M. Yoshida (Kyushu University)
Speakers:
K. Hikami (University of Tokyo)
N. Kawanaka (Osaka University)
A.N. Kirillov (Nagoya University and Steklov Institute, Russia)
A. Kuniba (University of Tokyo)
K. Mimachi (Kyushu University)
H. Murakami (Waseda University)
T. Nakanishi (Nagoya University)
M. Nishizawa (Waseda University)
M. Noumi (Kobe University)
T. Tokihiro (University of Tokyo)
M. Taneda (Kumamoto)
H. Terao (Tokyo Metropolitan University)
T. Terasoma (University of Tokyo)
K. Ueno (Waseda University)
H.F. Yamada (Hokkaido University)
Y. Yamada (Kobe University)
Schedule
August 23, Monday
August 24, Tuesday
August 25, Wednesday
August 26, Thursday
August 27, Friday
Schedule
(all talks in Main Lecture Hall 509 (except for Monday afternoon
in Room 111), refreshments in Room 522, computer
facilities Room 109)
Monday, August 23
10:3011:30 
Hiroshi Umemura, Painleve equations in the past
100 years, abstract 
11:3013:30 
lunch break 
13:3014:30 
Masaaki Yoshida, Uniformizations of pointconfiguration
spaces, abstract 
14:4515:45 
Anne Schilling, Levelrestricted rigged configurations,
abstract 
16:0017:00 
Atsuo Kuniba, Soliton cellular automata associated with finite
crystals, abstract 
Tuesday, August 24
9:3010:30 
Stephen C. Milne,
Transformations of U(n+1) multiple basic hypergeometric series,
abstract 
10:4511:45 
Etsuro Date (joint with S.S. Roan), On quotients of
the Onsager alagebra, abstract 
11:4513:30 
lunch break 
13:3014:30 
Masatoshi Noumi and Yasuhiko Yamada, A realization of Weyl
groups and combinatorics of special polynomials,
abstract 
14:4515:45 
Noriaki Kawanaka, A qCauchy identity for Schur functions
and complex reflection groups, abstract 
16:0017:00 
Michimoto Nishizawa (joint with Kimio Ueno), Integral solutions of
hypergeometric qdifference systems with q=1,
abstract 
17:1518:15 
Hitoshi Murakami (joint with Jun Mirakami), The colored Jones
polynomials and the simplicial volume of a knot,
abstract 
Wednesday, August 25
9:3010:30 
Stephen C. Milne (joint with Verne E. Leininger),
Some new infinite families of eta function identities,
abstract 
10:4511:45 
Paul Terwillinger (joint with Tatsuro Ito and Kenichiro Tanabe)
A qanalog of the Onsager algebra,
abstract 
11:4513:30 
lunch break 
13:3014:30 
Jan Felipe van Diejen, Combinatorial formulas for qspherical
functions associated to rankone (quantum) symmetric spaces,
abstract 
14:4515:45 
Tetsuji Tokihiro, Box and ball systems as ultradiscrete
soliton equations, abstract 
16:0017:00 
Tomohide Terasoma, Selberg integral and multiple
zeta values,
abstract

17:1518:15  Makoto Taneda, Representation of Umemura Polynomials,
abstract 
Thursday, August 26
9:3010:30 
Stephen C. Milne, Infinite families of exact sums of squares
formulas, Jacobi elliptic functions, continued fractions, and Schur
functions, abstract 
10:4511:45 
Nobushige Kurokawa, Multiple sine functions I.
Basic properties, abstract 
11:4513:30 
lunch break 
13:3014:30 
Kazuhiko Aomoto, Wu's equations and Quasi Hypergemetric
Functions, abstract 
14:4515:45 
Tomoki Nakanishi (joint with Atsuo Kuniba),
String solutions of the Bethe ansatz equation in the q=0 limit
and combinatorial character formula for affine quantum group,
abstract 
16:0017:00 
Katsuhisa Mimachi, A duality of the MacdonaldKoornwinder
polynomials and its application to the integral representations,
abstract 
17:1518:15 
Kazuhiro Hikami, Discrete Integrable Systems and
Quantum Dilogarithm Function, abstract 
Friday, August 27
9:3010:30 
Tetsuji Miwa, Combinatorics of $\widehat{sl}_2$ Spaces of
Coinvariants, abstract 
10:4511:45 
Nobushige Kurokawa, Multiple sine functions II.
