Title:  Asymptotic Geometry of non-abelian Hodge theory and  Riemann-Hilbert correspondence in a rank 3 case

Abstract:  I will report on current joint work with Miklos Eper, building on my earlier joint work with Takuro Mochizuki. I will describe the asymptotics near the end of the moduli space of the NAHT and RH maps, in the case corresponding to the affine  root lattice $\widetilde{E}_6$, i.e. structure group $SL(3)$ with three parabolic points in genus 0. We find a behavior similar to the Stokes phenomenon. The Lyapunov exponents on the sectors are governed by abelian integrals.

Title: Asymptotic behaviour of the Hitchin metric of the moduli space of Higgs 

bundles

Abstract: 

The moduli space of stable Higgs bundles of degree $0$ is equipped with the hyperk\"ahler metric, called the Hitchin metric. On the locus where the Hitchin fibration is smooth, there is also the hyperk\"ahler metric 

called the semi-flat metric, associated with the algebraic integrable systems with the Hitchin section. As predicted by Gaiotto, Moore and Neitzke, the difference between the metrics along the curve $(E,t\theta)$ $(t\geq 1)$ decays in an exponential way. There still remains the problem of refining the exponential order to the predicted order.

In this talk, we shall discuss the issue in the rank two case. We would like to explain that we can achieve it in the symmetric case, and what obscures in the non-symmetric case.

Title: Compactification of moduli spaces of connections and Okamoto--Painlev\'{e} pairs \\

Abstract:  

Okamoto introduced the space of initial conditions by resolution of the singularities of the foliations defined by the Painlev\'{e} equations. The space of initial conditions has nice compactifications. The pair of the compactification and the boundary divisor is called an Okamoto--Painlev\'{e} pair. On the other hand, we have another approach to study the Painlev\'{e} equations. This is isomonodromic approach. The subject of this talk is construction of the Okamoto--Painlev\'{e} pairs by isomonodromic approach.  Inaba--Iwasaki--Saito constructed a nice compactification of the moduli space of SL(2)-connections over the projective line with 4-regular singularities. They showed that this compactification gives the Okamoto--Painlev\'{e} pair of Painlev\'{e} VI. In this talk, Inaba--Iwasaki--Saito’s compactification is generalized. We construct compactifications of the moduli spaces of SL(2)-connections over the projective line with (generic unramified or generic ramified) irregular singularities. We show that our compactifications give the Okamoto--Painlev\'{e} pairs of other Painlev\'{e} equations. This talk is based on joint work in progress with Michi-aki Inaba.

Title: Oper and Canonical coordinates of the moduli spaces of parabolic connections and parabolic Higgs bundles

Abstract:  

In this talk, I will explain how we can itroduce  a system of coordinates on a dense Zarisiki open set of the moduli spaces of parabolic connections and parabolic Higgs bundles by using Oper and apparent singularity.  In rank 2 case, we can showed that these coordinates give  canonical coordinates with respect to  the natural algebraic symplectic structure introduced by Inaba-Iwasaki-Saito.  We will also explain the situation of higher rank case.  If time allow, we will  explain about the geometric structure of the moduli spaces by using these coordinates.  This talk is based on  joint works with Arata Komyo, Frank Loray and Szilard Szabo arXiv:2309.05012. 2023 and Szilard Szabo.

Title: Realization of $A^(1)^{*}_2$-surfaces as the moduli spaces of connections

Abstract:  An $A^(1)^{*}_2$-surface is one of the surfaces in Sakai's classification of Painlev\'{e} equations. There is a natural action of the affine Weyl group $W(E^(1)_6)$ on a family of $A^(1)*_2$-surfaces, and the action of the translation part gives rise to the difference Painlev\'{e} equations. In this talk, we derive $A^(1)*_2$-surfaces as compactifications of moduli spaces of rank three parabolic logarithmic connections on the projective line with three poles. In particular, I will introduce the explicit correspondence between stable parabolic logarithmic connections and points on an $A^(1)*_2$-surface. Moreover, I will explain the realization of the affine Weyl group in terms of connections.

