Schedule for: 16w5027 - Painleve Equations and Discrete Dynamics

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

On the basis of a joint work with Masahiko Ito, I report some recent progresses in the study of elliptic hypergeometric functions. I discuss in particular a class of elliptic interpolation functions and its application to the q-difference de Rham theory of elliptic hypergeometric integrals.

Littlewood reported in his preface to Hardy’s "Divergent Series” that Abel said divergent series were the invention of the devil. But such series arise commonly in the solutions of ODEs in asymptotic limits. The asymptotic description of transcendental solutions of the Painlevé equations has been a longstanding problem, which remains incomplete for many of these equations. We start with a review of these results before describing how such series occur in the solutions of the discrete Painlevé equations. In the latter part of the talk, I will focus on recent studies for additive discrete versions of Painlevé equations and a discrete analogue of the famous tritronquée solutions of the first Painlevé equation for a q-discrete equation. Joshi, N., and C. J. Lustri. "Stokes phenomena in discrete Painlevé I." In Proc. R. Soc. A, vol. 471, no. 2177, p. 20140874. The Royal Society, 2015. Joshi, N., Lustri, C. and Luu, S., 2016. Stokes Phenomena in Discrete Painlev\'e II. arXiv:1607.04494 Joshi N. and Takei, Y., 2016. Toward the exact WKB analysis of discrete Painlev\'e equations, RIMS Kˆokyuˆroku Bessatsu, to appear.

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painleve equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions. This talk is based on a joint work with Toshiyuki Mano (Math. Z. 2016).

The existence of Lax pairs associated to the Painlev\'e equations gives rise to a natural interpretation of their spaces of initial conditions as moduli spaces of differential equations. This suggests that one should develop the theory of such moduli spaces more generally, in particular for difference equations of various kinds (including elliptic) as well as for differential equations. I'll describe how to translate those moduli problems into natural moduli problems arising in noncommutative geometry, how (discrete) isomonodromy deformations arise naturally in that setting, and some of the consequences for the elliptic Painlev\'e equation and generalizations.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

The family of 2D-Toda tau functions of hypergeometric type that serve as generating functions for weighted Hurwitz numbers with polynomial weight generating functions have an associated family of spectral curves that are rational. The corresponding quantum spectral curves are given by a family of ODE's with rational coefficients whose monodromy is invariant under the deformations generated by the underlying KP flows. An alternative generating function for the weighted Hurwitz numbers is provided by the multicurrent correlators, which are expressible both as fermionic vacuum expectation values, and directly in terms of the tau function. The WKB series for the Baker function leads to a series of recursion relations between the weighted Hurwitz numbers, fitting within the general framework of the Topological Recursion program. (Based on joint work with A. Alexandrov, G. Chapuy and B. Eynard)

The Kontsevich integral is a matrix integral (aka "Matrix Airy function") whose logarithm, in the appropriate formal limit, generates the intersection numbers on $\mathcal M_{g,n}$. In the same formal limit it is also a particular tau function of the KdV hierarchy; truncation of the times yields thus tau functions of the first Painlev\'e\ hierarchy. This, however is a purely formal manipulation that pays no attention to issues of convergence. The talk will try to address two issues: Issue 1: how to make an analytic sense of the convergence of the Kontsevich integral to a tau function for a member of the Painlev\'e I hierarchy? Which particular solution(s) does it converge to? Where (for which range of the parameters)? Issue 2: it is known that (in fact for any $\beta$) the correlation functions of K points in the $GUE_\beta$ ensemble of size N are dual to the correlation functions of N points in the $GUE_{4/\beta}$ of size $K$. For $\beta=2$ they are self-dual. Consider $\beta=2$: this duality is lost if the matrix model is not Gaussian; however we show that the duality resurfaces in the scaling limit near the edge (soft and hard) of the spectrum. In particular we want to show that the correlation functions of $K$ points near the edge of the spectrum converge to the Kontsevich integral of size $K$ as $N\to\infty$. This line of reasoning was used by Okounkov in the $GUE_2$ for his "edge of the spectrum model". This is based on joint work with Mattia Cafasso (Angers).

In this talk, I show that a reduction from a 4-dimensional hypercube to a rhombic dodecahedron causes the reduction from ABS equations (partial difference equations) [ABS,Boll] to q-Painleve equations which are $A_4^{(1)}$-surface type in Sakai's classification [Sakai]. Moreover, I also present Lax pairs of the q-Painleve equations constructed by using this reduction. This work has been done in collaboration with Prof. Nalini Joshi and Dr Yang Shi and supported by an Australian Laureate Fellowship FL120100094 and grant DP130100967 from the Australian Research Council. [ABS]V.E. Adler, A.I. Bobenko, and Y.B. Suris. Classification of integrable equations on quad-graphs. The consistency approach. Comm. Math. Phys., 233(3):513-543, 2003. [Boll]R. Boll. Classification of 3D consistent quad-equations. J. Nonlinear Math. Phys., 18(3):337-365, 2011. [Sakai]H. Sakai. Rational surfaces associated with affine root systems and geometry of the Painleve equations. Comm. Math. Phys., 220(1):165-229, 2001.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The iteration of discrete birational dynamics yields a numbe recurrence relations, defined at first by rational expressions, but turning out to be producing only polynomials. This is a generalisation of the famous Laurent property verified for example by the Somos sequences. I will present a number of results on the phenomenon, in relation to integrability/singularity/geometry analysis.