Applications, abstract 
11:4513:30 
lunch break 
13:3014:30 
Hiroaki Terao, Flat connections arising from a family of arrangements,
abstract 
14:4515:45 
HiroFumi Yamada, Schur's Qfunctions and affine Lie algebras,
abstract 
16:0017:00 
Anatol Kirillov, Ubiquity of KostkaFoulkes polynomials,
abstract 
Abstracts
Kazuhiko Aomoto, August 26, 13:3014:30
Wu's equations and Quasi Hypergeometric Functions
Abstract: Among the class of quasi hypergeometric functions the ones
obtained by solving Wu's equations (which was discovered by Kazumoto
Iguchi) have particularly simple properties.
I shall discuss the properties of transformation formulas,
singularities and monodromy for them.
Etsuro Date (joint with S.S. Roan), August 24, 10:4511:45
On quotients of the Onsager alagebra
Abstract: We call an ideal of a Lie algebra closed if the quotient
by it does not contain central elements.
After classifying closed ideals of the Onsager algebra,
we study the structure of quotients by closed ideals. For that
purpose we use a realization of the Onsager algebra in a completion
of ${\bf C}[t]\otimes sl_2$.
Jan Felipe van Diejen, August 25, 13:3014:30
Combinatorial formulas for (q)spherical functions associated to rankone
(quantum) symmetric spaces
Abstract: This talk reports on joint work with A.N. Kirillov
regarding the study of rankone zonal spherical functions
and their qdeformations.
We present combinatorial representations
for the zonal spherical funcions on odddimensional
spheres and hyperboloids, as well as qdeformations
thereof.
The expansion coefficients in the combinatorial formulas
are interpreted as characters of irreducible representations
of the general linear group.
To derive the combinatorial formulas we use the inverse scattering
theory of reflectionless Jacobi operators.
Kazuhiro Hikami, August 26, 17:1518:15
Discrete Integrable Systems and Quantum Dilogarithm Function
Abstract: We study a quantization of discrete analogue of both
the KdVtype equation and the Toda field theory.
We shall construct the generating functions for the conserved
operators by use of the quantum dilogarithm function.
Noriaki Kawanaka, August 26, 14:4515:45
A qCauchy identity for Schur functions and complex reflection groups
Abstract: We consider a qanalogue of the FrobeniusSchur
indices for the irreducible
characters of imprimitive finite complex reflection groups. This leads to
a qanalogue of the classical Cauchy identity for Schur functions. Related
(partly conjectural) qidentities involving HallLittlewood, or Macdonald
symmetric functions will also be presented.
Anatol Kirillov, August 27, 16:0017:00
Ubiquity of KostkaFoulkes polynomials
Abstract: We will describe appearences of KostkaFoulkes
polynomials in the theory of finite abelian pgroups,
algebraic geometry, the representation theory of the general linear and
symmetric groups and statistical physics.
Atsuo Kuniba, August 23, 16:0017:00
Soliton cellular automata associated with finite crystals
Abstract:
I report on a joint work with G. Hatayama and T. Takagi.
We introduce a class of cellular automata associated with irreducible
finite dimensional representations of quantum affine algebras
$U'_q(\hat{\geh}_n)$. They exhibit soliton behavior whose scattering
is described conjecturally by the combinatorial Rmatrices of the
smaller algebra $U'_q(\hat{\geh}_{n1})$. For the $A^{(2)}_2$ example,
we derive piecewise linear evolution equations for Nsolitons. They
admit a tropical variable change into Todatype equations which are very
similar to the Y and Tsystems for transfer matrices.