Title:  Asymptotic solutions for linear odes with not-necessarily meromorphic coefficients 2

Abstract: We consider systems of linear ODEs with analytic coefficients on sectorial domains, which are asymptotically diagonal for large values of the independent variable |z|. Under some conditions, we show the existence and uniqueness of an asymptotic fundamental matrix solution on a big sectorial domain. This includes systems depending on parameters.

 An application are systems of ODEs with not-necessarily meromorphic coefficients, the leading diagonal term of the matrix coefficient being a generalized polynomial in z with real exponents (equations arising ODE/IM correspondence are of this kind).

 Another application is the classical case of ODEs with meromorphic coefficients. Our results reproduce (with a shorter proof) the main asymptotic existence theorems of Y. Sibuya [1962, 1968] and W. Wasow [1965], in their optimal refinements: namely, the sectors of validity of the asymptotics are maximal, and the asymptotic fundamental system of solutions is unique.

This is a joint work with G. Cotti  and D. Masoero (https://arxiv.org/pdf/2310.19739)

Title: Canonical representations of surface groups 

Abstract: The character variety parameterizing $r$-dimensional local systems on a genus $g$ surface with $n$ punctures has a natural action of the mapping class group of the surface. Algebraic solutions to the Painleve VI differential equation correspond to points on the character variety with finite orbit under this action in the case that $g = 0, n = 4,$ and $r = 2$.  We will describe joint work with Josh Lam and Daniel Litt where we classify points on the character variety with finite orbit under this mapping class group action in many more cases.

Title: Explicit formula, WKB approximation and quantization of irregular Riemann-Hilbert maps 

Abstract:  This talk focues on several aspects of meromorphic linear systems of ODEs. It first gives an introduction to the Stokes matrices of linear systems with a second order pole and the Jimbo-Miwa-Mori-Sato isomonodromy equation. It then derives the explicit formula of the Stokes matrices by studying the long time behaviour of the Jimbo-Miwa-Mori-Sato equation, which generalizes Jimbo's formula for Painleve VI to higher rank cases. Based on the formula, it studies the relation between the WKB approximation of the Stokes matrices and periods on the associated spectral curves. In the end, if time allowed, it also introduces the quantum Stokes matrices of an universal quantum ordinary differential systems with a k-th order pole, which gives rise to a quantization of the irregular Riemann-Hilbert maps at a k-th order pole, as a homomorphism between two associative algebras that quantize the Poisson structures on the de Rham and Betti spaces respectively.

Title: Shift operators and Riemann's problem

Abstract:  If two Fuchsian systems of ordinary differential equations  have the same monodromy, the local exponents of the two systems at each singular point differ by integers. We call one of the two systems a shifted system for the other system, and the tuple of the differences of the exponents a shift.Does a shifted system exist for an arbitrarily given system and arbitrarily prescribed shift? This problem naturally induces a generalization of Riemann's problem. We study this problem by the help of the theory of Lappo-Danilevsky.

Title: An affine Weyl group action on a $ 3 \times 3 $ Lax form for the $q$-$E_6^{(1)}$ Painlev\'{e} equation

Abstract:  The $q$-Painlev\'{e} equation of type $E_6^{(1)}$ was given as a $q$-difference system with an affine Weyl group symmetry of type $E_6^{(1)}$. In the previous work, we gave a new $3 \times 3$ matrix Lax form for the equation. Our goal is to understand its symmetry as actions on it. As a partial result, it was known that there were bi-rational actions on the Lax form which generated three kinds of affine Weyl groups of type $A_2^{(1)}$. In this talk, we show the actions and consider a relation between an affine Weyl group action of type $E_6^{(1)}$ and them. 

Title:  Asymptotic solutions for linear odes with not-necessarily meromorphic coefficients 1We consider systems of linear ODEs with analytic coefficients on sectorial domains, which are asymptotically diagonal for large values of the independent variable |z|. Under some conditions, we show the existence and uniqueness of an asymptotic fundamental matrix solution on a big sectorial domain. This includes systems depending on parameters.

 An application are systems of ODEs with not-necessarily meromorphic coefficients, the leading diagonal term of the matrix coefficient being a generalized polynomial in z with real exponents (equations arising ODE/IM correspondence are of this kind).

