・10:45 ー 11:45 Hidetaka Sakai
Title: On the bilinear equations of the Painlevé transcendents (Joint work with T. Hosoi)
Abstract: The sixth Painleve equation is a basic equation among the non-linear differential equations with three fixed singularities, corresponding to Gauss hypergeometric differential equation among the linear differential equations. It is known that 2nd order Fuchsian differential equations with three singular points are reduced to the hypergeometric differential equations. Similarly, for nonlinear differential equations, we would like to determine the equation from the local behavior around the three singularities. In this talk, the sixth Painleve equation is derived by imposing the condition that it is of type (H) at each three singular points for the homogeneous quadratic 4th-order differential equation.
・13:45 ー 14:45 Michi-aki Inaba
Title: Structure of the moduli space of rational connections and the isomonodromic deformation
Abstract: In this talk, we consider the moduli space of irregular singular connections on smooth projective curves. On this moduli space, we can define the isomonodromic foliation whose leaves are loci where the corresponding monodromy Stokes data are constant. The isomonodromic foliation is also determined by an algebraic subbundle of the tangent bundle of the moduli space. We regard this subbundle as the isomonodromy equation.
In a special case, the isomonodromy equations become Painlevé equations and the moduli space of connections becomes the space of initial conditions for Painlevé equations. In the logarithmic case, there is a moduli theoretic compactification of
the moduli space of connections and it is isomorphic to the rational surface which characterizes the Painlevé equations in Sakai's theory. In collaboration with Komyo, we try to extend this result to irregular singular cases. I will explain its brief idea.
・15:15 ー 16:15 Martin Guest
Title: A symplectic groupoid-theoretic approach to the space of solutions of the tt*-Toda equations
Abstract: This talk is a contribution to bridging the gap between the classical (function-theoretic) theory of solutions of Painlevé-type equations and the abstract (geometrical) theory of their moduli spaces. We discuss just one example, the tt*-Toda equations (tt* equations of Toda type), and focus on the relation between the space of monodromy data and the behaviour of solutions. This is a special example but it has the advantage that explicit calculations are facilitated by Lie theory. The talk is based on joint work with Alexander Its and Chang-Shou Lin, and with Nan-Kuo Ho, and also on an ongoing project with Alexander Its, Maksim Kosmakov, Kenta Miyahara, and Ryosuke Odoi.
Title: Variant of 2φ1 as a democratic q-hypergeometric equation
Abstract: In this talk, we study the variant of Heine's q-hypergeometric equation 2φ1 defined by Hatano-Matsunawa-Sato-Takemura (Funkcial. Ekvac., 2022). We give Euler type integral solutions and series solutions in terms of very-well-poised q-hypergeometric series 8W7 for the equation.
These solutions are obtained using the democratic viewpoint. Here the word ``democratic'' means that every point can be treated equally. We also present some generalization to multivariable cases. We discuss further generalizations if time permits. This talk is based on collaborations with Taikei Fujii (Kobe University).
・ 9:30 ー 10:30 Yoshitsugu Takei
Title: A system of difference-differential equations for hypergeometric functions and WKB analysis
Abstract: As is well-known, every Painlevé equation appears as the compatibility condition of a system of linear differential equations called the Lax pair. Furthermore, some discrete Painlevé equations also appear by considering shift operators with respect to parameters contained in the Lax pair. These facts suggest that some properties of Painlevé equations may be extracted from those of the Lax pairs and associated shift operators.
As a prototype of this approach, we discuss a system of difference-differential equations satisfied by the Gauss hypergeometric functions in this talk. Several important properties of hypergeometric functions can be deduced by considering the hypergeometric differential equation and the contiguity relations simultaneously. In the talk we explain the following two important quantities for hypergeometric functions are obtained in a natural and clear-cut manner through a system of difference-differential equations: (i) integral representations, and (ii) the so-called Voros coefficients, i.e., key ingredients for the description of global behaviors of solutions via the exact WKB analysis.
This is a joint work with Yumiko Takei (Ibaraki College).