We construct an explict solution in some sector, which leads to the
Nsoliton solution of the automaton through the
ultradiscretization.
Nobushige Kurokawa, August 26, 27, 10:4511:45
Multiple Sine Functions, I. Basic Properties, II. Applications
Abstract: In part I, I recall two ways to construct multiple
sine functions. The key properties are periodicity and algebraic
differential equations. Special values are very interesting. In part II,
these properties are applied to investigate special values of zeta
functions containing zeta(3), and to determine the gamma factors of
Selberg zeta functions. Some qanalogues will be reported also.
Stephen C. Milne, August 24, 9:3010:30
Transformations of U(n+1) multiple basic hypergeometric series
Abstract: The purpose of this talk is to survey the transformation
theory of U(n+1) multiple basic hypergeometric
series starting with the U(n+1) terminating
verywellpoised $_6\phi_5$ summation theorems. These
series were strongly motivated by L. C. Biedenharn and J.
D. Louck and coworkers mathematical physics research
involving angular momentum theory and the unitary groups
U(n+1), or equivalently A_n. They are directly related
to the corresponding Macdonald identities. This U(n+1)
or A_n theory has also been extended to the root
systems C_n and D_n. There are now many
applications of A_n and/or subsequently C_n and
D_n multiple basic hypergeometric series. These
include the following topics: A. N. Kirillov  quantum
groups; R. Gustafson  multidimensional beta and/or
Barnes integral evaluations; C. Krattenthaler
and I. Gessel  plane partition enumeration; S. Milne  analytic
number theory (sums of squares); S. Milne,
G. Lilly, G. Bhatnagar, C. Krattenthaler, and M. Schlosser 
multidimensional matrix inversions;
S. Milne and V. Leininger  new infinite families of eta
function identities; Y. Kajihara and M. Noumi  applications to raising
operators for Macdonald polynomials. As an introduction
to this area we discuss some of the main results and
techniques from the following outline of the development
of U(n+1) basic hypergeometric series.
The U(n+1) terminating verywellpoised $_6\phi_5$
summation theorems extend Rogers' classical onevariable
work and are central to our theory. They may be proved
using qdifference equations arising from the Lagrange
interpolation formula and partial fraction expansions.
The U(n+1) $_6\phi_5$ summation theorems may in turn be
specialized to obtain the U(n+1) extension of
Andrews' matrix formulation of the Bailey Transform. The
U(n+1) Bailey transform is then applied to the U(n+1)
$_6\phi_5$ summation theorems to derive the U(n+1)
terminating, balanced $_3\phi_2$ summation theorems,
whose special cases include U(n+1) qGauss summation
theorems, qChuVandermonde theorems and U(n+1)
qbinomial theorems. An analytic continuation argument
applied to a U(n+1) qbinomial theorem yields the
U(n+1) extension of Ramanujan's $_1\psi_1$ sum, and
Gustafson's U(n+1) $_6\psi_6$ summation turns out to be
the next higher dimensional version of the U(n+1)
$_1\psi_1$ sum. For example, the twodimensional U(3)
$_1\psi_1$ summation is equivalent to Bailey's classical
onedimensional $_6\psi_6$ summation. We also obtain
U(n+1) extensions of the Jacobi triple product
identity. The Bailey transform coupled with the
U(n+1) balanced $_3\phi _2$ summation theorems yields
the U(n+1) extension of Andrews' explicit
formulation of the Bailey Lemma, which upon
iteration gives several U(n+1) generalizations of
Watson's qanalogue of Whipple's transformation
formula. Special and limiting cases include the
nonterminating U(n+1) $_6\phi_5$ summation, the
U(n+1) extension of the terminating balanced $_8\phi_ 7$
summation theorem, and the U(n+1)
RogersRamanujanSchur identities. A classical
interchange of summation argument leads to the U(n+1)
$_{10}\phi_9$ transformation formulas. Important limiting
cases include the U(n+1) generalization of Bailey's
nonterminating extension of Watson's transformation.