We construct a q-analog of middle convolution. We can reduce the rank of the system by using this, when the system is irreducible, Fuchsian, and of rigidity index 2. In the terms of the middle convolution, we can consider a classification theory of Fuchsian q-difference equations.

In this talk we will show that on the level of monodromy manifolds the confluence of the Painleve differential equations corresponds to colliding two holes or two sides of the same hole in a Riemann sphere. This procedure gives rise to the notion of bordered cusp in a Riemann surface. We will introduce the concept of "SL2 decorated character variety" of a Riemann surface with bordered cusps and we will show that such decorated character varieties are endowed with a generalised cluster algebra structure. We will also provide a very explicit quantisation procedure.

In this talk I shall discuss the relationship between orthogonal polynomials with respect to semi-classical weights, which are generalisations of the classical weights and arise in applications such as random matrices, and integrable systems, in particular the Painlev\'e equations and discrete Painlev\'e equations. It is well-known that orthogonal polynomials satisfy a three-term recurrence relation. I will show that for some semi-classical weights the coefficients in the recurrence relation can be expressed in terms of Hankel determinants, which are Wronskians, that also arise in the description of special function solutions of Painleve equations. The determinants arise as partition functions in random matrix models and the recurrence coefficients satisfy a discrete Painleve equation. The semi-classical orthogonal polynomials discussed will include a generalization of the Freud weight and an Airy weight.

Topological recursion of Eynard and Orantin is known to produce solutions of Pain\-leve equations through the process of quantization of spectral curves. Recently, a similar quantization procedure is discovered for arbitrary Hitchin spectral curves. This time the topological recursion that is required for quantization is not the Eynard-Orantin type. It is a recursive system of PDEs, and the result of quantization turns out to be an "oper." The correspondence between the Hitchin spectral curve and the oper is exactly the same as the scaling limit construction conjectured by D. Gaiotto. This conjecture is recently solved in a joint paper of Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo, Andrew Neitzke, and myself. I will explain how the quantization fits into the WKB analysis of the quantum curve through the PDE recursion. The talk is based on a joint work with Olivia Dumitrescu.

We will derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbf C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel will be expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case principal minor expansion of the Fredholm determinant yields multivariate series representation for the tau function of the Garnier system obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ will provide an explicit series representation of the general solution to PainlevГ© VI equation. The talk is based on http://arxiv.org/abs/1608.00958.

We consider the gap probability for the Generalized Bessel process, a determinantal point process which arises as critical limiting kernel near the hard edge of the spectrum of a certain random matrix ensemble. We prove that such probability can be expressed in terms of the Fredholm determinant of a suitable Its-Izergin-Korepin-Slavnov integrable operator and linked in a canonical way to Riemann-Hilbert problem. Starting from the RH problem, we can construct a Lax pair and we can link the gap probability to the Painleve III hierarchy. Moreover, we are able to construct a system of two coupled Hamiltonians which can be hopefully identified with the 2-dimensional Garnier system LH(2+3). The talk is based on some previous results and an on-going project with Dr. Mattia Cafasso.

I will discuss the properties of a parameter-family of linear partial quad-graph equations and exhibit the rich integrable structure behind those equations: multidimensional consistency, Backlund transforms, multi-Lagrangian structure, associated continuous equations and periodic and scaling reductions, leading to discrete dynamics. All these properties can be lifted to the nonlinear case. Examples of the multicomponent case will be mentioned as well (if time allows).

This talk will describe my attempts to find an integral for a particular area preserving map. The most important dynamical characteristic of an integrable map is its Fomenko graph and the rotation functions on its edges. I will describe high precision numerics for the computation of the rotation function, with the detection of integrability in mind. For each rational value of the rotation function there exists a periodic orbit. Using discrete reversing symmetries and their fixed sets periodic orbits with large period can be efficiently computed. For the construction of local integrals I will use Birkhoff normal form. The globalisation of such an integral, however, remains difficult.

A method of constructing algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve given as a ramified double covering of $CP^1$, a meromorphic differential is constructed with the following property: the common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of the Painleve VI $(1/8,-1/8,1/8,3/8)$ equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. This approach is motivated by an observation of Hitchin connecting algebraic solutions of a Painleve VI equation to the Poncelet polygons in the plane. The research is partially supported by the NSF grant 1444147 and NSERC discovery grant.

