List of Talks
Szilard Szabo (Technical University of Budapest)
23, July 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Rank three representations of Painlev\'e systems: wild character
varieties and Fourier--Laplace transformation
Abstract :
We give explicit cubic equations for the wild character varieties corresponding to the rank $3$ representations of Painlev\'e equations, and compare them to the ones of their classical rank $2$ representations. We prove that Fourier--Laplace transformation induces a
hyperKaehler isometry between corresponding Hodge moduli spaces.
Arata Komyo (University of Hyogo)
9, July 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Description of generalized isomonodromic deformations of rank two
linear differential equations using apparent singularities
Abstract :
In this talk, we consider the generalized isomonodromic deformations
of rank two irregular connections on the Riemann sphere. We introduce
Darboux coordinates on the parameter space of a family of rank two
irregular connections by apparent singularities. By the Darboux
coordinates, we describe the generalized isomonodromic deformations as
Hamiltonian systems. This talk is based on the papers
[arXiv/2003.08045, 2205.14979].
Koji Hasegawa (Tohoku University, Sendai)
25, June 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Higher rank version of quantum discrete isomonodromic deformation equation
Abstract :
Painlev\'{e} Vi equation (PVI) is known to be the isomonodromic deformation equation for the rank two connection on P^1 with four poles. It is a non-autonomous Hamiltonian system with affine D_4^(1) Weyl group symmetry that transforms solutions to solutions, and the symmetry preserving discretization is known by Jimbo and Sakai. Their discrete PVI equation can be quantized in two ways and gives rise to the identical system as we explained before:1) to quantize the Weyl group action, and 2) to quantize the discrete Lax equation for the discretized isomonodoromic system. In this talk, we will briefly review these results and will proceed to the higher rank case including with the application to the higher rank Nekrasov partition function of the supersymmetric gauge theory. This talk will be based on the joint work with Awata, Kanno, Ohkawa, Shakilov, Shiraishi and Yamada [arxiv/2211.16772, 2309.15364] and ongoing works.
Joshua Lam (Humboldt University, Berlin)
11, June 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
On the converse to Eisenstein's last theorem
Abstract :
Shortly before his death, Eisenstein proved in 1852 that the power series expansion of an algebraic function has almost integral coefficients. In my talk, I will discuss a conjectural converse to this theorem. Concretely, this conjecture predicts when a power series solution to a differential equation, such as Painlevé VI, is algebraic.
I'll then present some evidence for this conjecture, namely, in the setting isomonodromy equations, a very natural class of PDEs including Painlevé equations and Schlesinger systems. This is joint work with Daniel Litt.
Samuel Bronstein (MPI Leipzig, Germany)
4, June 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Universal Variations of Hodge Structures and finite mapping class group orbits
Abstract :
We consider finite orbits of the mapping class groups of the punctured sphere in SL(2,C). A powerful theorem of Corlette and Donaldson gives us an alternative on these representations. Either they are "pullback", i.e. they come in an algebraic family of finite orbits, or they are rigid, and support a structure of universal variation of Hodge structures. We will describe the latter, and explain how it fits into the classification of finite orbits of the mapping class group. This is joint work with Arnaud Maret.
Ivan Tulli (University of Sheffield)
28, May 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Joyce structures, integrable systems and hyperkähler metrics
Abstract :
Joyce structures were introduced by T. Bridgeland in the
context of the space of stability conditions of a three-dimensional
Calabi-Yau category and its associated Donaldson-Thomas invariants, and
have also appeared in the context of isomonodromic systems associated
with Painlevé equations. Under certain non-degeneracy conditions, they
encode a complex hyperkähler structure on the tangent bundle of the base
of the Joyce structure. Taking inspiration from this work, I'll give a
description of (real) hyperkahler metrics compatible with an integrable
system structure that is very similar to the description of Joyce
structures. In particular, I will give examples of such hyperkähler
metrics where there is a precise analog of the Plebański function
appearing in the context of Joyce structures. This is work in progress,
which is a follow-up to https://arxiv.org/abs/2403.00548.
Mohammad Farajzadeh-Tehrani (University of Iowa)
21, May 2025 (Wed) 20:30～(JST)/13:30～(CEST/UTC+2) 6:30 (CDT/UTC-5)
The Geometric P=W Conjecture and Thurston's Compactification
Abstract :
The Geometric P=W conjecture predicts topological features of suitable
projective compactifications of character varieties associated to
Riemann surfaces, both closed and punctured. For SL(2,C) character
varieties, we explore the foundational relationship between this
conjecture and key aspects of Thurston's compactification of Teichmüller
space. We present a theorem in progress for the closed case and outline
recent explicit results confirming the conjecture in the punctured
setting. This talk is based on joint work with Charlie Frohman and
Ashwin A. Kutteri.
Arnaud Maret (University of Starsbourg)
14, May 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Hyperbolic triangle chains and finite mapping class group orbits
Abstract :
Algebraic solutions to Painlevé VI—and more generally, to rank 2 Schlesinger systems—are closely linked to finite mapping class group orbits in character varieties of genus-zero surface group representations into SL(2,C). In this talk, I will describe a method for constructing such finite orbits using elementary techniques from hyperbolic geometry. I will also outline how this approach naturally leads to a classification of all finite mapping class group orbits via an induction on the topology of the underlying punctured spheres. This is joint work with Samuel Bronstein.
Hiroshi Kawakami (Aoyama Gakuin University)
30, April 2025 (Wed) 17:30～(JST)/10:30～(CEST/UTC+2)
Toward a comprehensive theory of Painlev\'e-type equations with a focus on spectral types
Abstract :
Recently, research on higher-dimensional Painlev\'e-type differential equations has progressed, and particularly in the case where the phase space is four-dimensional, it can be said that we have obtained a comprehensive understanding of Painlev\'e-type differential equations. On the other hand, in the two-dimensional case, there exists a framework based on discrete Painlev\'e equations, within which the Painlev\'e differential equations are naturally positioned (Sakai's theory). Similarly, we aim to construct a framework based on discrete equations for higher-dimensional cases as well.
In this talk, I will present my research results on higher-dimensional Painlev\'e-type differential, difference, and q-difference equations, from the viewpoint of the deformation theory of linear equations.
Valerio Toledano Laredo (Boston)
26, March 2025 (Wed)
20:30--21:30 (JST) / 12:30--13:30 (CET/UTC+1) / 7:30--8:30 (Boston)
Title: Stokes phenomena, quantum groups and Poisson-Lie groups.
Abstract :
Quantum groups have long been known to be related to Conformal Field Theory through the monodromy of the Knizhnik-Zamolodchikov (KZ) equations, which are known to be a quantisation of the Schlesinger equations. I will explain a more recent version of this construction which relies on the dynamical KZ (DKZ) equations. Unlike their precursors, these have irregular singularities when the points z1,…,zn are infinitely far apart.
The corresponding Stokes matrices turn out to be the R-matrices of the quantum group. As the irregular parameter changes, these vary according to the Casimir equations, which are a quantisation of the IMD equations for irregular singularities of Poincaré rank 1.
In a parallel development, Boalch constructed the Poisson structure on the dual $G^*$ of a complex reductive group $G$ by using Stokes phenomena for such connections.
I will briefly review Boalch’s construction, and show that it can be obtained as a semiclassical limit of the construction of quantum groups from the DKZ equations.
Jean-Pierre Ramis （Université Toulouse III - Paul Sabatier）
19, March 2025 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
New approaches of confluences of character varieties of Painlevé equations via wild monodromy groupoids and affine del Pezzo surfaces.
Abstract :
During $2012$ spring, in a lecture at a conference in Wuhan university in China, I proposed a program around some (unformal) conjectures for the $9$ ``irregular" Painlev\'e equations ($P_J$, $J \neq VI$). In particular, I proposed the idea of non linear natural dynamics on the Okamoto varieties of initial conditions and conjectured that they are algebraized into rational dynamics on the character varieties by the (wild) Riemann-Hilbert coreespondance. The first decisive step was realized by Martin Klimes in $2016$ (cf. Painlv\'e seminar $2021$), with a very complete study of the confluence $P_{VI} \rightarrow P_V$. He proved in particular that the confluence of the corresponding character varieties $\mathcal{C}_{VI}$ and $\mathcal{C}_V$ (interpreted as affine cubic surfaces) can be expressed by a one parameter family of birational morphisms: $\psi_\kappa: \mathcal{C}_V \rightarrow \mathcal{C}_{VI,\kappa}$.
Later, with E. Paul, we recovered the $\psi_\kappa$ by ``duality", from a one parameter family of very simple natural morphisms $\phi_\kappa: \pi_1^{VI,\kappa} \rightarrow \pi_1^V$ of \emph{wild fundamental groupoids}. These groupoids are not the ``classical" wild groupoids used by several authors (as Philip Boalch). We grafted on these classical groupoids two ``algebraic tori of loops" and this is crucial.