This in turn leads to the nonterminating U(n+1)
extension of Bailey's balanced $_8\phi_ 7$ summation
theorem. The classical case of all this
work, corresponding to $A_1$ or equivalently U(2),
contains a substantial amount of the theory and
application of onevariable basic hypergeometric series.
Stephen C. Milne (joint with Verne E. Leininger), August 25, 9:3010:30
Some new infinite families of eta function identities
Abstract: Ever since Euler proved his expansion for
$\prod_{i=1}^\infty (1q^i)\equiv (q;q)_\infty$
mathematicians have been looking for other identities of
this form. In 1829, Jacobi utilized his triple product
identity to derive an elegant expansion for
$(q;q)_\infty^3$. Since then, expansions have been found
for $(q;q)_\infty^c$ for many values of c.
These included several infinite families of expansions
and a few exceptional cases.
In 1892 F. Klein and R. Fricke gave a result for
c=8. This was rediscovered by S. Ramanujan in 1916.
L. Winquist in 1969 proved a result for
c=10 but stated that this was first found by J.
Rushforth, then independently discovered by A. Atkin.
L. Winquist also noted that A. Atkin had formulae for
c=14 and c=26. The existence of these identities had
been suggested in 1955 by M. Newman. In
1972 F. Dyson gave his famous formula for
c=24 and stated that formulae corresponding to
c= 3,8,10,14,15,21,24,26,28,35,36,... had been
found, but noted that these ad hoc results had been
unified by I. Macdonald. In his landmark 1972 paper,
Macdonald related most of these expansions for
$(q;q)_\infty^c$ to affine root systems.
(This connection with Lie Algebras has been the
main focus of work since I. Macdonald.) A few
notable exceptions remained: c=2 found by Hecke and
Rogers, c=4 by Ramanujan, and c=26 by Atkin.
In this talk we discuss our derivation of new, more
symmetrical expansions for $(q;q)_\infty^{n^2+2n}$ by
means of our multivariable generalization of Andrews'
variation of the standard proof of Jacobi's
$(q;q)_\infty^3$ result. We also present examples of
our general expansion for $(q;q)_\infty^c$
where c=3,8,15,24. Our proof relies upon Milne's new
U(n) multivariable extension of the Jacobi triple
product identity. This result is deduced from a
U(n) multiple basic hypergeometric series
generalization of Watson's verywellpoised
$\_8\phi_7$ transformation. The derivation of our
$(q;q)_\infty^{n^2+2n}$ expansion also utilizes partial
derivatives and dihedral group symmetries to write the
sum over regions in nspace. We note that
our expansions for $(q;q)_\infty^{n^2+2n}$ are equivalent
to Macdonald's A_n family of etafunction identities.
In addition, we utilize various summation and
transformation formulas for U(n+1), equivalently A_n,
multiple basic hypergeometric series to derive new
infinite families of expansions for
$(q;q)_\infty^{n^2+2}$ and $(q;q)_\infty^{n^2}$, similar
products of these, and the corresponding powers of the
$\eta$function. (Recall that the eta function is defined
by $\eta(q):= q^{1/24}(q;q)_\infty$.) These additional
infinite families of expansions extend the list in
Appendix I of Macdonald's 1972 paper. All of this work
is motivated by Milne's U(n+1) multiple basic
hypergeometric series treatment of the Macdonald
identities for $A_{\ell}^{(1)}$.
Stephen C. Milne, August 26, 9:3010:30
Infinite families of exact sums of squares formulas,
Jacobi elliptic functions, continued fractions, and
Schur functions
Abstarct: In this talk we give several infinite families
of explicit exact formulas involving either
squares or triangular numbers, two of which
generalize Jacobi's (1829) 4 and 8 squares
identities to $4n^2$ or $4n(n+1)$ squares,
respectively, without using cusp forms.