 Another application is the classical case of ODEs with meromorphic coefficients. Our results reproduce (with a shorter proof) the main asymptotic existence theorems of Y. Sibuya [1962, 1968] and W. Wasow [1965], in their optimal refinements: namely, the sectors of validity of the asymptotics are maximal, and the asymptotic fundamental system of solutions is unique.

This is a joint work with G. Cotti  and D. Masoero (https://arxiv.org/pdf/2310.19739)

Title: Geometric local systems on very general curves

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 at least $\sqrt{g+1}$. We do so by proving a more general result that local systems on very general curves coming from geometry with infinite monodromy must have rank at least $2 \sqrt{g+1}$. 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 result of this talk is one of the key inputs in the proof of the main result from our previous talk.

Title: $K$-theoretic wall-crossing formulas and multiple basic hypergeometric series

Abstract:  

We study $K$-theoretic integrals over famed quiver moduli via wall-crossing phenomena. We study the chainsaw quiver varieties, and consider generating functions defined by two types of $K$-theoretic classes. In particular, we focus on integrals over the handsaw quiver varieties of type $A_{1}$, and get functional equations for each of them. We also give explicit formulas for these partition functions. In particular, we obtain geometric interpretation of transformation formulas for multiple basic hypergeometric series including the Kajihara transformation formula, and the one studied by Langer-Schlosser-Warnaar and Halln\"{a}s-Langman-Noumi-Rosengren.

Title: Wall-crossing in mirror symmetry and beyond 

Abstract:  In joint work with Bernd Siebert, I have constructed mirrors to log Calabi-Yau and Calabi-Yau manifolds. These can be constructed from "wall structures", which tell us how to glue together standard pieces to obtain the mirror. For us, wall structures are determined by so-called "punctured Gromov-Witten invariants" of Abramovich, Chen, Gross and Siebert. However, similar structures also show up in other contexts, such as in spaces of Bridgeland stability conditions. I will survey recent results

Title: Instanton partition function and q-KZ equation

Abstract: We clarify the relation between the 5d Nekrasov partition function and the q-Knizhnik-Zamolodchikov equation.  This talk is based on the collaboration with Awata, Hasegawa, Kanno, Ohkawa, Shakirov and Shiraishi.

Title: A system of difference-differential equations and an integral representation of solutions for the hypergeometric functions

Abstract:  We consider a system of difference-differential equations satisfied by the Gauss hypergeometric function. By using the Laplace transform with respect to parameters we get an integral representation of solutions in the form of the triple complex integral. By using the gauge transformation the well-known integral representation of solutions is also obtained. Similar results hold for confluent hypergeometric functions as well. 

Title: Twisted meromorphic connections

Abstract:  A twisted meromorphic connection on a compact Riemann surface is a meromorphic connection which is, in some sense, equivariant under some finite group action. In this talk we show that some twisted meromorphic connections appear as the Fourier-Laplace transforms of meromorphic connections on the Riemann sphere with orthogonal/symplectic structure group.

Title:Stokes structure of difference modules. 

Abstract:  We introduce a category of filtred sheaves on a circle to describe the Stokes phenomenon of linear additive difference equations at infinity. We will explain a kind of Riemann-Hilbert correspondence for difference equations with a mild singularity (and how to generalize it to the wild case if time permits). 

Title: Dynamics of the Painleve VI foliation and the Riemann-Hilbert problem.

Abstract: We investigate interplay between the dynamics of Painleve VI foliation

and the problem of realizing a monodromy representation by Heun equation.

This is a joint work in progress with Gabriel Calsamiglia and Titouan Serandour.

Title: Connection Problems on $q$-linear equations associated to the Painlev\'{e} equations.

Abstract:  We study connection formulae of some $q$-linear equations.  One is on basic hypergeometric equations and the second is on $q$-linear equations which appear in the Lax equations of $q$-Painlev\'{e}equations.  The character variety of the $q$-Painlev\'{e} VI equation is a degree four del Pezzo surface, known as the Segre surface. We will study special solutions of the $q$-Painlev\'{e} equation by means of the character varieties.