(See the longer abstract
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Shun Shimomura (Keio University)
5, March 2025 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Boutroux ansatz for
the degenerate third Painlev\'e transcendents
Abstract :
The degenerate third Painlev\'e equation P3d has quite different properties
from those of the complete third Painlev\'e equation; for
example, P3d admits algebraic solutions written by the cube root.
For a general solution of P3d we show the Boutroux ansatz, that is,
present an asymptotic elliptic expression near the point at infinity.
Applying some changes of variables to P3d and to the related isomonodromy
linear system, we rewrite them in the forms suitable for our purpose, and
examine the manifold of monodromy data.
For this linear system we first solve the direct monodromy problem by WKB
analysis, and in the next step for given monodromy data we find an asymptotic
form of the desired solution as solution to the inverse monodromy problem,
the validity of which is guaranteed by Kitaev's justification scheme.
Consequently we obtain an asymptotic representation in terms of
the Weierstrass pe-function in cheese-like strips along generic directions.
Publ.~RIMS Kyoto Univ. 60 (2024), arXiv:2207.11495
Mohamad Alameddine (St Etienne)
26, February 2025 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Twisted meromorphic connections of rank-2 on the Riemann sphere and the Painlevé I hierarchy
Abstract :
The geometry of the Painlevé I hierarchy is known to be governed by isomonodromic deformations of twisted meromorphic connections with one ramified pole located at infinity on the Riemann sphere. In this talk, the rank-2 setting of this problem is reviewed and after deforming the base of times the explicit twisted Hamiltonian is presented. The Painlevé I hierarchy is obtained for specific values of the order of the pole after reduction of the twisted Hamiltonian.
Marta Dell'Atti (University of Warsaw)
12, February 2025 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Exploring quartic Hamiltonian systems of Painlevé and quasi-Painlevé type
Abstract :
We investigate non-autonomous Hamiltonian systems that are quartic in the dependent variables, focusing on the local behavior of solutions around movable singularities. These singularities include simple poles (Painlevé type) and algebraic poles (quasi-Painlevé type), with the systems encompassing both cases where singularities are of the same type and cases where they are of mixed types. Using a geometric approach, we assign a surface type to each system by constructing the space of initial conditions (for Painlevé systems) or its analogue as a defining manifold (for quasi-Painlevé systems). The classification is based on the initial base points in the extended phase space (the projective complex plane) and their multiplicities, which arise from the coalescence of four simple base points in the generic case. Through successive degeneration (by setting certain Hamiltonian coefficient functions to zero) and further coalescence of base points, we derive all possible sub-cases of quartic Hamiltonian systems with the quasi-Painlevé property. These sub-cases are characterized by Newton polygons associated with the polynomial Hamiltonians, the types of singularities, and the corresponding surface types. This multi-dimensional description enables a comprehensive classification of the systems. As particular examples, we recover systems equivalent to known Painlevé equations. The talk is based on a joint work with T. Kecker (https://doi.org/10.48550/arXiv.2412.17135).
Duong Dinh (University of Pennsylvania)
5, February 2025 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Opers with apparent singularities from lambda-connections.
Abstract :
On a Riemann surface X, the moduli space of (rank-2 bundles, sub-line bundle) is a
convenient auxiliary moduli space to investigate other moduli spaces associated with rank-2
bundles. I will discuss how the moduli space of lambda-connections on X can be naturally
modelled by an affine bundle on this moduli space, and describe a natural Poisson rational map from this bundle to a parameter space of PSL(2, C)-opers with a fixed number of apparent
singularities. Furthermore, in both moduli spaces of Higgs bundles and flat connections, collision of these apparent singularities with specified behavior of the residue parameters has a limit in lower stratum defined by the C*-action. If time permits, I will also discuss some subtleties related to wobbly bundles and a prediction of new Lagrangians in the moduli space of flat connections.
Olivier Marchal (Université Jean Monnet Saint-Etienne )
29, January 2025 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Isomonodromic deformations, quantization and exact WKB
Abstract :
In this talk, I will review the theory of isomonodromic deformations of meromorphic connections on $\mathfrak{gl}_2$ and the underlying symplectic structure. In particular, I will explain how to obtain explicit formulas for the Hamiltonian systems and the Lax pairs. Next, I will explain how one can formally reconstruct these results using the quantization of the classical spectral curve using topological recursion. Finally, I will explain the current challenges and results to upgrade this formal reconstruction to an analytic one focusing on the genus zero case where one can use Borel resummation of WKB solutions.
Galina Filipuk (University of Warsaw)
8, January 2025 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
The Painleve Equivalence Problem for a Constrained 3D System:
an analytic approach
Abstract :
We propose an analytic and geometric approach to study Painleve equations appearing as constrained systems of three first-order ordinary differential equations.
We illustrate this approach on a system of three first-order differential equations arising in the theory of semi-classical orthogonal polynomials. We show that it can be restricted to a system of two first-order differential equations in two different ways on an invariant hypersurface.
We build the space of initial conditions for each of these restricted systems and verify that they exhibit the Painleve property from a geometric perspective.
Utilising the Painleve identification algorithm we also relate this system to the Painleve VI equation and we build its global Hamiltonian structure.
Finally, we prove that the autonomous limit of the original system is Liouville integrable, and the level
curves of its first integrals are elliptic curves, which leads us to conjecture that the 3D system itself also possesses the Painleve property without the need of restricting it on the invariant hypersurface.
The talk will be based on the paper joint with M. Graffeo, G. Gubbiotti and A. Stokes, which is available at https://arxiv.org/pdf/2411.01657
Nianzi Li (Tsinghua University)
27, November 2024 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Metric asymptotics on the irregular Hitchin moduli space
Abstract :
In this talk, we consider the moduli space of rank-two Higgs bundles with irregular singularities over the projective line. Along a curve of certain type, we show that Hitchin's hyperkähler metric is asymptotic to a simpler semi-flat metric at an exponential rate, based on the foundational works of Fredrickson, Mazzeo, Mochizuki, Swoboda, Weiss, and Witt. In our gluing construction of the harmonic metric, we introduce a new building block for weakly parabolic singularities with trivial flags. In dimension four, we explicitly compute the asymptotic limit of the semi-flat metric, which is of type ALG or ALG*. Joint work with Gao Chen.
Andreas Hohl (Chemnitz University of Technology)
13, November 2024 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Galois descent for generalized monodromy data
Abstract :
A linear differential equation on a complex variety generally determines
a set of topological data. In the case of irregular singularities, these
are usually called "Stokes data", generalizing the classical notion of
monodromy. It is an important question to ask whether such data, which
are a priori defined over the complex numbers, admit a structure over
some smaller field.
In this talk, I will explain some descent results in the context of
irregular singularities, and in particular for hypergeometric systems.
For this, we will use the framework of the irregular Riemann-Hilbert
correspondence of D'Agnolo-Kashiwara in order to study irregular
singular points. This is partly based on joint work with D. Barco, M.
Hien and C. Sevenheck.
Tatsuki Kuwagaki (Kyoto University)
23, October 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
On the generic existence of WKB spectral network/Stokes graph
Abstract :
For a given hbar-differential equation on a Riemann surface, its spectral network/Stokes graph together with Voros coefficients conjecturally controls the connection formula of exact WKB solutions. In this talk, we discuss the generic existence of spectral network/Stokes graph for a large class of hbar-differential equations.
Volodya Roubtsov (Angers University)
17, July 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
Multi-particle Painleve Hamiltonians - Non-commutativity, Reduction and Duality
Abstract :
My talk splits into two related parts: the first is an overview of my results on isomonodromy representation of «multiparticle» Inozemtsev-Takasaki Painleve equations (joint with M. Bertola and M. Cafasso, CMP 2018). The second part presents an analog of the Ruijsenaars duality for these multiparticle Painleve-Calogero systems (joint with I.Gaiur, Revista Math. Iberoamericana, 2024).
Kanehisa Takasaki (Kyoto Univ.(Emeritus) /OCAMI)
3, July 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
Multi-particle Painleve Hamiltonians - Non-commutativity, Reduction and Duality
Abstract :
My talk splits into two related parts: the first is an overview of my results on isomonodromy representation of «multiparticle» Inozemtsev-Takasaki Painleve equations (joint with M. Bertola and M. Cafasso, CMP 2018). The second part presents an analog of the Ruijsenaars duality for these multiparticle Painleve-Calogero systems (joint with I.Gaiur, Revista Math. Iberoamericana, 2024).
