeycombs and Vinnikov Curves
Abstract: The multiplicative version of Horn's problem asks what the possible critical values of three matrices $A$, $B$ and $C$ are, given $ABC=1$. A Vinnikov curve is a projective plane curve which can be written as $\det(xX+yY+zZ)=0$ with $X$, $Y$ and $Z$ positive definite Hermitian matrices; Vinnikov has solved the problem of classifying such curves. We will relate these two problems to each other and pose tropical versions of them. This allows us to reprove Knutson and Tao's criterion for the solvability of Horn's problem in terms of honeycombs and illuminate the origins of honeycombs.

Taichiro Takagi (National Defense Academy, Kanagawa, Japan)
Geometric crystal and tropical R-matrix
Abstract: A tropical R is a totally positive birational map which intertwines the action of "Kashiwara operators" in a geometric crystal. In this talk I will explain about an explicit example of a tropical R. (Joint work with Atsuo Kuniba, Masato Okado, Yasuhiko Yamada, based on IMRN 2003-48(2003), pp.2565-2620)

Yasuhiko Yamada (Kobe University)
Tropical affine Weyl group representation of type $E^{(1)}_n$
Abstract: We give a tropical (=subtraction free birational) representation of affine Weyl group of type $E^{(1)}_n$ ($n=6,7,8$). Its relation to the discrete Painlev\'e equations is discussed.

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