Research Inspiration from Mathematical Sciences |
Research Institute for Mathematical
SciencesKyoto
University |

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),
abstractM. Kashiwara, (RIMS, Kyoto University),
abstractA. Kirillov (RIMS, Kyoto University),
abstractA. Knutson (University of California, Berkeley, USA),
abstractS. Manida (St.Petersburg State University, Russia),
abstractG. Mikhalkin (University of Toronto, Canada),
abstractE. Miller (University of Minnesota, USA),
abstractT. Nakashima (Sophia University, Tokyo),
abstractM. Noumi (Kobe University),
abstractM. Okado (Osaka University),
abstractD. Speyer (University of California at Berkeley, USA),
abstractT. Takagi (National Defense Academy, Kanagawa, Japan),
abstract O. Viro (Uppsala University, Sweden),
abstractY. Yamada (Kobe University),
abstract |

## P r o g r a m |

time | August 9 (Mon) |
August 10 (Tue) |
August 11 (Wed) |
August 12 (Thu) |
August 13 (Fri) |
---|---|---|---|---|---|

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"**

*Abstract:*

**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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Last updated July 1, 2004