Research Inspiration from
Mathematical Sciences
Kyoto University
Research Inspiration from Mathematical Sciences |
Research Institute for Mathematical
Sciences Kyoto University |
Speakers |
Program |
Abstracts |
RIMS |
Kansai Seminar House |
Weather in Kyoto |
Welcome to Kyoto |
Japan Travel Guide
O r g a n i z e r s: |
| M. Kashiwara, A.N. Kirillov, T. Miwa |
Invited Speakers: |
|
A. Braverman (Harvard University, USA), abstract P. Caldero (University Claude Bernard Lyon I, France), abstract H. Derksen (University of Michigan, Ann Arbor, USA), abstract J.-F. van Diejen (University of Talca, Chile), abstract C. Dong (University of California at Santa Cruz, USA), abstract B. Feigin (Landau Inst. for Theoretical Physics, Russia, and RIMS, Kyoto University), abstract E. Frenkel (University of California at Berkeley, USA), abstract M. Jinzenji (Hokkaido University), abstract A. Kirillov (RIMS, Kyoto University, and Steklov Institute, St.Petersburg, Russia), abstract A. Malkin (Yale University, USA), abstract H. Miyachi (Science University of Tokyo), abstract S. Naito (University of Tsukuba), abstract A. Nakayashiki (Kyushu University), abstract M. Noumi (Kobe University), abstract T. Otofuji (Nihon University), abstract Y. Saito (University of Tokyo), abstract M. Semenov-Tian-Shansky (University of Bourgogne, France, and Steklov Institute, St.Petersburg, Russia), abstract A. Takahashi (RIMS, Kyoto University), abstract C. Teleman (Cambridge University, United Kingdom), abstract H. Terao (Tokyo Metropolitan University), abstract M. Wakimoto (Kyushu University), abstract M. Zabrocki (York University, Canada), abstract |
P r o g r a m |
The Workshop is to be held at Kansai Seminar House where accommodation and meals (according to the program below) for all speakers and most participants will be provided.
| time | August 12 (Sun) |
August 13 (Mon) |
August 14 (Tue) |
August 15 (Wed) |
August 16 (Thu) |
|---|---|---|---|---|---|
| 8:00 - 9:00 | Breakfast | Breakfast | Breakfast | Breakfast | |
| 9:30 - 10:30 | Miyachi | Naito | Otofuji | Dong | |
| 10:45 - 11:45 | Braverman | Frenkel | Semenov- Tian-Shansky |
Zabrocki | |
| 12:15 - 13:00 | Registration | Lunch | Lunch | Lunch | Lunch |
| 13:15 - 14:15 | Wakimoto | Feigin | Discussions | Discussions | Noumi |
| 14:30 - 15:30 | Caldero | Discussions | Discussions | Discussions | van Diejen |
| 15:45 - 16:45 | Terao | Discussions | Discussions | Discussions | Kirillov |
| 17:00 - 18:00 | Nakayashiki | Discussions | Discussions | Discussions | Discussions |
| 18:00 - 18:45 | Dinner | Dinner | Dinner | Dinner | Dinner |
| 19:00 - 20:00 | Reception | Malkin | Teleman | Jinzenji | Bonfire |
| 20:15 - 21:15 | Reception | Saito | Derksen | Takahashi | Bonfire |
Daily Program |
|
August 12 Sunday |
|
| 12:30 - 13:00 | Registration |
| 13:15 - 14:15 | Minoru Wakimoto (Kyushu University): N=2 superconformal modules with half-modular properties |
|
|
| 14:30 - 15:30 | Philippe Caldero (University Claude Bernard Lyon I, France): Adapted algebras for the Berenstein-Zelevinsky conjecture |
|
|
| 15:45 - 16:45 | Hiroaki Terao (Tokyo Metropolitan University): Multiderivations of Coxeter arrangements |
|
|
| 17:00 - 18:00 | Atsushi Nakayashiki (Kyushu University): On the space of local operators of SU(2) invariant Thirring model |
| 18:00 - 19:00 | Dinner |
| 19:00 - 21:00 | Reception
|
| August 13 Monday |
|
| 8:00 - 9:00 | Breakfast |
| 9:30 - 10:30 | Hyohe Miyachi (Science University of Tokyo): On some crystallized decomposition numbers |
|
|
| 10:45 - 11:45 | Alexander Braverman (Harvard University, USA): Drinfeld's compactifications and periodic Kazhdan-Lusztig polynomials |
| 12:15 - 13:00 | Lunch |
| 13:15 - 14:15 | Boris Feigin (Landau Inst. for Theoretical Physics, Russia,
and RIMS, Kyoto University): to be announced |
| 14:30 - 18:00 | Discussions |
| 18:00 - 18:45 | Dinner |