The Schur function form of these infinite families
of identities are analogous to the $\eta$function
identities of Macdonald. We also utilize a special case
of our methods to give a proof of the two KacWakimoto
conjectured identities involving representing a positive
integer by sums of $4n^2$ or $4n(n+1)$ triangular
numbers, respectively. These results, depending on new
expansions for powers of classical theta functions, arise
in the setting of Jacobi elliptic functions, associated
continued fractions, regular Cfractions, Hankel or
Tur\'anian determinants, Fourier series, Lambert series,
inclusion/exclusion, Laplace expansion formula for
determinants, and Schur functions. Our 16 and
24 squares identities were originally obtained
via multiple basic hypergeometric series,
Gustafson's $C_{\ell}$ nonterminating
$_6\phi_5$ summation theorem, and Andrews'
basic hypergeometric series proof of Jacobi's
2, 4, 6, and 8 squares identities. We
have (elsewhere) applied symmetry and Schur
function techniques to this original approach to
prove the existence of similar infinite families
of sums of squares identities for $n^2$ or
n(n+1) squares, respectively. Our sums of
more than 8 squares identities are not the
same as the formulas of Mathews (1895), Glaisher
(1907), Ramanujan (1916), Mordell (1917, 1919),
Hardy (1918, 1920), Kac and Wakimoto (1994), and
many others.
Katsuhisa Mimachi, August 26, 16:0017:00
A duality of the MacdonaldKoornwinder polynomials and
its application to the integral representations
Abstract: A formula representing a duality
of the MacdonaldKoornwinder polynomials
will be presented. The formula induces
several integral representations of the
MacdonaldKoornwinder polynomials, of
HeckmanOpdam's Jacobi polynomials of type $BC_m,$
etc.
Tetsuji Miwa, August 27, 9:3010:30
Combinatorics of $\widehat{sl}_2$ Spaces of Coinvariants
Abstract: I will talk some results obtained in collaboration
with B. Feigin, R. Kedem, S. Loktev and E. Mukhin
on the $\widehat{sl}_2$ coinvariants.
Hitoshi Murakami (joint with Jun Murakami), August 24, 17:1518:15
The colored Jones polynomials and the simplicial volume of a knot
Abstract: We show that the set of colored Jones polynomials
and the set of generalized Alexander polynomials defined by Akutsu, Deguchi
and Ohtsuki intersect nontrivially. Moreover it is shown that the
intersection is (at least includes) the set of Kashaev's quantum
dilogarithm invariants for links. Therefore Kashaev's conjecture can be
restricted as follows: The colored Jones polynomials determine the
hyperbolic volume for a hyperbolic knot. Modifying this, we propose a
stronger conjecture: The colored Jones polynomials determine the simplicial
volume for any knot. If our conjrcture is true, then we can prove that
a knot is trivial if and only if and only if all of its Vassiliev invariants
are trivial.
Tomoki Nakanishi (joint with Atsuo Kuniba), August 26, 14:4515:45
String solutions of the Bethe ansatz equation in the q=0 limit
and combinatorial character formula for affine quantum group
Abstract: We study a class of meromorphic solutions (string solutions)
of the XXZ Bethe ansatz equation. We show the string solutions are
obtained by solving the linear congruence equation (string equation).
The counting of the offdiagonal solutions of the string equation
leads us to an interesting combinatorial character formula for finite
dimensional modules of affine quantum group. The formula is related
to the character formula by Kirillov and Reshetikhin.
Michimoto Nishizawa (joint with Kimio Ueno), August 24, 16:0017:00
Integral solutions of hypergeometric qdifference systems with q=1
Abstract: By using the double sine function, we construct two
kinds of integral solutions of hypergeometric qdifference system with q=1 .
One is the Barnes type and the other is the Euler type. We consider
some properties of these solutions.
Masatoshi Noumi (joint with Yasuhiko Yamada), August 24, 13:3014:30
A realization of Weyl groups and combinatorics of special polynomials
Abstract: We will discuss a realization of Weyl groups as groups
of birational transformations and combinatorial properties of certain cocycles
of Weyl groups. These cocycles give a generalization of special
polynomials arising from the Painleve equations, to arbitrary root systems.