Filip Zivanovic (Stony Brook)
26, June 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
Floer-theoretic filtration on Painlevé Hitchin systems
Abstract :
We classify equivariant C*-actions on moduli spaces of Higgs bundles corresponding to the Painlevé equations. We deduce the Floer-theoretic filtrations on the cohomology of these spaces, introduced by Ritter--Zivanovic
We compare this filtration with the P=W and the filtration obtained by multiplicities of the irreducible components of the nilpotent cone, for all the 2-dimensional Higgs moduli. This is a joint work with Szilárd Szabó.
Elba Garcia-Failde (IMJ-PRG, Sorbonne Univ.)
19, June 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
Topological recursion as a mean to quantise spectral curves
Abstract :
For some decades, deep connections have been forming among enumerative geometry, complex geometry, intersection theory and integrability. Topological recursion is a universal procedure that helps building these connections. It associates to some initial data called spectral curve, consisting of a Riemann surface and some extra data, a doubly indexed family of differentials on the curve, which often encode some enumerative geometric information, such as volumes of moduli spaces, matrix model correlation functions and intersection numbers. After an introduction to topological recursion and its relation to different topics, I will focus on the integrability side of the story. The quantum curve conjecture claims that one can associate a differential equation to a spectral curve, whose solution can be reconstructed by the topological recursion applied to the original spectral curve. I will present this problem in some simple cases and comment on some of the technicalities that arise when proving the conjecture for algebraic spectral curves of arbitrary rank, like having to consider non-perturbative corrections. As an example, we will reconstruct the whole two-parameter solutions of Painlevé equations via topological recursion. The last part will be based on joint work with B. Eynard, N. Orantin and O. Marchal.
Veronica Fantini (IHES)
5, June 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
Wall-crossing formulas and Maurer-Cartan equation
Abstract :
Wall-crossing formulas (WCF) usually allow to compute the so called BPS invariants of certain supersymmetric field theories. For instance, the Kontsevich-Soibelman WCFs compute the BPS degeneracies in 4d supersymmetric field theories, equivalently, they compute numerical Donaldson-Thomas invariants on 3 dimensional Calabi–Yau categories. Similarly, BPS indexes in 2 dimensional supersymmetric gauge theories can be computed in terms of Stokes automorphisms. In the so called coupled 2d-4d systems, Gaiotto, Moore and Neitzke showed that BPS degeneracies can be computed using WCFs given by combining the 2d and 4d ones. The aim of this talk is to present the formalism of WCFs in some examples and discuss the relationship between WCFs for coupled 2d-4d systems and solutions of the Maurer-Cartan equation that governs deformations of holomorphic pairs.
Jean Douçot (University of Lisbon)
29, May 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
On the Fourier transform of Stokes data of irregular connections
Abstract :
There is a notion of Fourier transform for connections with irregular singularities on the Riemann sphere, inducing a kind of duality between connections with in general different ranks, numbers of singularities and irregular types. It plays an important role in many contexts, underlying among other things the existence of several different Lax pairs for a given Painlevé-type equation. Since connections with irregular singularities on curves admit a topological description in terms of generalised monodromy data, or Stokes data, the question of describing the Stokes data of the Fourier transform in terms of those of the initial connection arises naturally and has attracted quite some interest recently. In this perspective, I will explain how, by translating some results of T. Mochizuki in the framework of Stokes local systems, one can obtain in a large class of cases a topological algorithm to determine the Stokes data of the Fourier transform in a fully explicit way. This yields isomorphisms between the corresponding wild character varieties, which we conjecture are symplectic. This is joint work with Andreas Hohl (arXiv:2402.05108).
Jean-Baptiste Teyssier (Paris)
24, April 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
Stokes structures in homotopy theory
Abstract :
Stokes structures are algebraic structures that can be extracted from the asymptotic analysis of solutions for flat bundles at singular points. Due to works of Deligne-Malgrange in dimension 1
and Mochizuki in higher dimension, they were shown to realize the irregular Riemann-Hilbert correspondence for flat bundles. In this talk, we will explain how recent advances in stratified homotopy theory provide new perspectives on Stokes structures and their moduli in any dimension. This is joint work with Mauro Porta.
Hélène Esnault (Freie Universitat Berlin)
10, April 2024 (Wed) 17:30～(JST)/10:30～(CET/UTC+1)
Survey on Rigid Local Systems
Abstract :
We’ll describe results obtained mostly with Michael Groechenig in the last 6 years which are centred on
Simpson’s conjecture according to which rigid local systems in complex geometry
should be of geometric origin.
Victoria Hoskins (Radboud Universiteit, Nijmegen)
27, March 2024 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Cohomology and motives of moduli spaces of Higgs bundles
Abstract :
The geometry of moduli spaces of Higgs bundles on a curve is extremely
rich due to its interaction with various related moduli spaces and its
symplectic structure. I will survey some key results concerning the
cohomology of moduli spaces of Higgs bundles and then I will explain
joint work with Simon Pepin Lehalleur on describing the motive of the
Higgs moduli space. I will explain how we use this description to prove
a motivic version of mirror symmetry for the dual Langland's groups SLn
and PGLn, which was first conjectured on the level of cohomology by
Hausel and Thaddeus.
Koji Hasegawa (Tohoku Univ.)
20, March 2024 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Quantization of discretized Painlev´e/Garnier system via affine quantum group
Abstract :
Painlev´e equations have rich symmetry known as ”B¨acklund
transformations” which in fact generate affine Weyl groups.
In the VIth case, the group is of type D_4^(1) and there is a symmetry-
preserving time discretization by Jimbo and Sakai.
We will quantize this, i.e. construct a non-commutative version of
their discrete VIth equation.
There are two ways to quantize: the symmetry point of view and the
monodromy-preserving point of view.
It turns out that these two approaches provide the same equation.
Daisuke Yamakawa (Tokyo Univ. of Science)
6, March 2024 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Symmetries of quiver schemes
Abstract :
This talk is based on joint work with Ryo Terada.
Motivated by works of the speaker and Geiss-Leclerc-Schroer, Hausel-Wong-Wyss associated a family of affine schemes,
called quiver schemes, to each finite quiver equipped with a collection of positive integers
indexed by the vertices (called multiplicities). Such a quiver determines a symmetrizable (possibly non-symmetric)
generalized Cartan matrix, and in the multiplicity-free case,
the matrix is symmetric and the quiver schemes are usual quiver varieties in the sense of Nakajima.
It is well-known that phase spaces of the Painleve equations of type II, III, IV, V and VI
are two-dimensional examples of quiver schemes. In this talk we construct Weyl group symmetries of quiver schemes,
as generalization of reflection functors and Okamoto's symmetries.
Gabriel Calsamiglia (Universidade Federal Fluminense, Niteroi, Brazil)
21, February 2024 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Isoperiodic foliations in moduli spaces of mermorphic forms with simple poles
Abstract :
The equivalence relation "having the same periods" defines a regular holomorphic foliation in the moduli space of meromorphic differentials with
at worst simple poles on complex curves of fixed genus and number of poles. We will present some results that allow to describe the closure of each leaf,
in terms of the topological properties of the set of periods in the complex plane.
Changgui Zhang (Univ. of Lille)
20, December 2023 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
On the connexion formula of a pantograph equation
Abstract :
One calls a pantograph equation the following $q$-difference differential
equation
$$
y'(x)=ay(qx)+by(x)
$$
where $0 < q < 1$ and $ a b \not= 0$. Under the initial condition $y(0)=1$, one finds the power series solution
$$
y_0(x)=1+\sum_{n \geq 1} \frac{(b+a)...(b+aq^{n-1})}{n!}x^n,
$$
that is an entire function. Beside that, given any complex number $\nu$
such that $a\,q^\nu+b=0$, the series
$$
g_\nu(x)=x^\nu\left(1+\sum_{n\ge" 1}\frac{(-\nu)(-\nu+1)...(-\nu+n-1)}{(1-q)(1-q^2)...(1-q^n)}\,q^{n(n+1)/2}\,(b\,x)^{-n}\right)
$$
satisfies the above functional equation. The main question we will talk
about in this seminar is the following: How to express $y_0$ using the
system $\{g_{\nu}\mid a\,q^\nu+b=0\}$ ? Some related problems will also
be considered during the talk.
Reference:
C.Z. Analytical study of the pantograph equation using Jacobi theta functions, Journal of Approximation Theory,
Volume 296, December 2023, 105974.
