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

Speakers** |**
Program** |**
Abstracts** |**
RIMS** |**
Weather in Kyoto** |**
Welcome to Kyoto** |**
Japan Travel Guide** |**

## O r g a n i z e r s: |

S.Mori, A.N. Kirillov, S. Okamoto |

## Invited Speakers: |

A. Bondal (IPMU, Tokyo University, Japan)
abstractE. Gorsky (Stony Brook University, USA),
abstractB. Kim (KIAS, South Korea),
abstract S. Iwao (Rikkyo University, Tokyo, Japan)
abstractA. Kuniba (Tokyo University, Japan),
abstractA. Molev (University of Sydney, Australia),
abstractA. Nobe (Chiba University, Japan),
abstractS. Payne (Yale University, USA),
abstractJ. Shiraishi (Tokyo University, Japan),
abstractA. Silantyev (Tokyo University),
abstractO. Viro (Stony Brook University, USA),
abstractS. Yanagida (Kobe University, Japan),
abstract |

## P r o g r a m |

Prof. Atsuo Kuniba "Tropical Bethe ansatz and ultradiscrete integrable systems"

In the lectures elementary exposition will be given on the following topics:

1. Box-ball systems and ultradiscre soliton equations

2. Crystals, combinatorial R and Yang-Baxter equation

3. Tropical (combinatorial) Bethe ansatz and related topics

4. Action-angle variables and inverse scattering tranforms

5. Application to periodic box-ball system

February 6-10, 10:30 -- 11:30, Room 420, RIMS

Prof. Alexander Molev "Classical Lie algebras and
Yangians

We start by developing matrix techniques for constructing families of Casimir elements

for classical Lie algebras. The constructions rely on the Schur-Weyl duality involving

the symmetric group and the Brauer algebra. Then we introduce the Yangians and review

their basic properties. The properties will be applied to derive relations between the

families
of Casimir elements and find their eigenvalues in the irreducible
representations.

Another application of the matrix techniques is a construction of explicit
generators of

the centers of the vertex algebras associated with
the affine Kac-Moody algebras.

February 10, 14:30--15:30; February 13,15 , 10:30 -- 11:30, Room 420, RIMS

Prof. Oleg Viro "Basic Concepts of Tropical Geometry"

1. An introduction to topological aspects of real algebraic geometry

2. Finite type invariants for real algebraic curves

3. Hyperfields

4. Real and complex tropical geometries

5. Tropical varieties as amoebas

February 20-24, 10:30 -- 11:30, Room 111, RIMS

Prof. Sam Payne "Tropical Geometry of Curves"

1. A review of Riemann-Roch and Brill-Noether Theory for curves

2. Divisors, linear equivalence, and Riemann-Roch for graphs

3. Specialization from curves to graphs

4. A tropical proof of the Brill-Noether Theorem

5. Frontiers of tropical Brill-Noether theory

time | February 13 (Mon) |
February 14 (Tue) |
February 15 (Wed) |
February 16 (Thu) |
February 17 (Fri) |
---|---|---|---|---|---|

10:30 - 11:30 | Viro | Yanagida | Viro | Yanagida | Gorsky |

11:40 - 12:40 | Yanagida | Kim | Payne | Viro | Viro |

12:50 - 14:20 | Lunch | Lunch | Lunch | Lunch | Lunch |

14:30 - 15:30 | Shiraishi | Bondal | Silantyev | Gorsky | Yanagida |

15:45 - 16:45 | Iwao | Nobe | No lectures | Molev | Gorsky |

## A b s t r a c t s |

**Eugene Gorsky**
(Stony Brook University, USA)

**Quantum Geometry of Torus Knots,
DAHA representations and plane curve singularities. **

** Abstract:**
A theorem of Y. Berest, P. Etingof and V. Ginsburg states that finite
dimensional irreducible representations of a type A_n rational Cherednik
algebra are classified by one rational number m/n. It turns out that such
representation is tightly related to different invariants of the plane
curve singularity x^m=y^n, and, conjecturally, to the Khovanov-Rozansky
homology of the corresponding torus knot. I will describe some of these
relations and, in particular, explain the surprising symmetry between m
and n. The talk is based on a joint project with A. Oblomkov, J. Rasmussen
and V. Shende.