| 19:00 - 20:00 | Anton Malkin (Yale University, USA): Induction of singular quiver varieties |
|
|
| 20:15 - 21:15 | Yoshihisa Saito (University of Tokyo): On elliptic Lie algebras |
| August 14 Tuesday |
|
| 8:00 - 9:00 | Breakfast |
| 9:30 - 10:30 | Satoshi Naito (University of Tsukuba): Twining characters, Path models, and crystal bases |
|
|
| 10:45 - 11:45 | Edward Frenkel (University of California at Berkeley, USA): Representations of p-adic and quantum affine groups |
| 12:15 - 13:00 | Lunch |
| 13:00 - 18:00 | Discussions |
| 18:00 - 18:45 | Dinner |
| 19:00 - 20:00 | Constantin Teleman (Cambridge University, United Kingdom): The Strong Macdonald conjecture and geometric applications |
|
|
| 20:15 - 21:15 | Harm Derksen (University of Michigan, Ann Arbor, USA): Quiver representations and Littlewood-Richardson coefficients |
| August 15 Wednesday | |
| 8:00 - 9:00 | Breakfast |
| 9:30 - 10:30 | Takashi Otofuji (Nihon University): Quantum cohomology algebra of infinite-dimensional flag manifolds |
|
|
| 10:45 - 11:45 | Michael Semenov-Tian-Shansky (University of Bourgogne, France,
and Steklov Mathematical Institute, St.Petersburg, Russia): q-deformed quantum Toda lattice: a hint to representation theory of noncompact quantum groups |
| 12:15 - 13:00 | Lunch |
| 13:00 - 18:00 | Discussions |
| 18:00 - 18:45 | Dinner |
| 19:00 - 20:00 | Masao Jinzenji (Hokkaido University): Gauss-Manin system and the virtual structure constants |
|
|
| 20:15 - 21:15 | Atsushi Takahashi (RIMS, Kyoto University): BPS invariants on Calabi-Yau 3-fold |
| August 16 Thursday |
|
| 8:00 - 9:00 | Breakfast |
| 9:30 - 10:30 | Chongying Dong (University of
California at Santa Cruz, USA): Vertex operator algebras and reductive Lie algebras |
|
|
| 10:45 - 11:45 | Michael Zabrocki (York University, Canada): q-analogs in Hopf algebra structures |
| 12:15 - 13:00 | Lunch |
| 13:15 - 14:15 | Masatoshi Noumi (Kobe University): q-Painleve equations arising from a discrete version of the modified KP hierarchy |
|
|
| 14:30 - 15:30 | Jan Felipe van Diejen (University of Talca, Chile): Modular hypergeometric sums |
|
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| 15:45 - 16:45 | Anatol Kirillov (RIMS, Kyoto University, and Steklov
Mathematical Institute, St.Petersburg, Russia): Generalized saturation conjecture |
| 16:45 - 18:00 | Discussions |
| 18:00 - 19:00 | Dinner |
| 19:00 - 22:00 | Bonfire |
A b s t r a c t s |
Alexander Braverman (Harvard University, USA)
Philippe Caldero (University Claude
Bernard Lyon I, France)
Harm Derksen (University of Michigan, Ann Arbor, USA)
Jan-Felipe van Diejen (University of Talca, Chile)
Chongying Dong (University of California at Santa Cruz,
USA)
Edward Frenkel (University of California at
Berkeley, USA)
Masao Jinzenji (Hokkaido University)
Anatol N. Kirillov (RIMS, Kyoto University, and
Steklov Institute, St.Petersburg, Russia)
Anton Malkin (Yale University, USA)
Hyohe Miyachi (Science University of Tokyo)
Satoshi Naito (University of Tsukuba)
Atsushi Nakayashiki (Kyushu University)
Masatoshi Noumi (Kobe University)
Yoshihisa Saito (University of Tokyo)
Michael Semenov-Tian-Shansky (University of Bourgogne,
France, and Steklov Mathematical Institute, St.Petersburg, Russia)
Atsushi Takahashi (RIMS, Kyoto University)
Constantin Teleman (Cambridge University, United Kingdom)
Hiroaki Terao (Tokyo Metropolitan University)
Takashi Otofuji (Nihon University)
Minoru Wakimoto (Kyushu University)
Michael Zabrocki (York University, Canada)
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Drinfeld's compactifications and periodic Kazhdan-Lusztig polynomials
Abstract: The purpose of this talk is to introduce certain
compactifications of the moduli space of parabolic bundles on a smooth
projective curve over an algebraically closed
field (these compactifications were originally defined by Drinfeld).