Anne Schilling, August 23, 14:4515:45
Levelrestricted rigged configurations
Abstract:
Recently, in collaboration with A.N. Kirillov and M. Shimozono,
we established a statisticpreserving bijection between
LittlewoodRichardson tableaux and rigged configurations.
In this talk we review this bijection and its main properties,
and then show that it respects levelrestriction. More precisely,
there is a natural way to impose a levelrestriction on the set
of LittlewoodRichardson tableaux. The image of this restricted set
under the bijection can be described explicitly.
As a corollary, fermionic formulas for the levelrestricted
KostkaFoulkes polynomials and for branching functions
of affine type A cosets can be derived/proved.
Makoto Taneda, August 25, 17:1518:15
Representation of Umemura Polynomials
Abstract: Umemura presented the special polynomials associated with
solutions of the sixth Painleve equation. Noumi, Okada and Okamoto gave
the conjectural representation of these special polynomials
using an interesting combinatorial interpretation.
We will show that these conjectures are true.
Hiroaki Terao, August 27, 13:3014:30
Flat connections arising from a family of arrangements
Abstract:A combinatorially stable family of arrangements
determines a flat connection on the moduli space of the family.
When the family is universal, the connection is completely determined
by the combinatorial type of the arrangements. We study the exlplicit
form of the connection matrices presented with respect to $\beta$
nbcbases.
Tomohide Terasoma, August 25, 16:0017:00
Twisted Ehrhart polynomials
Abstract: Let $f(x_1, \dots , x_n) \in \bold C [x_1, \dots ,x_n]$ be
a nondegenerate Laurant polynomail with respect to its Newton
polytope $\Delta$. For a sufficiently generic rational number
$\alpha_0 ,\dots , \alpha_n \in \frac{1}{d}\bold Z$, the
twisted cohomology group $H^n(\bold C^{\times}V(f), \bold C(\eta))$
with the coefficient in the local system $\bold (\eta)$ of the
multivalued function $\eta = f^{\alpha_0}x_1^{\alpha_1}\cdots x_n^{\alpha_n}$
has a natural $\bold Q(\zeta_d)$Hodge structure:
$H^n(\bold C^{\times}V(f), \bold C(\eta)) = \oplus_{i=1}^n
H^{i,j}.$
It is called a hypergeometric Hodge structure. The generating function
$p(t) = \sum_{i=0}^n (\dim H^{i,j}) t^i$ is called the Hodge polynomial
of $H^n(\bold C^{\times}V(f), \bold C(\eta))$ and it is expressed in
terms of twisted Ehrhart polynomial of the Newton polytope $\Delta$.
It is know that there exists many nontirival identity for hypergeomtric
Hodge structure which is a generalization of Gauss multiplication formula.
This identity arise from algebraic correspondeces. In this lecture,
we will discuss about the identity of twisted Ehrhart polynomial arising
from these generalized Gauss multiplication correspondences.
Paul Terwillinger (joint with Tatsuro Ito and Kenichiro Tanabe),
August 25, 10:4511:45
A qanalog of the Onsager algebra
Abstract: The Onsager algebra is a certain infinite dimensional
complex Lie algebra that has been used to study the
two dimensional Ising and chiral Potts model. Perk and Davies
showed the Onsager algebra has a presentation involving
two generators
$A$, $A^*$ satisfying the relations
$[A,[A,[A,A^*]]] = 16[A,A^*]$
$[A^*,[A^*,[A^*,A]]] = 16[A,A^*]$
where [ , ] denotes the Lie bracket.
The above relations are known as the DolanGrady relations.
Later Roan showed the Onsager algebra
is isomorphic to a certain subalgebra
of the affine Lie algebra $A_1^{(1)}$.
In this talk, we define an algebra T which
can be viewed as a qanalog of the Onsager algebra.
Let $\fld$ denote any field,
and let $\beta, \gamma, \gamma^*,\varrho, \varrho^*$ denote scalars in $\fld$.