** Shinsuke Iwao **
(Rikkyo University, Tokyo, Japan)

**Inverse scattering method for integrable piecewise linear maps **

** Abstract:**
Many integrable piecewise linear maps are obtained from well-known
discrete integrable systems, such as the discrete KdV equation, the
discrete Toda equation and the discrete KP equation, through
so-called "ultradiscretization". In this talk, I introduce the
Box-Ball system, which is an integrable piecewise linear map obtained
from the discrete KdV equation. The Box-Ball system has many
"integrable" characteristics, for example a sufficient number of
conserved quantities. Moreover, under some boundary conditions, the
general solutions of the Box-Ball system are expressed as a sum of
theta functions associated with a tropical curve. I will explain the
method to solving the Box-Ball system with tropical geometric
techniques.

**Bumsig Kim**
(KIAS, South Korea)

**Quasimap Theory for Fano toric varieties **

** Abstract:**
We will show that Quasimap theory coincides with
GW theory when the target is a smooth Fano projective toric variety.
This is a joint work with I. Ciocan-Fontanine.

**Alexander Molev ** (University of Sydney, Australia)

**Feigin-Frenkel center for classical types**

** Abstract:**
For each simple Lie algebra g consider the corresponding affine vertex algebra V(g) at the critical level.
The center of this vertex algebra is a commutative associative algebra whose structure was described by
a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas
for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of
explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine
vertex algebras V(g) associated with the simple Lie algebras g of classical types. The approach is
based on the Schur-Weyl duality and leads to explicit constructions of commutative
subalgebras of the universal enveloping algebras U(g[t]) and U(g).

** Atsushi Nobe ** (Chiba University, Japan)

**Tropical Jacobian of a hyperelliptic curve**

** Abstract:**
I plan to discuss a realization of addition in Jacobian of a tropical hyperelliptic curve in terms of the intersection with a curve and its applications to ultradiscrete integrable systems.

**Sam Payne** (Yale University, USA)

**Nonarchimedean geometry, tropicalization, and metrics on curves
**

** Abstract:**
I will discuss the relationship between the nonarchimedean
analytification of an algebraic variety and the tropicalizations of
its various embeddings in toric varieties, with attention to the
metrics on both sides in the special case of curves. This is joint
work with Matt Baker and Joe Rabinoff.

**Junichi Shiraishi** (Tokyo University, Japan)

**Quantum Algebraic Approach to Refined Topological Vertex
**

** Abstract:**
We establish the equivalence between the refined topological vertex of
Iqbal-Kozcaz-Vafa
(or Awata-Kanno) and a certain representation theory of the quantum
algebra introduced by Miki.
Our construction involves trivalent intertwining operators associated
with triples of the bosonic Fock modules. Resembling the topological
vertex, a triple of vectors is attached to each intertwining operator,
which satisfy the Calabi-Yau and smoothness conditions.
(Joint work with H. Awata and B. Feigin. arXiv:1112.6074)

**Alexey Silantyev**(Tokyo University,Japan)

**Generalized Macdonald-Ruijsenaars systems and Double Affine Hecke Algebra
**

** Abstract:**
The Macdonald-Ruijsenaars systems can be considered as commutative algebras of difference operators. One way to prove their integrability is to present these commutative difference operators via so-called Cherednik-Dunkl operators, which belong to the Double Affine Hecke Algebra (DAHA). We use Cherednik-Dunkl operators to obtain generalized (deformed) Macdonald-Ruijsenaars systems. We construct appropriate submodules of DAHA, which were obtained by Kasatani for some special cases. Considering Cherednik-Dunkl operators in the corresponding factor-representation we derive the generalized (deformed) Macdonald-Ruijsenaars systems, obtained by Sergeev and Veselov for the A series. This is a joint work with Misha Feigin.

**Oleg Viro** (Stony Brook University, USA)

**Hyperfields and tropical geometry **

** Abstract:**
I plan to discuss basic notions related to
hyperfields (fields with multivalued addition),
constructions that give rise to hyperfield, and,
in particular, degenerations (dequantizations),
tropical geometry and its extensions over tropical degenerations of the
real and complex fields.

**Shintarou Yanagida** (Kobe University, Japan)

**Fourier-Mukai transforms and wall-crossing of Bridgeland's stability conditions **

** Abstract:**
The behavior of Gieseker stability under Fourier-Mukai transform
is an important subject in the study of moduli
of semi-stable sheaves on surfaces.
The main object of this talk is the translation of this subject
into the language of Bridgeland's stability conditions.
I will start with the introduction of Bridgeland's stability conditions.
Focusing on the case of abelian surfaces,
various concrete examples will be explained.
If there is enough time,
I will also explain some results on the structure of moduli spaces,
following the collaborations
with Hiroki Minamide and Kota Yoshioka
(arXiv:1106.5217, 1111.6187).

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Last updated February 10, 2012