# Workshop Programme

## for period 30 March - 3 April 2009

### Geometric Aspects of Discrete and Ultra-discrete Integrable Systems

30 March - 3 April 2009

Timetable

Monday 30 March | ||||

10:15-11:00 | Satsuma, J (Aoyama Gakuen) |
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Properties of an extended ultradiscrete systems | ||||

We propose an extension of ultradiscretization and apply it to nonlinear inte-grable systems. We also discuss on the properties of the extended ultradiscrete systems and their solutions. |
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11:30-12:15 | Tokihiro, T (Tokyo) |
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Box-Ball system and Toda equation | ||||

A Box-Ball system (BBS) is constructed by a limiting procedure called ultra-discretization from the discrete KdV equation or the Toda lattice equation. In this talk, the relation between the BBS with the periodic boundary condition (PBBS) and discrete Toda equation. In particular, we show the relation between the conserved quantities of PBBS and a spectral curve of the Toda equation. Fundamental cycle of PBBS and initial value problem of PBBS are also discussed. |
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14:15-15:00 | Adler, V (Landau Institute) |
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Classification of discrete integrable equations of Hirota type | ||||

One of possible denitions of integrability of discrete equations is based on the notion of multidimensional consistency. We apply this approach for classification of integrable 3-dimensional equations of Hirota, or discrete KP-type. It is proven, under rather general and natural assumptions, that the list of such equations is exhausted by Hirota equation itself and its three modications. |
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15:30-16:15 | van der Kamp, P (La Trobe) |
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Initial values problems for lattice equations, and integrability of periodic reductions of integrable cases | ||||

I will describe geometrically a way of constructing initial values problems for lattice equations dened on arbitrary stencils. These also provide normal forms for dierence elimination algorithms. By imposing a periodicity condition, the initial value problems yield mappings, or correspondences. If the lattice equation is integrable one expects the periodic reductions to be integrable as well. In a general setting, a lax pair for a lattice equation can be used to construct integrals for the derived mappings. And we can show certain characteristics to grow polynomially as opposed to exponentially for non-integrable systems. |
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16:15-17:00 | Szablikowski, B (Glasgow) |
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An unification of field, lattice and q-deformed soliton systems as integrable evolution equations on regular times scales | ||||

An unied theory of the construction of the bi-Hamiltonian nonlinear evolution hierarchies such as eld, lattice and q-discrete soliton hierarchies, will be presented. I will give a brief review of the concept of time scales, including definitions of -derivative and -integral. A construction of the bi-Hamiltonian structures for integrable systems on regular time scales will be presented. The main result consists on the denition of the trace functional on an algebra of -pseudo-dierential operators, valid on an arbitrary regular time scale. I will illustrate the theory by -dierential counterparts of AKNS and Kaup-Broer hierarchies. The talk will be based on the article: arXiv:0810.0766. (This is joint work with Maciej Blaszak and Burcu Silindir.) |

Tuesday 31 March | ||||

09:30-10:15 | Berenstein, A (Oregon) |
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From geometric crystals to crystal bases | ||||

The goal of my talk is to construct crystal bases (for irreducible modules over semisimple Lie algebras) by means of geometric crystals. Geometric and unipotent crystals have been introduced a few years ago in a joint work with David Kazhdan as a useful geometric analogue of Kashiwara crystals. More recent observations (based on the recent joint paper with David Kazhdan) show that geometric crystals, in addition to providing families of piecewise-linear parametrizations of crystal bases, also reveal such hidden combinatorial structures as 'crystal multiplication' and 'central charge' on tensor products of crystal bases. |
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10:15-11:00 | Nakashima, T (Sophia) |
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Epsilon systems of geometric crystals | ||||

An epsilon system is a set of rational functions on a geometric crystal, by definition, which satises certain relations and conditions for the actions of the geometric crystal operators. We shall introduce their basic properties, e.g., product structures and invariance for tropical R-maps, and see some application and observation. |
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11:30-12:15 | Kedem, R (Illiois, Urbana-Champaign) |
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Quantum affine algebras, cluster algebras, and proof of the positivity conjecture | ||||

Q-systems and T-systems as cluster algebras on the one hand, and the solution of the resulting discrete integrable systems on the other. Among other things, this proves the positivity conjecture of Fomin and Zelevinsky for those systems, as well as providing explicit expressions for the q-characters. [This is joint work with Philippe Di Francesco.] |
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14:15-15:00 | Okado, M (Osaka) |
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KR crystals and combinatorial R-matrices | ||||

Recently the existence of crystal bases for KR modules was shown for all nonexceptional ane types and their combinatorial structures were claried. Our next problem is to calculate the image of the combinatorial R-matrix for any pair of KR crystals. As a rst step, we calculate it for any classical highest weight element in the tensor product of KR crystals Br;k B1;l for type D(1) n ;B(1) n ;A(2) 2n..1. The notion of -diagrams is eectively used for the identication of classical highest weight elements in B1;l Br;k. This is a joint work with Reiho Sakamoto. |
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15:30-16:15 | Inoue, R (Susuka) |
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Tropical geometry and integrable cellular automata I Tropical spectral curve and isolevel set | ||||

We propose a method to study the integrable cellular automata via the tropical algebraic geometry. First we review the theory of tropical curves and Jacobi varieties introduced by Mikhalkin and Zharkov, and apply it to the spectral curve and the isolevel set of the ultra-discrete Toda lattice (UD-Toda) with periodic boundary condition. Next we introduce the interesting close relation between the periodic UD-Toda and the box and ball system. |
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16:15-17:00 | Takenawa, T (Tokyo Uni of Marine Science) |
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Tropical geometry and integrable cellular automata II Bilinear form and Tropical Fay's identity | ||||