Gabriele Rembado (IMAG, Univ. of Montpellier)
6, December 2023 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Moduli spaces of wild connections: deformations & quantisations
Abstract :
Moduli spaces of meromorphic connections on (principal bundles over) Riemann
surfaces have a rich geometric structure. In the logarithmic case, they
encompass the complex character varieties of pointed Riemann surfaces,
which assemble into flat Poisson/symplectic fibre bundles upon deforming
the surface; after geometric/deformation quantisation, this yields (projectively)
flat vector bundles over the base space of deformations, famously including
the Knizhnik--Zamolodchikov connection from 2d conformal field theory---
whose semiclassical limit is the Schlesinger system, i.e. in particular
the sixth Painleve equation. In this talk we will aim at a review of part
of this story, and then present recent work about extensions involving
deformations & quantisations of moduli spaces of irregular singular
(`wild') meromorphic connections. These are joint work with (P. Boalch,
J. Doucot, M. Tamiozzo) & (G. Felder, R. Wentworth).
Masa-Hiko Saito (Koube Gakuin Univ./Kobe Univ.)
29, Novemver 2023 (Wed) 17:30～(JST)/9:30～(CET/UTC+1)
Canonical coordinates for moduli spaces of parabolic Higgs bundles and parabolic connections on curves.
Abstract :
We will consider moduli spaces $M_{H}$ (resp. $M$ ) of parabolic Higgs
bundles (resp. parbolic connections) on a curve $C$ of genus $g \geq 2$
with a certain fixed formal structures at singularities of Higgs fields
(resp. connections). Then $M_{H}$ and $M$ can be constructed as algebraic
varieties with natural symplectic sturctures induced by the deformation
theory. For a suitable Zariski dense open subset of $M_H$, we can introduce
coordinate systems $(p_i, q_i)$ and show that $M_H$ is birational to a
Hilbert schemes of points of a total space of a line bundle $L$ of $C$.
In rank 2 case, on a certain Zariski dense open subset $M^{0}$ of $M$ we
can introduce canonical coordinates $(p_i, q_i)$ and we can define the
map from $M^0$ to a symetric products of twisted line bundle $\Omega(L,
c)$. Then, we can show that $M^0$ is birational to the symmetric products
and the map is symplectic. This talk is based on a joint paper with Arata
Komyo, Frank Loray and Szilard Szabo (arXiv:2309.05012) and a joint preprint
with Szilard Szabo.
Kohei Iwaki (U. Tokyo)
15, Novemver 2023 (Wed) 17:30～(JST)/9:30～(CTE/UTC+1)
Non-linear Stokes phenomenon for Painleve τ-function and topological recursion.
Abstract :
I will propose a conjectural statement on the Stokes phenomenon for Painleve τ-function and the topological recursion partition function.
Our proposal is based on the exact WKB theoretic approach to the direct monodromy problem of the isomonodromic quantum curve.
This talk is based on a joint work with Marcos Marino: ArXiv:2307.02080 [hep-th].
Takafumi Matsumoto (Kobe U.)
8, Novemver 2023 (Wed) 17:30～(JST)/9:30～(CTE/UTC+1)
Moduli space of rank three logarithmic connections on the projective line with three poles.
Abstract :
We describe the moduli space of rank three parabolic logarithmic connections on the projective line with three poles for any local exponents.
In particular, we show that the family of moduli spaces of rank three parabolic $\phi$-connections on the projective line with three poles, which are compactifications of moduli spaces of parabolic logarithmic connections,
is isomorphic to the family of $A^{(1)*}_2$-surfaces in Sakai's classification of Painlev\'e equations
Through this description, we investigate the relation between the apparent singularities and underlying parabolic bundles.
Yota Shamoto (WIAS, Waseda U.)
18, October 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Stokes structure of mild difference module.
Abstract :
P. Deligne introduced the notion of a Stokes-filtered local system, or Stokes structure,
to describe the asymptotic behavior of solutions of linear differential equations
in one complex variable.
We shall present the analogous concept for linear difference equations and
formulate the Riemann-Hilbert correspondence under a mild condition on formal structure.
If time permits, we will also discuss the further directions.
Ryo Ohkawa (OCAMI/RIMS)
11, October 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Non-stationary difference equation and affine Laumon space.
Abstract :
We describe the relation of the non-stationary difference equation proposed by Shakirov and
the quantized discrete Painleve VI equation.
The original equation can be factorized as a coupled system for a pair of
functions $(\mathcal{F}^{(1)}, \mathcal{F}^{(2)})$,
which is a consequence of the identification of the Hamiltonian as a translation element in the extended affine Weyl group.
Recently we showed that the affine Laumon partition function gives a solution to Shakirov’s equation.
Motivated by these results, we introduce a wall-crossing formula for framed quiver moduli as a method to study these functions systematically.
This is based on joint work with Awata, Hasegawa, Kanno, Shiraishi, Shakirov and Yamada.
Thiago Fassarella (UFF, Niteroi, Brasil)
27, September 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Logarithmic connections over elliptic curves.
Abstract :
We look at moduli spaces of logarithmic connections with fixed spectral data over elliptic curves.
We show that these moduli spaces have a covering whose members are easily described, and change of
coordinates are determined by elementary transformations.
Using this description, we see that the spectral data is detected by the symplectic structure.
This is a joint work with Frank Loray and Alan Muniz.
Xiaomeng Xu (BICMR, Peking Univ.)
6, September 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Monodromy problem and the boundary condition for some isomonodromy deformation equations.
Abstract :
The talk emphases interrelations between integrable billiards, extremal polynomials, Riemann surfaces, potential theory, and isomonodromic deformations.
We introduce and study dyThis talk focuses on the isomonodromy deformation equation of a meromorphic linear system with Poncare rank 1. Such an equation naturally arises from the theory of
Frobenius manifolds, stability conditions, Poisson-Lie groups and so on,
and can be seen as a higher rank generalization of the sixth Painleve equation.
This talk first gives a parameterization of the asymptotics of generic solutions of the isomonodromy equation at a critical point, and then give the explicit formula of
the monodromy/Stokes matrices of the associated linear problem via the parameterization, as well as a connection formula between two differential critical points.
It can be seen as a generalization of Jimbo's work for the sixth Painleve equation to the higher rank case.
If time allows, the talk discusses the WKB approximation of the nonlinear isomonodromy equation.
Vladimir Dragovic (The University of Texas at Dallas)
30, August 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Chebyshev dynamics, isoharmonic deformations, and constrained Schlesinger systems.
Abstract :
The talk emphases interrelations between integrable billiards, extremal polynomials, Riemann surfaces, potential theory, and isomonodromic deformations.
We introduce and study dynamics of Chebyshev polynomials on several intervals. We introduce a notion of isoharmonic deformations and study their isomonodromic properties.
We formulate a new class of constrained Schlesinger systems and provide explicit solutions to these systems.
One line of motivation for this research goes back to the works of Hitchin in the mid 90’s. The talk is based on joint results with Vasilisa Shramchenko.
Ilia Gaiur (IHES)
23, August 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Higher Bessel Functions: product formulas, Landau-Ginzburg type models and corresponding integer sequences.
Abstract :
Product formulas for the standard Bessel functions as well as special functions of the same type have been known since the works of N.Sonine and L.Gegenbauer from 1880.
Such formulas may be found in various contexts and recently appeared in the works of Golyshev-Mellit-Rubtsov-van Straten
and Kontsevich-Odesskii. In my talk I will report on the progress with V. Rubtsov and D. van Straten on
the multiplication kernels for higher rank versions of the Bessel equation.
I will discuss connections of these kernels with the periods of the Landau-Ginzburg superpotentials,
integrality, linear differential equations, their isomonodromy and quantization.
Daniel Panazzolo (University of Haute-Alsace)
12, July 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Analytic classification of germs of doubly-resonant vector fields in dimension 3; applications to Painlev\'e equation (following A. Bittmann).
Abstract :
In 2016, Bittmann obtained formal and analytic classification of some germs
of vector fields whose low jet is given by $x^2 d/dx + \lambda (y_1 d/d y_1 - y_2 d/d y 2)$.
These germs appear in Boutroux's compactification of the first Painlev\'e equation.
Kento Osuga (University of Tokyo)
28, June 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Quantisation via the Q-top recursion and the Nekrasov-Shatashivili limit.
Abstract :
Topological recursion has become known as a useful notion to compute a variety of invariants in enumerative geometry.
It turns out that the structure of topological recursion is closely related to the Virasoro algebra in the self-dual limit.
Then, an interesting question arises: is there any recursion relevant to the Nekrasov-Shatashivili limit?
In this talk I will introduce such a recursion, which I call the Q-top recursion, and explain why the Q-top recursion is indeed relevant in
the Nekrasov-Shatashivili limit. Then, I will discuss how the Q-top recursion can be utilised to construct the corresponding quantum curve.
If time permits, I will mention an intriguing coincidence between a deformation condition and a quantisation condition of the Q-top recursion.
