Research Inspiration from Mathematical Sciences	Research Institute for Mathematical Sciences Kyoto University

International Workshop on Tropical and Quantum Geometries

February 13-17, 2012

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) abstract E. Gorsky (Stony Brook University, USA), abstract B. Kim (KIAS, South Korea), abstract S. Iwao (Rikkyo University, Tokyo, Japan) abstract A. Kuniba (Tokyo University, Japan), abstract A. Molev (University of Sydney, Australia), abstract A. Nobe (Chiba University, Japan), abstract S. Payne (Yale University, USA), abstract J. Shiraishi (Tokyo University, Japan), abstract A. Silantyev (Tokyo University), abstract O. Viro (Stony Brook University, USA), abstract S. Yanagida (Kobe University, Japan), abstract

P r o g r a m

Master Classes

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

Workshop

Tentative lectures schedule

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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