I will explain how one can think about Drinfeld's compactifications as certain
global models for the (at the moment) non-existing "partial semi-infinite
flag manifolds". In particular, I will explain that one can realize the
"periodic polynomials" introduced recently by Lusztig in terms of stalks
of some natural perverse sheaves on Drinfeld's compactifications.
This is a joint work with M.Finkelberg, D.Gaitsgory and I.Mirkovic.
Adapted algebras for the
Berenstein-Zelevinsky conjecture
Abstract: Let
$G$ be a simply connected semi-simple complex Lie group and fix a
maximal unipotent subgroup $U^-$ of $G$. Let $q$ be an
indeterminate and let's denote by $\B^*$ the dual canonical basis
of the quantized algebra $\C_q[U^-]$ of regular functions on
$U^-$. A.Berenstein and A.Zelevinsky conjecture that two elements
of $\B^*$ $q$-commute if and only if they are multiplicative,
i.e. their product is an element of $\B^*$ up to a power of $q$.
For all reduced decomposition ${\tilde w_0}$ of the longest
element of the Weyl group of $\g$, we associate a subalgebra
$A_{\tilde w_0}$, called adapted algebra, of $\C_q[U^-]$ such
that
1) $A_{\tilde w_0}$ is a $q$-polynomial algebra which
equals $\C_q[U^-]$ up to localization,
2) $A_{\tilde w_0}$ is
spanned by a subset of $\B^*$,
3) the Berenstein-Zelevinsky
conjecture is true on $A_{\tilde w_0}$.
Quiver Representations and Littlewood-Richardson Coefficients
Abstract: I will discuss quiver representations and a
theorem about generators
of rings of semiinvariants for quivers. Applied to the triple flag
quiver, the theorem implies the result of Knutson and Tao about
the saturation of Littlewood-Richardson coefficients. Using quiver
representations, one can also describe the faces (of arbitrary
dimension) of the cone of nonzero Littlewood-Richardson coefficients.
Modular hypergeometric sums
Abstract: Recenlty Frenkel and Turaev introduced a modular
hypergeometric generalization of the celebrated very-well-poised balanced basic
hypergeometric ${}_8\Phi_8$ sum. In this talk various multidimensional
generalizations of the Frenkel-Turaev sum are discussed.
Vertex operator algebras and reductive Lie algebras
Abstract: A VOA is of CFT type if the negative weight
space is zero and
weight zero subsapce is one-dimensional. The weight one subspace
in this case forms a Lie algebra. In this talk we present three
results concerning VOAs of CFT type:
1. The weight one subspace is a reductive Lie algebra.
2. The rank of the reductive Lie algebra is less than or equal to the central
charge of the VOA.
3. Apply the results in 1 and 2 to the classification of holomorphic VOAs
of small central charges.
Representations of p-adic and quantum affine groups
Abstract: It is well-known that the category of
unipotent representations of p-adic GL_n over the field of complex
numbers is equivalent to the category of finite-dimensional
representations of the corresponding affine
Hecke algebra, which is in turn equivalent to a subcategory of the
category of finite-dimensional representations of the quantum affine
algebra of gl(N) for large N, via Jimbo's generalization of the Schur
functor (with q=p). Recently, Vigneras has considered unipotent
representations of p-adic GL_n over a field of characteristic l>0, prime
to p. This category turns out to be related to the category of
finite-dimensional representations of quantum affine gl(N), where q is a
root of unity of order equal to the order of p mod l. In this talk I will
discuss this correspondence, and also review recent results (obtained
jointly with E.Mukhin) on the Hopf algebra structure of the Grothendieck
ring of finite-dimensional representations of quantum affine gl(N), as N
goes to infinity, both when q is generic and a root of unity.