Sorin Dumitrescu (Universite Côte d'Azur)
14, June 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Holomorphic sl(2,C) differential systems on compact Riemann surfaces and curves in compact quotients of SL(2, C)
Abstract :
We explain the strategy of a recent result that constructs holomorphic
sl(2,C)-differential systems over some Riemann surfaces $\Sigma_g$ of genus $g
\geq 2$ such that the image of the associated monodromy homomorphism is
some cocompact Kleinian subgroup $\Gamma \subset SL(2,{\bf C})$.
As a consequence, there exist holomorphic maps from $\Sigma_g $ to the quotient $ SL(2, {\bf C})/ \Gamma $ , that do not factor through any elliptic curve. This answers
positively a question asked by Huckleberry and Winkelmann, also raised
by Ghys. This is a joint work with Indranil Biswas (TIFR, Mumbai), Lynn
Heller (BIMSA, Beijing) and Sebastian Heller (BIMSA, Beijing).
Katsunori Iwasaki (Hokkaido Univ.)
31, May 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Hypergeometric groups and automorphisms of K3 surfaces
Abstract :
A hypergeometric group is a matrix group modeled on the monodromy
group of a generalized hypergeometric differential equation. We use hypergeometric
groups to produce non-projective K3 surface automorphisms of positive entropy,
especially automorphisms with Siegel disks.
Irina Bobrova (HSE Univ.)
17, May 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
<Non-Abelian Painleve equations
Abstract :
Non-abelian extensions of various integrable systems are one of the interests of modern mathematical physics.
Due to the close connection between integrable models and the Painleve equations,
the latter provide a good example of this phenomena.
In the present talk, we will discuss classification problems related to non-abelian generalisations
for the continuous and discrete Painleve equations.
This talk is based on a series of papers joint with Vladimir Sokolov and an ongoing project
with Vladimir Retakh, Vladimir Rubtsov, and Georgy Sharygin.
Michi-aki Inaba (Kyoto Univ.)
10, May 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Canonical 2-form on the moduli space of ramified connections
Abstract :
This talk is based on the paper
``Moduli space of factorized ramified connections and generalized isomonodromic deformation''
SIGMA Symmetry Integrability Geom. Methods Appl. 19 (2023), Paper No. 013.
In this talk, I will explain a rough idea to construct the algebraic symplectic form on the moduli space of connections
including ramified irregular singularities. On a family of moduli spaces of connections, we can construct the generalized isomonodromic subbundle
of the tangent bundle. Using this tool, we can lift the relative symplectic form
to a total 2-form which we call the generalized isomonodromic 2-form.
Yosuke Ohyama (Tokushima Univ.))
5, April 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
Global theory on q-Painleve equations
Abstract :
We study a q-analog of character variety of q-Painleve equations.
Jimbo found that the character variety of the sixth Painleve equation is
The Fricke cubic in 1982. We study a q-analog of the sixth Painleve equation
And the character variety is the Segre surface.
This is a jointed work with J.-P. Ramis and J, Sauloy.
Harini Desiraju (The Univeristy of Sydney)
29 March 2023 (Wed) 17:30～(JST)/10:30～(UTC+2)
The solutions of isomonodromic equations on a torus in terms of Fredholm determinants and consequences
Abstract :
In this talk, I will show the construction of the isomonodromic tau-function as a Fredholm determinant starting from the Lax pair.
I will then use the determinant to obtain a correspondence between the symplectic forms on the initial value and monodromy spaces.
This is based on the recent works with Fabrizio Del Monte and Pasha Gavrylenko.
Jacques Sauloy (Cantoperdic)
1 March 2023 (Wed) 17:30～(JST)/9:30～(UTC+2)
Various forms of the "Riemann-Hilbert-Birkhoff correspondence for
q-difference equations"
Abstract :
In 1913, Birkhoff proposed a new solution (different of that of Hilbert and Plemelj) to what he called
the "Riemann problem" and used it to provide a unified framework for (linear complex analytic)
differential, difference and q-difference equations. Starting from Birkhoff, I shall describe various
ways of associating "intrinsic transcendental invariants" (as Birkhoff said) to q-difference equations:
geometric, galoisian, cohomological ... up to a mysterious sheaf theoretical formulation by Kontsevitch
and Soibelman. Most of that will rest on work by the Ramis school.
Jean-Baptiste Teyssier (Sorbonne University)
15 February 2023 (Wed) 17:30～(JST)/9:30～(UTC+2)
Cohomological boundedness for flat bundles and applications
Abstract :
We will advertise some consequences of the conjectural existence of moduli spaces for flat bundles
with prescribed behavior at infinity.
If time permits, we will give the ingredients of unconditional proofs in the surface case.
This is joint work with Haoyu Hu.
Toshiyuki Mano (University of the Ryukyus)
1 February 2023 (Wed) 17:30～(JST)/9:30～(UTC+2)
Flat structures on solutions to the sixth Painlev\'e equation
Abstract :
We will talk about some relationship between a geometric notion called ``flat structure (without metric)''
and solutions to the sixth Painlev\'e equation.
In our formulation, a flat structure is constructed based on ``a space of Okubo-Saito potentials''.
Typical examples of spaces of Okubo-Saito potentials include
1) a space of period integrals of (K. Saito's) primitive form,
2) the dual space $V^*$ of the standard representation space $V$ of a well-generated unitary reflection group $G$.
Then, there exists a correspondence between three dimensional generically semi-simple flat strucures
and generic solutions to the sixth Painlev\'e equation.
This correspondence enable us to introduce invariants of solutions to the sixth Painlev\'e equation.
Claude Sabbah (Ecole Polytechnique)
18 January 2023 (Wed) 17:30～(JST)/9:30～(UTC+2)
Rigid irreducible meromorphic connections in dimension one
Abstract :
I will illustrate the Arinkin-Deligne-Katz algorithm for rigid irreducible meromorphic bundles
with connection on the projective line by giving motivicity consequences similar
to those given by Katz for rigid local systems.
Szilard Szabo (Alfred Renyi Inst./
Budapest Univ. of Technology and Econ.))
11 January 2023 (Wed) 17:30～(JST)/9:30～(UTC+2)
<Quantum spectra of birationally equivalent surfaces
Abstract :
We discuss the spectrum of the quantum connection associated to a smooth projective surface, and in particular the effect of blow-ups on this spectrum. Joint work with Adam Gyenge.
Emmanuel Paul (Univ Toulouse III)
4 January 2023 (Wed) 17:30～(JST)/9:30～(UTC+2)
Several dynamics on the Painleve V character variety
Abstract :
The leaves of the Painleve foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections.
Therefore they are the fibers of the Riemann-Hilbert correspondence which send each connection on its monodromy data,
and this correspondence induces a conjugation between the dynamics of the foliation and a dynamic on a space of representations of some fundamental groupoid (a character variety).
This one can be identified to a family of cubic surfaces through trace coordinates. We describe here the dynamics on the character variety related to the Painleve V equation.
We have here to consider irregular connections, and representation of wild groupoids.
We describe and compare all the dynamics which appear on this wild character variety: the tame dynamics,
the confluent dynamics, the canonical sympletic dynamics and the wild dynamics.
Tatsuki Kuwagaki, (Kyoto Univ.)
23 November 2022 (Wed) 17:30～(JST)/9:30～(UTC+2)
Sheaf quantization and Riemann--Hilbert correspondence
Abstract :
Sheaf quantization (SQ, for short) is a topological version of quantization of Lagrangian submanifold.
One can regard SQ as an enhancement of the notion of constructible sheaves.
Therefore, one can expect that a setup, in which constructible sheaves have played a role,
can be renewed with SQ.
In this talk, I will start with an introduction to SQ and then explain an SQ-version of RH-correspondence.
WKB analysis plays a central role in the latter formulation.
Oleg Lisovyy（Univ. Tours) )
16 November 2022 (Wed) 17:30～(JST)/9:30～(UTC+2)
Perturbative connection formulas for Heun equations
Abstract :
Connection formulas relating Frobenius solutions of linear ODEs at different Fuchsian singular points can be expressed
in terms of the large order asymptotics of the corresponding power series. I will show that for the Heun equation and
some of its confluent versions, the series expansion of the relevant asymptotic amplitude in a suitable parameter can
be systematically computed to arbitrary order. This allows to check a recent conjecture of
Bonelli-Iossa-Panea-Tanzini expressing the Heun connection matrix in terms of quasiclassical Virasoro conformal blocks.
I will also discuss a potential relation of this conjecture to the extended Jimbo-Miwa-Ueno 1-form.
Martin Guest (Waseda University)
9 November 2022 (Wed) 17:30～(JST)/9:30～(UTC+2)
Hamiltonian aspects of the tt*-Toda equations
Abstract :
The tt*-Toda equations can be regarded as a higher-rank generalization of the (real) PIII(D8) equation.