・10:45 ー 11:45 Kohei Iwaki
Title: Topological recursion, holomorphic anomaly and Painlevé I
Abstract: The topological recursion/WKB analogue of the Kyiv formula for Painlevé I was presented in the speaker's paper https://doi.org/10.1007/s00220-020-03769-2. In this talk, I will discuss the B-model picture of the Kyiv formula, including its connections to holomorphic anomaly equations and quasi-modular forms. This talk is based on a joint work with N. Iorgov, O. Lisovyy and Y. Zhuravlov.
・13:45 ー 14:45 Marta Mazzocco
Title: Segre surfaces for the Painlevé equations
Abstract: TBA
・15:15 ー 16:15 Davide Dal Martello
Title: Painlevé VI, symmetries, and clusters
Abstract: The sixth Painlevé equation (PVI) admits a native sl_2(C)-Fuchsian isomonodromy representation. Taking the multiplicative middle convolution of a higher Teichmüller coordinatization for the corresponding Fuchsian monodromy group, we give Okamoto’s symmetry of PVI a realization on the representation space in the language of cluster X-mutations. The explicit mutation formula admits dual characterizations in geometric terms of the colored associahedron and star-shaped fat graphs, expanding the cluster state of the art for PVI.
・16:30 ー 17:00 Benedetta Facciotti
Title: An example of Stokes data and real decorated blowups
Abstract: This talk will focus on the study of a specific example of a meromorphic connection on a vector bundle of rank 3 on the projective line. More precisely, it will be shown how the Stokes data arising from the higher order pole can be computed exploiting the decorated real oriented blowup of the surface involved and Fock-Goncharov theory.
・ 9:30 ー 10:30 Harini Desiraju
Title: Elliptic orthogonal polynomials and their integrability
Abstract: Elliptic orthogonal polynomials are a family of special functions that satisfy certain orthogonality condition with respect to a weight function on an elliptic curve. I will present a generalized framework using Riemann-Hilbert analysis to study such polynomials and obtain associated discrete and continuous integrable systems. As a quick check, for the simplest weight function, I will show that the elliptic formulation of the sixth Painlevé equation can be recovered.
・10:45 ー 11:45 Yuma Mizuno
Title: Geometry of q-Painlevé equations and cluster structures
Abstract: This talk presents an approach to understanding the geometry and symmetry of q-Painlevé equations from the perspective of cluster algebras. Specifically, I will explain the mechanism by which the quiver and mutation combinatorics arises from the geometry of rational surfaces associated with q-Painlevé equations.
・13:45 ー 14:45 Andrii Naidiuk
Title: Perturbative connection formulas for Heun equations
Abstract: We recover the predictions of CFT regarding the connection formulae for the family of Heun equations. Two methods of derivation of the connection coefficients are employed: the theorem of Schäfke and Schmidt that allows for a rigorous proof in some cases, as well as the analysis of the asymptotic behavior of the Floquet solutions. Both methods yield closed formulas in terms of continued fractions which can be compared with the results obtained from CFT by means of Zamolodchikov conjecture. We will review two simple cases of the application of these methods: the reduced confluent Heun equation and the Mathieu equation.
・15:15 ー 16:15 Oleg Lisovyy 1
Title: Many-faced Painlevé I (Part 1)
Abstract: I will explain how the Fourier transform structure of the Painlevé I tau function arises from an extension of the Jimbo-Miwa-Ueno differential. We will then discuss how the main building block of this representation - the Painlevé I partition function - is related to irregular CFT, topological recursion and holomorphic anomaly equation. The main emphasis will be on the algebraic construction of the rank 5/2 Virasoro irregular states and associated conformal blocks.
・16:30 ー 17:00 Haruki Nakagawa
Title: The degeneration limit of Virasoro conformal blocks and Painlevé tau functions
Abstract: The 2-dimensional conformal field theory was established by Belavin, Polyakov, and Zamolodchikov. It is well known that the tau function of the sixth Painlevé equation is expanded in terms of the conformal blocks, defined as the expectation of the vertex operators. I will talk about the degenerate scheme of the vertex operators with Virasoro algebra. Furthermore, we prove the series expansion formula of the tau functions of the fourth Painlevé equation. This is joint work with Hajime Nagoya.
