Schedule: Nov 30 (Thu) 9:30 - Dec 2 (Sat), 2017
Venue: Room B301, Graduate School of Science, Kobe University, Japan
Speakers
Program and Abstracts (PDF file)
The banquet will take place at an izakaya near Rokko-Michi station in the evening on November 30 (reservation required).
Supported by
Contact
Title: Non-commutative Painlevé equations and systems of Calogero type
Abstract: All Painlevé equations can be written as the motion of a particle under
a time dependent potential, and as such they admit a natural generalisation
to the case of several particles with an interaction of Calogero type (rational,
trigonometric or elliptic). In this talk, I will show that these many-particles
Hamiltonian systems admit an isomonodromic formulation, thus answering
to a question raised by Takasaki. After explaining the general theory,
I will focus on some examples and applications related to the second Painlevé
equation.
This is a joint work with Marco Bertola and Vladimir Rubtsov.
Title: Block Toeplitz determinants and integrable hierarchies
Abstract: The aim of this talk is to explain how one can use block Toeplitz determinants,
and in particular their asymptotics for large $N$, to effectively compute
the tau function of integrable hierarchies.
This talk will be a summary of results I obtained in collaboration with
Chaozhong Wu, Di Yang, Ann du Crest de Villeneuve, Pavlo Gavrylenko and
Oleg Lisovyy.
Yoshishige Haraoka (Kumamoto University, Japan)
Title: Katz theory and KZ equation
Abstract: KZ equation is found by Knizhnik and Zamolodchikov as a differential equation satisfied by $n$-point correlation functions in CFT, and has been intensively studied from the viewpoint of representation theory. In this talk, we extend the Katz theory on rigid local systems to KZ equation. We define the additive and multiplicative middle convolutions for KZ equation and apply them to get several remarkable results.
Title: On the monodromy representation of the confluent KZ equation
Abstract: In this talk, we discuss the monodromy representation of
the confluent KZ equation. First we see that the representation of the
framed braid group,
which is the semi-direct product of the braid group and a free abelian
group,
appears as the monodromy representation.
Then, based on the integral representation of solutions,
we construct the representations of the framed braid groups homologically
by using integral cycles.
Title:Tau functions as Widom's constants
Abstract: I am going to explain how to associate a tau function to
the Riemann-Hilbert problem set on a union of non-intersecting smooth
closed curves with generic jump matrix. The main focus will be on the
case of one circle, relevant to the analysis of Painlevé VI
equation, its degenerations to Painlevé V and III as well as its
extension to Fuji-Suzuki-Tsuda system. The tau functions in question
will be defined as block Fredholm determinants of integral operators
with integrable kernels. I will show that they can also be represented
as combinatorial sums over tuples of Young diagrams. For
Riemann-Hilbert problems of isomonodromic origin, these sums coincide
with dual Nekrasov-Okounkov partition functions of certain
supersymmetric gauge theories.
Title: Potential vector fields associated with solutions to Painlevé equations
Abstract: In this talk, we introduce flat coordinates and potential vector fields
on the spaces of variables of isomonodromic deformations of (generalized)
Okubo systems and discuss its consequences. I would like to also treat
algebraic and analytic studies on solutions to the sixth Painlevé
equation in terms of potential vector fields if time permits.
Title: On Painleve/Gauge Theory Correspondence
Abstract: In this talk, I would like to overview the correspondence between Painlevé
equations and four-dimensional rank-one $\mathcal{N}=2$ theories, mostly
from the viewpoint of the latter theories. In addition to the SU(2) gauge
theory with $N_f = 0,1,2,3$ and $4$ which correspond to Painleve III3, III2, III1, V, and VI, we will find that three $\mathcal{N}=2$ rank-one superconformal
theories of Argyres-Douglas type correspond to Painlevé I, II and
IV by studying the Seiberg-Witten curves of the former and the connection
associated with the latter isomonodromic problem. Based on this correspondence
we provide long-distance expansions at various canonical rays for all Painlevé
functions in terms of magnetic and dyonic Nekrasov partition functions
for $\mathcal{N}=2$ gauge theories and Argyres-Douglas theories at self-dual
Omega background or equivalently in terms of $c=1$ irregular conformal
blocks.
Title:Simply-laced quantum connections
Abstract: In this talk, we construct a family of flat connections generalising the
KZ connection. This family is obtained via deformation quantisation of certain
time-dependent Hamiltonian systems controlling the isomonodromy
deformations of meromorphic connections on the Riemann sphere: the
simply-laced isomonodromy systems.
In a first talk we will describe the construction of the simply-laced quantum connections, and in a second one we will compare them with other quantum connections already known in the literature.
Title: A higher order generalization of the Painlevé VI equation with $W(A_{2n+1}^{(1)})$ symmetry
Abstract: In this talk, we propose a higher order generalization of the Painlevé
VI equation from a viewpoint of an affine Weyl group symmetry of type $A$.
Firstly we derive a generalized $P_{\rm VI}$ system from a Drinfeld-Sokolov
type integrable hierarchy by a similarly reduction. Next we state some
properties of our system: Lax pairs, an affine Weyl group symmetry and
a particular solution in terms of the generalized hypergeometric function.
If time permits, we discuss its $q$-analogue and associated $\tau$-functions.