They are a system of "integrable" nonlinear pde (both in the zero-curvature/Lax pair sense, and the isomonodromy sense).
Their (smooth) solutions have rich connections to geometry, e.g. Frobenius manifolds and quantum cohomology.
There is a version of these equations for any complex semisimple Lie algebra,
and the monodromy data (Stokes data) exhibits interesting links with Lie theory.
In particular there is a Lie-theoretic description of the moduli space of (all) solutions.
We shall discuss some symplectic/Hamiltonian aspects of this description.
Masato Wakayama (NTT Institute for Fundamental Mathematics)
19 October 2022 (Wed) 17:30～(JST)/10:30～(UTC+2)
Quantum Interaction and Number Theory, Representation Theory - modular
forms a bit beyond, infinite symmetric group, Fuchsian ODE
Abstract :
In this talk, we will overview various spectral structure of the quantum
Rabi model (QRM) and its asymmetric version (AQRM), which are widely
studied most fundamental models of light-matter interactions. Further,
we will present their ``covering" model NCHO (Non-commutative harmonic
oscillator), which has rich but mysterious arithmetic properties, such
as in a closed relation with modular forms elliptic curves, (a slightly
extended) Eichler forms and corresponding cohomology. Targets of the
materials are the discrete path integral (infinite series)
representation of the heat kernel (and partition functions) of the QRM
in relation with representation theory of infinite symmetric group,
special values of the spectral zeta functions of NCHO (and QRM), and a
hidden symmetry for AQRM. In addition, certain open problems/conjectures
will be given.
Titouan Serandour (Univ Rennes/ENS Lyon)
<12 October 2022 (Wed) 17:30～(JST)/10:30～(UTC+2)
The monodromy of meromorphic projective structures
Abstract :
Painleve VI foliation is the isomonodromic
foliation of the space of complex projective structures on the
Riemann sphere CP1 with four singularities and one branch point. A
compact complex projective structure P is a curve locally modeled
on CP1. To such a geometric object is associated an algebraic one
: a representation of its fundamental group into the automorphism
group of CP1, up to conjugacy. This is defined through the
monodromy of the analytic continuations of charts of P. The
monodromy map, which (roughly speaking) to a projective structure
on a fixed compact oriented surface S associates its monodromy, is
neither surjective nor injective. However, Dennis A. Hejhal proved
in 1975 that it is locally injective. During my PhD thesis
supervised by Frank Loray, I have generalised Hejhal's theorem to
meromorphic projective structures. This talk will introduce the
subject and present the technics used in the proof.
Yuma Mizuno (Chiba University)
5 October 2022 (Wed) 17:30～(JST)/10:30～(UTC+2)
q-Painleve systems, toric surfaces, and cluster Poisson varieties
Abstract :
A space of initial conditions of q-Painleve systems is constructed as a
composition of blowups from a toric surface at smooth points on the
toric boundary. We see that although the choice of toric surfaces and
blowup points are not unique, they are related by a composition of
simple operations called mutations. From this point of view, we explain
the relationship between q-Painleve systems and the theory of cluster
algebras. This talk is based on arXiv:2008.11219.
Yoshishige Haraoka (Kumamoto University)
28 September 2022 (Wed) 17:30～(JST)/10:30～(UTC+2)
Dynamical system on KZ type equations and deformation
Abstract :
A KZ type equation is a Pfaffian system given by a logarithmic 1-form singular along
the diagonal. We have shown that Katz' middle convolution operation can be extended to an
operation for KZ type equations. Another important operation for KZ type equations is a restriction to a singular locus.
These operations define a dynamical system of the moduli space of KZ type equations.
We study this dynamical system. In particular, we study the relation among middle convolution and restriction to a
singular locus, which brings an extended notion of deformation. We also study a restriction to a critical regular locus.
In studying the dynamical system, we get several algebraic solutions of deformation
equations and also non-rigid equations with integral representations of solutions.
Milena Radnovic (The University of Sydney)
26 August 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
Asymptotic behaviour of Painleve transcendents in the Okamoto space
Abstract :
We study dynamics of solutions of Painleve equations in the initial value space as the independent variable approaches singularity.
Our main results describe the repeller set, show that the number of poles and zeroes of general solutions is unbounded,
and that the complex limit set of each solution exists and is compact and connected.
Aaron Landesman (Harvard University)
8 July 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
Geometric local systems on very general curves and isomonodromy
Abstract :
What is the smallest genus h of a non-isotrivial curve over the generic genus g curve? In joint work with Daniel Litt, we show h is more than
$\sqrt{g}$ by proving a more general result about local systems on sufficiently general curves.
As a consequence, we show that local systems on a sufficiently general curve of geometric origin are not Zariski dense in
the character variety parameterizing such local systems. This gives counterexamples to conjectures of Esnault-Kerz and Budur-Wang.
The main input in the proof is an analysis of stability properties of flat vector bundles under isomonodromic deformation.
The result is also a key step in classifying algebraic solutions to a higher genus analog of the Painleve VI equation,
which Daniel Litt has spoken about previously in this seminar.
Julio Rebelo (University of Toulouse)
24 June 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
Dynamics of groups of birational automorphisms of cubic surfaces and Fatou/Julia decompositions for Painleve 6.
Abstract :
We study the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces
SA,B,C,D={(x,y,z)∈C3:x2+y2+z2+xyz=Ax+By+Cz+D}, where A,B,C, and D are complex parameters.
This dynamical system arises naturally in the study of character varieties and
it also describes the monodromy of the famous Painleve 6 differential equation.
We explore the Fatou/Julia dichotomy for this group action (defined analogously to
the usual definition for iteration of a single rational map)
and also the Locally Discrete / Non-Discrete dichotomy (a non-linear
version from the classical discrete/non-discrete dichotomy for subgroups of Lie groups).
The interplay between these two dichotomies allows us to prove several results about the topological
dynamics of this group. Moreover,
we show the coexistence of non-empty Fatou sets and Julia sets with non-trivial interior for a large set of parameters.
This is joint work with Roland Roeder.
Daniel Litt (University of Georgia)
10 June 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
Canonical Representations of Surface Groups
Abstract :
We consider representations of surface groups whose conjugacy class has finite orbit under the mapping class group of,
or equivalently algebraic isomonodromic deformations over the moduli space of curves.
We give a complete classification of such representations when --
they are precisely the representations with finite image.
This answers questions of Junho Peter Whang and Mark Kisin.
The proof relies on (mixed and nonabelian) Hodge theory and takes input from the Langlands program.
Pieter Roffelsen (The University of Sydney )
13 May 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
On a space of connection matrices associated with $q$-Painlev\'e VI
Abstract :
The linear problem associated with a classical Painlev\'e equation gives rise to a
bijective correspondence between solutions of the ODE and points on a monodromy
manifold. In turn, such a monodromy manifold can be identified with an affine cubic
surface so that each point on this surface represents a unique solution of the
Painlev\'e equation at hand. In this talk, I will discuss an analogue correspondence
for the $q$-difference Painlev\'e VI equation, obtained in a recent work with Nalini
Joshi. The associated monodromy manifold is a space of connection matrices which
appear in the classical papers by Birkhoff and his students on linear $q$-difference
equations. I will show how this manifold can be identified with the affine
intersection of two quadrics in $\mathbb{C}^4$ and detail some of its properties in
relation to $q$-Painlev\'e VI.
Arata Komyo (Kobe University )
6 May 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
Moduli space of irregular rank two parabolic bundles over the Riemann
sphere and its compactification
Abstract :
We consider rank 2 (quasi) parabolic bundles over the Riemann sphere with
an effective divisor, which is not necessary reduced. We investigate the
moduli spaces of parabolic bundles. In particular, we are interested in
the cases where the degree of the effective divisor is 5. In these cases,
the dimensions of the moduli spaces are 2. For these cases, we describe
the moduli spaces of parabolic bundles in detail. This description is
related to geometry of some weak del Pezzo surfaces. This is a joint work
with Frank Loray and Masa-Hiko Saito (arXiv:2203.10816).
Jean Douçot (Université de Genève)
15 April 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
Diagrams and the classification of wild character varieties
Abstract :
Wild character varieties are moduli spaces of generalized monodromy data of meromorphic connections with irregular singularities,
whose explicit presentations involve products of Stokes matrices. Remarkably, it happens that wild character varieties associated to
connections with different ranks, number of singularities, and irregular types may be isomorphic, which raises the question of their classification.
An approach to this problem consists in associating to a connection a diagram encoding its singularity data,
such that the same diagram can be read in several different ways as coming from different connections. I will review known instances of this story,
present a recent generalization to the case of connections with an arbitrary number of (possibly twisted)
irregular singularities, and discuss examples involving
Painlevé-type equations where different Lax pairs for the same equation
can be seen as different readings of the same diagram. This is based on arXiv:2107.02516.