・18:00 ー Social Dinner
・ 9:30 ー 10:30 Hajime Nagoya
Title: Bilinear equations for irregular conformal blocks and Painlevé tau functions
Abstract: Using the embedding of a direct sum of two Virasoro algebras into a sum of Majorana fermion and Neveu-Schwartz algebra, Bershtein and Shchechkin showed in 2015 that a conjectured series expansion in terms of the Liouville conformal blocks with central charge c=1 for the Painlevé VI tau function satisfies the bilinear equation. In this talk, we introduce an irregular vertex operator for the Neveu-Schwartz algebra and give a decomposition theorem of irregular vertex operators of the Neveu-Schwartz algebra into irregular vertex operators of two Virasoro algebras. The decomposition theorem yields bilinear equations for irregular conformal blocks. For example, we discuss bilinear equations for irregular conformal blocks of type (02) and the (quantum) Painlevé IV tau function.
・10:45 ー 11:45 Pavlo Gavrylenko 1
Title: Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general q-Painlevé III₃ tau functions
Abstract: We reformulate the q-difference linear system corresponding to the q-Painlevé equation of type A₇⁽¹⁾' as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant constructed from the jump matrix of this Riemann-Hilbert problem and prove that it satisfies bilinear relations equivalent to P(A₇⁽¹⁾'). We also find the minor expansion of this Fredholm determinant in an explicit factorized form. Finally, we solve the connection problem for these isomonodromic tau functions, finding their global behavior in this way. The talk will be based on the paper https://arxiv.org/abs/2501.01419.
・13:45 ー 14:45 Oleg Lisovyy 2
Title: Many-faced Painlevé I (Part 2)
Abstract: I will explain how the Fourier transform structure of the Painlevé I tau function arises from an extension of the Jimbo-Miwa-Ueno differential. We will then discuss how the main building block of this representation - the Painlevé I partition function - is related to irregular CFT, topological recursion and holomorphic anomaly equation. The main emphasis will be on the algebraic construction of the rank 5/2 Virasoro irregular states and associated conformal blocks.
・15:15 ー 16:15 Yasuhiko Yamada
Title: Higher rank affine Laumon partition function and q-KZ equation
Abstract: A q-difference equation which generalizes the Shakirov’s equation is formulated. The generalized equation is expected to characterize the higher rank affine Laumon partition function. We show some evidence for this conjecture. The relation to 3d R-matrix (the tetrahedron) is also discussed. This talk is based on the joint work with Awata, Hasegawa, Kanno, Ohkawa, Shakirov and Shiraishi.
・16:30 ー 17:00 Satoshi Tsuchimi
Title: A solution of the type (A2+A1)^(1) q-Painlevé equation in terms of mock modular forms
Abstract: We present a solution of the (A2+A1)^(1) q-Painlevé equation in terms of the μ-function. The μ-function introduced by Zwegers in the most fundamental object in the study of mock theta functions. In this talk, we show that the mock modular forms and the τ-function of discrete integrable systems are closely related.
・ 9:30 ー 10:30 Pavlo Gavrylenko 2
Title: Zeros of isomonodromic tau functions, blow-up relations, and holomorphic anomaly equation
Abstract: Isomonodromic tau functions have explicit expressions as sums of c=1 conformal blocks (or Nekrasov functions), known as the Kyiv formulas, discovered by Gamayun, Iorgov, and Lisovyy. The zeros of these tau functions are described by classical, or c=∞ conformal blocks, also identified with the Painlevé action on the trajectory. We analyze an expansion of the tau function around its zero and combine it with the genus expansion of the conformal block. Modular properties of the tau function are well-defined and imply the modular properties of the conformal block. Namely, we prove that the conformal block satisfies the so-called holomorphic anomaly equation as a function of E₂ quasimodular form. The talk will be based on the paper https://arxiv.org/abs/2410.17868.
・10:45 ー 11:45 Toshiyuki Mano
Title: Flat structure and Painlevé transcendents
Abstract: TBA