Gauss-Manin System and the Virtual Structure Constants
Abstract: In this talk, we discuss some application of ODE
to enumerative problems on rational curves in projective hypersurfaces.
In particular, we clarify the correspondence between the virtual structure
constants and Givental's differential equation.
Generalized saturation conjecture
Abstract: Saturation Conjecture, now theorem by
A.Knutson and T.Tao, 1998, was
a final step in the proof of longstanding Horn Conjecture about
the spectrum of sum of two hermitian matrices. Since the original
proof by A.Knutson and T.Tao, several new proofs and generalizations
have been given by H.Derksen and J.Weyman, A.Schofield, and others.
In my talk I will explain a new proof of the saturation theorem
which is based on the theory of rigged configurations, parabolic
Kostka polynomials, and use some ideas from Bethe's ansatz and
Corner transfer matrix theory. I will also state several
conjectures which are further generalizations of the saturation theorem
and Fulton's conjecture.
Induction of singular quiver varieties
Abstract: I'll describe an induction procedure for singular quiver
varieties of ADE type which allows one to give a geometric construction of
the tensor category of finite-dimensional representations of a reductive
algebraic group over C. More elementary, one obtains a generalization of
Hall polynomials to ADE case. The proof uses geometric crystals associated
to quiver varieties.
On some crystallized decomposition numbers
Abstract: I'd like to talk about some recent results on crystallized
decomposition numbers defined by Leclerc-Thibon. We can get
some closed formulas for certain vectors of the canonical bases
of the Fock space representation of $U_v(\hat{gl}_n)$.
(This result is a joint work with B. Leclerc.)
On the other hand, the origin of these formulas comes from
$\ell$-modular representations of $GL_m(F_q)$. (This result
is derived from a joint work with A. Hida.) I'd like to
explain their connections, too.
Twining characters, Path models, and crystal bases
Abstract: We talk about a combinatorial approach to a
twining character formula for a Demazure module over a Kac-Moody
algebra (this is a joint work with Dr. D. Sagaki). In this
approach, we use the path model of a Demazure module and the
global crystal base of a quantum Demazure module, in order to
describe the basis elements of the module fixed by a diagram
automorphism in terms of the orbit Lie algebra.
On the space of local operators of SU(2) invariant Thirring model
Abstract: I will present a huge family of local operators of SU(2)
invariant Thirring model in the form of form factors.
In this case form factors of local operators are
characterized as the infinite sets of solutions of
sl_2 qKZ equation at level zero satisfying certain
conditions. Up to now there are two important method to construct
form factors, Smirnov's and Lukyanov's. These two constructions
are very different. We generalize the results of Smirnov
and discuss the relation of it with the latter construction.
This is a joint work with Y. Takeyama.
q-Painleve equations arising from a discrete version of
the modified KP hierarchy
Abstract: I will discuss a class of q-Painleve equations with
affine Weyl group symmetry which arise from a q-version
of the modified KP hierarchy.
This talk is based on a joint work with K.Kajiwara and Y.Yamada.
On elliptic Lie algebras
Abstract:
I will introduce a new class of infinite dimensional Lie algebras so-called
Elliptic Lie algebras. This algebra was originally defined by K.Saito and
D.Yoshii. I will discuss some class of representations and compute their
characters.
This talk is based on a joint work with H.Awata, A.Kato, Y.Shimizu and
A.Tsuchiya.
q-deformed quantum Toda lattice: a hint to representation theory
of noncompact quantum groups
Abstract: The q-deformed quantum Toda lattice may be
studied by a combination of the methods of representation theory and of the
Quantum Inverse Scattering Method. The key
phenomenon which is under scrutiny is the role of the modular duality concept
(first discovered by L.Faddeev) in the representation theory of noncompact
semisimple quantum groups. Explicit formulae for the Whittaker vectors are
presented in terms of the double sine functions and the wave functions of the
N-particle q-deformed open Toda chain are expressed by means of a multiple
integral of the Mellin-Barnes type.
BPS invariants on Calabi-Yau 3-fold
Abstract: I shall explain a mathematical definition of
``BPS invarants" of Calabi-Yau 3-fold
based on the joint work with S.Hosono and M.-H.Saito.