We define
$T=T(
\beta, \gamma, \gamma^*,
\varrho, \varrho^*)$
to be the associative $\fld$algebra with identity generated by two symbols
$A$, $A^*$
subject to the relations
$0 = [A,A^2A^*\beta AA^*A + A^*A^2 \gamma (AA^*+A^*A)
\varrho A^*],$
$0 = [A^*,A^{*2}A\beta A^*AA^* + AA^{*2} \gamma^* (A^*A+AA^*)
\varrho^* A],$
where [r,s] means rssr.
If one sets $\beta = 2$, $\gamma = 0$, $\gamma^*=0$, $\varrho=16$,
$\varrho^*=16$, in the above relations, one gets essentially
the DolanGrady relations.
To understand the algebra T,
we consider the finite dimensional irreducible Tmodules on which
$A$ and $A^*$ act as semisimple linear transformations.
One type of Tmodule of this sort is given by what we
call a Leonard pair.
Let V denote a vector space over $\fld$ that has
finite positive dimension. By a Leonard pair on V, we
mean an ordered pair $A,A^*$
consisting of linear transformations from V to V
that satisfy both conditions
(i) there exists a basis for V with respect to which the matrix
representing $A$ is diagonal, and the matrix representing $A^*$ is
irreducible tridiagonal,(ii) there exists a basis for V with
respect to which the matrix representing $A^*$ is
diagonal, and the matrix representing $A$ is irreducible tridiagonal.
(A tridiagonal matrix is said to be irreducible whenever all entries
immediately above and below the main diagonal are nonzero).
The Leonard pairs have been completely classified by Terwilliger, and they
turn out to be closely related to the qRacah polynomials.
It is an intriguing fact that there exist
finite dimensional irreducible Tmodules on which
$A$ and $A^*$ act as semisimple linear transformations,
but are not Leonard pairs.
It is an open problem to classify these modules,
and the main goal of the present talk is to explain what is known
so far along this line.
Tetsuji Tokihiro, August 25, 14:4515:45
Box and ball systems as ultradiscrete soliton equations
Abstract: A soliton cellular automaton, which represents a movement
of finite number of balls in any array of boxes, is investigated.
Its dynamics is described by two kinds of ultradiscrete equations
obtained from the degenerating formula of KP hierarchy. The combinatorial
rules for soliton interactions and factorization property of the
scattering matrices (YangBaxter relation) are proved by means of
inverse ultradiscretization. The interpretation with the aid of
combinatorial Rmatrix lattice model is also presented.
Hiroshi Umemura, August 23, 10:3011:30
Painleve equations in the past 100 years
Abstract: We review developments in the research of the
Painleve equations. The Painleve equations are a hundred years old.
They were almost forgoten during several decades. Today they revive
and we can admet that they are more interesting mathematical objects
than expected in the early years of the century. For example, recent
discovery of their relation with the combinatorics
reveals a new feature of the Painleve equations.
HiroFumi Yamada, August 27, 14:4515:45
Schur's Qfunctions and affine Lie algebras
Abstract: This talk deals with the weight vectors of the basic
representations of
the twisted affine Lie algebra of type D^(2)_{l+1}. The weight vectors
are expressed in terms of Schur's Qfunctions. The up and down motion along
a string of the fundamental imaginary root is described as a combinatorial
game on an abacus. As a consequence one can determine the homogeneous
polynomial solutions to the reduced BKP hierarchies as the maximal weight
vectors. The results obtained will be applied to the case of A^(1)_1.
The embedding of A^(1)_1 into D^(2)_4 gives rise to a realization of
the basic representation of A^(1)_1. The isomorphism between the
"standard" and
this "nonstandard" realizations of the basic representation induces
a "funny" identity of Schur's Sfunctions and Qfunctions indexed by
certain partitions.
In fact this identity reflects a nature of decomposition matrices of the
modular
spin representations of the symmetric groups with characteristic 2.
Masaaki Youshida, August 23, 13:3014:30
Uniformizations of pointconfiguration spaces
Abstract: Some examples of (real and complex) pointconfiguration
spaces (number of points less then 6) will be maniacly studied.
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