Giordano Cotti (SISSA, Trieste)
8 April 2022 (Fri) 17:30～(JST)/10:30～(UTC+2)
Quantum differential equations, isomonodromic deformations, and derived
categories
Abstract :
The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X.
It is an ordinary differential equation in the complex domain which encodes information
of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore,
the asymptotic and monodromy of its solutions conjecturally rules also the topology and complex geometry of X.
These differential equations were introduced in the middle of the creative impetus for mathematically rigorous foundations of
Topological Field Theories, Supersymmetric Quantum Field Theories and related Mirror Symmetry phenomena.
Special mention has to be given to the relation between qDE's and Dubrovin-Frobenius manifolds,
the latter being identifiable with the space of isomonodromic deformation parameters of the former.
The study of qDEÃÔ represents a challenging active area in both contemporary geometry and mathematical
physics: it is continuously inspiring the introduction of new mathematical tools,
ranging from algebraic geometry, the realm of integrable systems, the analysis of ODEÃÔ,
to the theory of integral transforms and special functions. This talk will be a gentle
introduction to the analytical study of qDEÃÔ, their relationship
with derived categories of coherent sheaves (in both non-equivariant and equivariant settings),
and a theory of integral representations for its solutions. The talk will be a
survey of the results of the speaker in this research area.
Galina Filipuk (University of Warsaw, Poland)
18 March 2022 (Fri) 17:30～(JST)/9:30～(UTC+1)
Hamiltonians of the Painlevé and quasi-Painlevé equations
Abstract :
In this talk I will expalain recent results on the Hamiltonians of the
Painlevé and quasi-Painlevé equations using the geometric approach of K.
Okamoto and H. Sakai. The talk will be based on several papers and
preprints joint with A. Dzhamay, A. Ligeza, A. Stokes and T. Kecker.
Masatoshi Noumi (Kobe University)
25 February 2022 (Fri) 17:30～(JST)/9:30～(UTC+1)
Remarks on Umemura polynomials
Abstract :
The Umemura polynomials refer to a sequence of special polynomials
associated with certain algebraic solutions of the sixth Painleve
equation.
After explaining remarkable combinatorial properties of Umemura
polynomials, I formulate a conjecture regarding an explicit formula
for generalized Umemura polynomials with two discrete parameters.
Anton Shchechkin (Skoltech and HSE University, Moscow)
18 February 2022 (Fri) 17:30～(JST)/9:30～(UTC+1)
Folding transformations for q-Painlevé equations
Abstract :
Folding transformation of the Painlevé equations is an
algebraic (of degree greater than 1) transformation between solutions
of different equations. In 2005 Tsuda, Okamoto and Sakai classified
folding transformations of differential Painlevé equations. These
transformations are in correspondence with automorphisms of affine
Dynkin diagrams.
We give a complete classification of folding transformations of the
q-difference Painlevé equations, these transformations are in
correspondence with certain subdiagrams of the affine Dynkin diagrams
(possibly with automorphism). The method is based on Sakai's approach
to Painlevé equations through rational surfaces.
The talk is based on joint work with Mikhail Bershtein, arXiv 2110.15320.
28 January 2022 (Fri) 17:30～(JST)/9:30～(UTC+1)
Analytic description of non-generic isomonodromy deformations and some applications.
Abstract :
We will give an overview of results on isomonodromic deformations of a class of irregular systems which are non-generic, in the sense that the deformation parameters,
which are the eigenvalues of the leading matrix at an irregular singularity, vary in a domain containing a "coalescence" set (a set where some eigenvalues merge).
This has a counterpart in the coalescence of Fuchsian singularities when a Laplace transform is applied to the irregular system. The point of view will be classical, namely analytic. Some applications will be sketched, which include Dubrovin-Frobenius manifolds and Painlevé equations.
14 January 2022 (Fri) 17:30～(JST)/9:30～(UTC+1)
Fractional analysis of linear differential equations on the Riemann sphere
Abstract :
Middle convolutions are quite effective to analyze linear differential equations on the Riemann sphere in cooperate with other operations such as gauge transformations, extensions to several variables and restrictions to curves, unfoldings and confluences etc.
An analysis of the solutions to the equations using these operations will be explained.
In particular, versal unfoldings, integral representations of the solutions, connection problems (Stokes coefficients) etc. will be discussed.
Indranil Biswas (Tata Institute of Fundamental Research)
17 December 2021 (Fri) 17:30～(JST)/9:30～(UTC+1)
Holomorphic connections on the trivial holomorphic bundle over Riemann surfaces.
Abstract :
The lecture is based on works arXiv:2104.04818, arXiv:2008.11483, arXiv:2003.06997,
arXiv:2002.05927 done with S. Dumitrescu, L. Heller, S. Heller, J. P. dos
Santos and T. Mochizuki. We investigate the properties of the holomorphic
connections on the trivial holomorphic bundle on a Riemann surface and
more generally on a compact Kaehler manifold.
3 December 2021 (Fri) 17:30～(JST)/9:30～(UTC+1)
Painlevé equations, foliations and neighborhoods of curve.
Abstract :
We will discuss two problems about transcendental features
of Painlevé foliations in relationship with neighborhoods of compact
curves in complex surfaces. One, about neighborhoods of rational curves,
is a work in progress with Maycol Falla Luza. The second one, about
neighborhoods of elliptic curves, is a work in progress with Gibran Espejo
and Laura Ortiz, using recent classification obtained by Frederic Touzet,
Sergei Voronin and the author.
19 November 2021 (Fri) 17:30～(JST)/9:30～(UTC+1)
Isomonodromic deformations and degenerations of irregular singularities
Abstract :
Isomonodromic (that is, integrable) deformations of connections with irregular singularities in dimension one are well understood away from turning points of the parameter space.
In general, at the turning points, the theorem of Kedlaya-Mochizuki is needed to understand the local behaviour of the Stokes structure,
but it breaks the notion of deformation. Motivated by understanding boundaries of Frobenius manifolds, Cotti,
Dubrovin and Guzzetti have analyzed some simple turning points and shown vanishing of certain entries of the Stokes matrices
at the neighbourhood of these turning points. The talk will give a different point of view on these results.
5 November 2021 (Fri) 17:30～(JST)/9:30～(UTC+1)
Irregular conformal blocks and Painlevé equations
Abstract :
We review series representations of tau functions of
Painlevé equations and their relations to irregular conformal blocks,
which are defined as expectation values of vertex operators for
Virasoro algebra on irregular Verma modules. A conjectural
combinatorial formula for a three point irregular conformal block is
given. Toward proving that series representations of tau functions of
Painlevé equations in terms of irregular conformal blocks satisfy
bilinear equations, irregular vertex operators for a super Virasoro
algebra(Neveu-Schwarz-Ramond algebra) are presented.
29 October 2021 (Fri) 17:30～(JST)/10:30～(UTC+2)
Isomonodromy and Painlevé type equations. Search and Case studies.
Abstract :
We present a method to construct certain families $\mathcal{M}$ of connections on the projective line.
The fibres of the Riemann--Hilbert morphism $RH:\mathcal{M}\rightarrow \mathcal{R}$, where $\mathcal{R}$ denotes the family of analytic data, should be parametrized by a variable $t$. This produces a Lax pair and Painlevé type equations.
The analytic classification of singularities of connections will be presented, because this is essential background for the construction of
$\mathcal{M}$ and $\mathcal{R}$. The method produces besides the classical Painlevé equations, new families of Painlev\'e type equations.
8 October 2021 (Fri) 17:30～(JST)/10:30～(UTC+2)
Moduli of quasi-parabolic logarithmic connections of rank 2, and construction
of a twistor space
Abstract :
We look at the moduli space of rank 2 logarithmic lambda-connections with quasi-parabolic
structure on a curve. Up to the action of a groupoid of local gauge transformations, there is a
Riemann-Hilbert correspondence. This in turn leads to the construction of the Deligne-Hitchin
twistor space such that harmonic bundles give preferred sections. The relative
tangent bundle along a preferred section has a mixed twistor structure where the
weight two piece parametrizes the deformations of local monodromy transformations at
the singularities.