In particular, for any projective morphism $f:X\to Y$
of normal projective varieties,
it is shown that there exists a natural $sl_2\times sl_2$ action on
the intersection cohomology group of $X$.
Some evidences for the Gopakumar-Vafa formula,
the equivalence between BPS and Gromov-Witten invariants,
will be given.
Relation with BPS algebra will also be discussed.
The Strong Macdonald conjecture and geometric applications
Abstract: We shall give a quick sketch of the proof of the
"Strong Macdonald
Constant Term conjecture" of Feigin and Hanlon, and discuss one
of its geometric applications, to wit, the computation of the
coherent sheaf cohomology of the cotangent bundle of the moduli
of holomorphic G-bundles over a Riemann surface, and the
cohomology of the sheaf of differential operators of the same
The proof of the conjecture is joint work with Fishel and
Grojnowski; the application is joint work with E. Frenkel.
The work is based on ideas of Hitchin, Feigin and Beilinson.
Multiderivations of Coxeter arrangements
Abstract: Let $V$ be an $\ell$-dimensional Euclidean space.
Let $G \subset O(V)$ be a finite irreducible orthogonal
reflection group. Let ${\cal A}$ be the corresponding Coxeter
arrangement. Let $S$ be the algebra of polynomial functions on $V.$
For $H \in {\cal A}$ choose $\alpha_H \in V^*$ such that $H = {\rm
ker}(\alpha_H).$ For each nonnegative integer $m$, define the derivation
module
$D^{(m)}({\cal A}) = \{ \theta \in {\rm Der}_S ~|~\theta(\alpha_H) \in S
\alpha^m_H\}$.
The module is known to be a free $S$-module of rank $\ell$ by K. Saito
(1975) for $m=1$ and L. Solomon-H. Terao (1998) for $m=2$.
The main result of this paper is that this is the case for all $m$.
Moreover we explicitly construct a basis for $D^{(m)} (\cal A)$.
Their degrees are all equal to $mh/2$ (when $m$ is even) or are equal to
$((m-1)h/2) + m_i (1 \leq i \leq \ell)$ (when $m$ is odd).
Here $m_1 \leq \cdots \leq m_{\ell}$ are the exponents of $G$ and $h=
m_{\ell} + 1$ is the Coxeter number.
The construction heavily uses the primitive derivation $D$ which plays
a central role in the theory of flat generators by K. Saito
(or equivalently the Frobenius manifold structure
for the orbit space of $G$.) Some new results concerning
the primitive derivation $D$ are obtained in the course of proof of the
main result.
Quantum cohomology algebra of infinite-dimensional flag manifolds
Abstract: I will discuss a generalization of Givental-Kim's
result on the quantum cohomology algebra of finite-dimensional flag manifolds
and the relation with Toda lattices
to the infinite-dimensional case. I will discuss also a trial for a
quantum Schubert calculus of those spaces.
This is a joint work with Martin Guest.
N=2 superconformal modules with half-modular properties
Abstract: In the construction of W-algebras associated to affine
superalgebras via the quantized Drinfeld-Sokolov reduction, it turns out that
the W-algebra of sl(2|1)^ is the direct sum of the centerless
Virasoro algebra and the N=2 superconformal algebra.
This picture takes, of course, care of the usual N=2 discrete
series representations which are obtained from sl(2|1)^-modules
of boundary levels, and moreover, using admissible sl(2|1)^-modules
in general, reveals series of representations of the N=2
superconformal algebra whose characters are "half" of modular
functions in the sense that, for each N=2 superconformal module
belonging to this series, there exists another module such that
the sum of characters of these two is a modular function.
It is well known that the central charges of discrete series
representations of the N=2 superconformal algebra are c(m)=3m/(m+2)
where m are non-negative integers. The central charges of these
half-modular series of representations are just equal to c(m) with
m being rational numbers such that m+2 >0.
q-analogs in Hopf algebra structures
Abstract: Using only operations that exist in any graded Hopf
algebra, we introduce a formula for taking a q-analog of homomorphisms of
the Hopf algebra. This quantization seems to be important in the theory
of symmetric functions as it gives rise to
Hall-Littlewood and Macdonald polynomials from
formulas for better known symmetric functions.
By changing the Hopf algebra, we will show that this
q-analog also gives rise to other q-coefficients using the same
sort of algebraic construction.
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