The ultra-discrete Toda lattice (UD-Toda) is essentially equivalent to the integrable Box and Ball system, and considered to be a fundamental object in ultra-discrete integrable systems. In this talk, we construct the general solution of the UD-Toda with periodic boundary condition, by using the tropical theta function and the bilinear form. For the proof, we introduce a tropical analogue of Fay's trisecant identity for the tropical spectral curves of the periodic UDToda. As a result, we can also prove that the general isolevel set of the periodic UD-Toda is isomorphic to the tropical Jacobian. |

Wednesday 1 April | ||||

09:30-10:15 | Veselov, A (Loughborough) |
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Yang-Baxter and braid dynamics | ||||

I am going to discuss the Yang-Baxter maps in connection with actions of the braid groups. These two closely related themes are often mixed up, so I would like to emphasize the dierence between them using an interesting example of the braid dynamics related to Markov numbers. This example had recently appeared in dierent contexts in the theory of Frobenius manifolds, Painleve-VI equation and Teichmueller spaces. |
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10:15-11:00 | Kakei, S (Rikkyo) |
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Yang-Baxter maps arising from the BKP hierarchy | ||||

The BKP hierarchy is a hierarchy of soliton equations associated with the spin representation of B1. Starting from the discrete BKP hierarchy, we will construct several 1+1 dimensional discrete soliton equations and discuss its relations to Yang-Baxter maps. |
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11:30-12:15 | Willox, R (Tokyo) |
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Crystal-like structures arising from the dKP hierarchy | ||||

1+1 dimensional time evolutions obtained from the discrete KP hierarchy can be re-interpreted as certain types of Yang-Baxter maps. As these maps are known to be related to geometric crystals, it seems natural to ask whether such crystals also arise within the context of the dKP hierarchy. It will be shown that this is indeed the case and that a particular class of Darboux transformations for the linear problem of the dKP hiearchy gives rise to crystal-like structures. |

Thursday 2 April | ||||

09:30-10:15 | Kuniba, A (Tokyo) |
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Classical and quantum aspects of ultradiscrete solitons | ||||

grable systems such as solitons, action-angle variables crystal base and combinatorial Bethe ansatz led to the inverse scattering transform and a complete solution of the initial value problem of the generalized box-ball systems on both innite and periodic lattices in terms of ultradiscrete/tropical analogues of the tau and the Riemann theta functions. |
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10:15-11:00 | Noumi, M (Kobe) |
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Some determinantal identities for generalized hypergeometric functions | ||||

In this talk, I propose some determinantal identities for generalized hypergeometric functions nFn..1, relating determinants of dierent sizes. I will explain an origin of these identities, and also how one can derive this class of identities in the scope of duality of dierential equations. |
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11:30-12:15 | Kajiwara, K (Kyushu) |
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Ultradiscretization of solvable chaotic maps and the tropical geometry | ||||

We consider a certain one-dimensional solvable chaotic map arising from the duplication formula of elliptic function, which is a generalization of the logistic map. Applying the ultradiscretization, we obtain the tent map and its general solution simultaneously. We then discuss the tropical geometric interpretation of the tent map; it arises from the duplication map of a certain tropical plane biquadratic curve. If time permits, I will mention on the map arising from the m-th multiplication formula of the elliptic function, and recent result on a certain two-dimensional map. |
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14:15-15:00 | Weston, R (Heriot-Watt) |
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Form-factors and dynamical structure factors in the massless XXZ model | ||||

Form factors are useful in the computation of dynamical correlation functions of lattice models. A key application is to quasi-one-dimensional crystals whose dynamical structure factors are directly measurable by neutron scattering experiments. I shall describe the calculation of exact form-factors for the massless phase of the XXZ quantum spin chain. |
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15:30-16:15 | Yamada, Y (Kobe) |
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Lax formalism for the elliptic difference Painleve equation | ||||

A Lax formalism for Sakai's elliptic dierence Painleve equation is presented. The construction is based on the geometry of the curves on P1 P1 and desribed in terms of the point congurations. |
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16:15-17:00 | Iwao, S (Tokyo) |
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Abelian integrals over complex curves and lattice integrals over tropical curves | ||||

The lattice integral over tropical curve is an analog of the Abelian integral over complex curve. Recently, it was found that these two integrals coincide with each other through the method of ultradiscretization. In this talk, we will introduce the application of the lattice integral to the Box-ball systems. |

Friday 3 April | ||||

09:30-10:15 | Schief, WK (TU Berlin) |
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Discrete Laplace-Darboux sequences, Menelaus' theorem and the pentagram map | ||||

The discrete analogue of the classical notion of Laplace-Darboux sequences of conjugate nets was introduced by Doliwa. In view of the associated classication problem, we take a fresh look at this topic and re-formulate the theory in terms of the multi-ratio condition associated with Menelaus' theorem. As an application, we demonstrate that the pentagram map analysed in detail by Ovsienko, Schwartz and Tabachnikov may be regarded as a particular Laplace-Darboux sequence leading to a (non-standard) discrete version of the Schwarzian Boussinesq equation. |
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10:15-11:00 | Nijhoff, F (Leeds) |
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Elliptic solutions of ABS lattice equations | ||||

11:30-12:15 | Korff, C (Glasgow) |
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Crystals and noncommutative Schur functions | ||||

conformal eld theory. The structure constants of this ring are known to be dimensions of moduli spaces of generalized -functions, the main result is that they can be expressed as matrix elements of Schur functions dened over a noncommutative alphabet. The letters in the alphabet are closely related to Kashiwara's crystal operators of the kth-symmetric tensor product of Uqcsu(n). |