Hidetaka Sakai (The University of Tokyo)
1 October 2021 (Fri) 17:30～(JST)/10:30～(UTC+2)
Discrete Hamiltonians of discrete Painleve equations
(joint work with T. Mase and A. Nakamura)
Abstract :
We express discrete Painleve equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painleve equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
10 September 2021 (Fri) 17:30～(JST)/10:30～(UTC+2)
Genus two curves associated with the autonomous 4-dimensional
Painlevé-type systems
Abstract :
The 4-dimensional Painlevé-type systems are tangible (in a sense) yet
give nontrivial higher-dimensional analogs of the 2-dimensional
Painlevé systems. In this talk, we study the genus two curves
associated with the autonomous versions of the 4-dimensional
Painlevé-type systems. As an application of the result, we can find a
linear problem starting from a nonlinear problem. We will mention
some connections to work on "Generalized Hitchin systems on rational
surfaces" by Eric Rains. This talk is partially based on joint work
with Eric Rains.
7 July 2021 (Wed) 17:30～(JST)/10:30～(UTC+2)
Moduli varieties of twisted local systems.
Abstract :
Complex analytic orbi-curves give rise to natural examples of twisted local systems, for which the fundamental group acts non-trivially on the coefficients.
In this talk, we construct moduli varieties of twisted local systems, and prove that these are affine varieties over the complex numbers,
whose (strong) topology can be studied through an appropriate version of the non-Abelian Hodge correspondence to the case of nonconstant coefficients.
This partially answers a question of Carlos Simpson on the meaning of the Dolbeault moduli space in the nonconstant case.
Takafumi Matsumoto (Kobe University)
23 June 2021 (Wed) 17:30～(JST)/10:30～(UTC+2)
Birational geometry of moduli spaces of rank 2 logarithmic connections.
Abstract :
We study the birational structure of moduli spaces of rank 2 logarithmic connections on smooth projective curves.
We generalize a previous result by F. Loray and M. -H. Saito in the projective line case.
Our approach is to analyze the underlying parabolic bundles and apparent singularities of the parabolic connections.
Guy Casale (IRMAR, Université de Rennes 1)
9 June 2021 (Wed) 17:30～(JST)/10:30～(UTC+2)
Algebraic relations between solutions of differential equations.
Abstract :
In 2004, K Nishioka showed that if y_1, ... , y_n are solutions of the first Painlevé equation
such that the transcendence degree of the extension of C(t) by y_1,y'_1, ... , y_n,y'_n is strictly
less than 2n, then there exist i<j such that y_i = y_j.
This result was extended to other Painlevé equations by J.Nagloo and A.Pillay in 2017 using Hsushovski-Skolovic trichotomy theorem in Model Theory of differential field of characteristic 0.
In this talk, I will explain how the Galois pseudogroup defined independently by B. Malgrange and H. Umemura can be used as an alternative to the trichotomy theorem.
26 May 2021 (Wed) 17:30～(JST)/10:30～(UTC+2)
Quantum representation of Weyl group $W(E_8^{(1)})$
Abstract :
We study a quantum (non-commutative) representation of the affine Weyl
group of type $E_8^{(1)}$.
The representation is given by birational actions on two variables $x, y$ with $q$-commutation relation $yx=qxy$.
We also construct a lift of the representation including the tau variables.
The Weyl group actions on tau variables are described by interesting quantum polynomials $F(x,y)$.
We give a characterization of the polynomials using their singularity structures as the $q$-difference operators.
As an application, the quantum mirror curve for 5d E-string is rederived
by the Weyl group symmetry.
This talk is base on the joint work with S. Moriyama, arXiv.:2104.06661[math.QA].
12 May 2021 (Wed) 17:30～(JST)/10:30～(UTC+2)
Confluent approach to Fifth Painlevé equation
Abstract :
The Fifth Painlevé equation (PV) is obtained from the
Sixth one (PVI) by confluence. In principle, this allows to transfer knowledge
from PVI to PV, however, to do this, one needs to be able to deal with
divergence. For PVI a great deal of information can be obtained from its
nonlinear monodromy group, the elements of which act on solutions by analytic
continuation along loops, and which has a well known representation through
the Riemann-Hilbert correspondence as an explicit action on the character
variety of the (linear) monodromy data. For PV the analogical object is
the ``nonlinear wild monodromy pseudogroup'' which expresses not only the
nonlinear monodromy but also the nonlinear Stokes phenomenon at the singularity
at infinity. The goal of the talk is to show how one can obtain the corresponding
action of this pseudogroup on the "wild character variety" of
the (linear) monodromy and Stokes data by studying the confluence. To do
this I will try to explain how the confluence PVI -> PV works on both
sides of the Riemann-Hilbert correspondence.
28 April 2021 (Wed) 17:30～(JST)/10:30～(UTC+2)
Deformation of moduli spaces of meromeprhic connections on the
Riemann sphere via unfolding of irregular singularities
Abstract :
It is well known that irregular singular points of
differential equations are obtained by the confluence of some regular
singular points and then
some analytic properties of these irregular singular points can be related to
that of regular singular points via this confluence procedure.
In this talk, I will explain a construction of flat families of
moduli spaces of meromeprhic connections on the Riemann sphere in
which generic fibers are moduli spaces of regular singular connections
and specializations of deformation parameters correspond to the
confluence of their regular singularities. Then I will show that every
moduli space of connections with unramified irregular singularities
has this kind of deformation.
7 April 2021 (Wed) 17:30～(JST)/10:30～(UTC+2)
WKB Filtrations and the Singularly Perturbed Riccati Equation
Abstract :
We study meromorphic $\hbar$-connections on vector bundles over a Riemann surface.
These are families of vector bundles with meromorphic connections, parameterised by a small complex parameter $\hbar$,
which degenerate to a Higgs bundle in the limit as $\hbar \to 0$. I will show under rather general assumptions that (at least in rank two) the vector bundle has canonical flat filtrations whose limits as $\hbar \to 0$ in a halfplane converge to the eigendecomposition of the corresponding limiting Higgs bundle. These filtrations,
which we call the WKB filtrations,
are defined over certain open subsets given by real-flows of a certain holomorphic vector field obtained from the limiting Higgs field. The key to the construction of WKB filtrations is the ability to find exact solutions to a singularly perturbed Riccati equation.
The name is derived from the fact that, in the special case where the $\hbar$-connection arises from a Schrödinger equation,
the WKB filtrations are generated by local exact WKB solutions.
We also show that near each pole of the connection,
the WKB filtration (at every fixed nonzero value of $\hbar$) coincides with the local Levelt filtration determined by growth rates of flat sections as they are parallel transported into the singular point. As a result,
the WKB filtration is a very special filtration on the vector bundle which converges to the Higgs bundle eigendecomposition as $\hbar \to 0$ and also to the eigendecomposition of the connection’s principal part as we parallel transport into the pole.
24 March 2021 (Wed) 17:30～(JST)/9:30～(UTC+1)
A generalization of the $q$-Garnier system and its Lax form
Abstract :
We propose a birational representation of an extended affine Weyl group of type $(A_{mn-1}+A_{m-1}+A_{m-1})^{(1)}$.
It provides a generalization of Sakai's $q$-Garnier system as a group of translations.
The affine Weyl group is formulated in two ways.
One is a cluster mutation and the other is a Lax form with $mn\times mn$ matrices.
If time permits, we discuss a particular solution in terms of a $q$-hypergeometric function.
Szilard Szabo (Budapest Univ. of Technology and Economics)
10 March 2021 (Wed) 17:30～(JST)/9:30～(UTC+1)
Asymptotic analysis of non-abelian Hodge theory in rank 2
Abstract :
First we explain how recent results on the asymptotic solutions
of Hitchin's equations on a curve allow one to prove Simpson's
Geometric P = W conjecture in the Painlevé 6 case. In the second
part of the talk we outline the main ideas of our ongoing work on
the extension of this result to the Garnier case with 5 parabolic
points.
10 February 2021 (Wed) 17:30～(JST)/9:30～(UTC+1)
Title : q-connection problems on hypergeometric and Painlevé equations
Abstract:
We study connection problems of q-Painlevé equations. This problem is divided
Into three parts:
1. Study q-connection problems on hypergeometric equations
2. Study q-connection problems on linearized equations of the Painlevé equations
3. Consider q-connection problems on the Painlevé equations
>
27 January 2021 (Wed) 17:30～(JST)/9:30～(UTC+1)
Title : Quantum Painlevé monodromy manifolds and Sklyanin-Painlevé algebra
Abstract :
In this talk, I will discuss the quantisation of the Painlevé monodromy manifolds as a special class of quantum del Pezzo surfaces.
In particular I will introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties.
This algebra contains as limiting cases the generalised Sklyanin algebra,
Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.
>13 January 2021 (Wed) 17:30～(JST)/9:30～(UTC+1)
Title : Painlevé equations on weighted projective spaces
Abstract : The weight of a differential equation is defined through the Newton diagram
of the equation. It gives the weighted projective space, that is the natural
compactification of the phase space of a differential equation.
In this talk, I show how to construct a weighted projective space,
analysis of the Painlevé equation using the geometry of a weighted projective space,
and related topics.