We give a construction of completely integrable (2m)-dimesnional Hamiltonian systems with m cubic integrals in involution. Applying to the corresponding quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at birational (2m)-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $R^{2m}$, and possess m independent integrals of motion, which are perturbations of the original integrals. Thus, these maps are completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original m-tuples of commuting vector fields, the m-tuples of maps commute and share the invariant symplectic structure and m integrals of motion.

We study symplectic properties of monodromy map of second order linear equation with meromorphic potential having only simple zeros on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^* Mcal_{g,n}$ implies under an appropriately defined monodromy map the Goldman Poisson structure on the corresponding character variety, thereby extending the recent results of the paper of M.Bertola, C.Norton and the author from the case of holomorphic to meromorphic potentials.

My work is motivated by a list of 40 types of 4-dimensional Painleve-type equations derived from isomodromic deformation and degeneration process (Sakai, Kawakami--Naka\-mura--Sakai, Kawakami). My goal is to characterize these systems in a geometrical way. I deal with the integrable systems derived as the isospectral limit of these Painleve-type equations and consider the degeneration of the theta divisors of their Liouville tori in two different ways.

The pentagram map was introduced by Richard Schwartz in 1992, and is now one of the most renowned discrete integrable systems which has deep connections with many different subjects such as projective geometry, integrable PDEs, and cluster algebras. In this talk I will explain how the pentagram map interacts with polygons inscribed or circumscribed about conic sections.

Abstract: From the seminal papers of Witten and Kontsevich we know that the intersection theory on the moduli spaces of complex curves is described by a tau-function of the KdV integrable hierarchy. Moreover, this tau-function is given by a matrix integral and satisfies the Virasoro constraints. Recently, an open version of this intersection theory was introduced. I will show that this open version can also be naturally described by a tau-function of the integrable hierarchy (MKP in this case), and the matrix integral, Virasoro and W constraints for the open case are also simple deformations of the closed ones. However, it is not clear how to deform the first Painleve hierarchy, satisfied by the Kontsevich-Witten tau-function.

We present a Lax pair for the sixth Painleve equation arising as a continuous isomonodromic deformation of a system of linear difference equations with an additional symmetry structure. We call this a symmetric difference-differential Lax pair. We show how the discrete isomonodromic deformations of the associated linear problem gives us a discrete version of the fifth Painleve equation. By considering degenerations we obtain symmetric difference-differential Lax pairs for the fifth Painleve equation and the various degenerate versions of the third Painleve equation.

Meet in foyer of TCPL to participate in the BIRS group photo. Please don't be late, or you will not be in the official group photo! The photograph will be taken outdoors so a jacket might be required.

We will talk about a generalized $q$-analogue of methods introduced by Arinkin and Borodin. In particular we will explain how the one-interval gap probability function for the $q$-Hahn orthogonal polynomial ensemble can be expressed through a solution of the asymmetric $q$-Painlev\'e V equation, which requires a new derivation of a $q$-difference equation of Sakai's hierarchy of type $A_{2}^{(1)}.$ Our approach is based on the analysis of $q$-connections on $\mathbf P^1$ with a particular singularity structure.

In a recent paper [1], we classified the critical behaviour of solutions of the discrete Painleve equation $q-P(A_1)$ around the origin and infinity. Even though this equation is ranked higher than q-PVI and hence PVI in Sakai's classification scheme [2], the critical behaviours seem to resemble those found for the sixth Painleve equation. The generic expansions near the two critical points each contain two arbitrary q-constants, and relating them explicitly constitutes the $q-P(A_1)$ connection problem. A well known method of attack is given by the isomonodromic deformation method, which we apply to an associated linear system formulated by Yamada [3]. It turns out that the connection problem of the Yamada system factorises asymptotically, both as the PainlevВґe variable approaches the origin and infinity, into two copies of a linear system associated with the continuous dual q-Hahn polynomials. The connection matrix of the limiting system can be calculated explicitly, which allows us to solve the direct monodromy problem for the critical expansions of solutions of $q-P(A_1)$. In particular this reduces the $q-P(A_1)$ connection problem to solving an equation involving q-elliptic functions. References [1] N. Joshi and P. Roffelsen, Analytic solutions of $q-P(A_1)$ near its critical points, arXiv:1510.07433 [nlin.SI]. [2] H. Sakai, Rational Surfaces Associated with Affine Root Systems and Geometry of the Painleve Equations, Commun. Math. Phys. 220 (2001). [3] Y. Yamada, Lax Formalism for q-Painleve Equations with Affine Weyl Group Symmetry of Type $E_n^{(1)}$ , Int. Math. Res. Not. IMRN 2011 (2011).

About seventy years after the original discovery, the six Painleve equations have reappeared in two settings, namely self-similar solutions of integrable hierarchies of ODEs, and random-matrix theory. This talk reports on the case of Painleve VI, proposing an isomonodromic counterpart to Sato's operator, and related Darboux transformations.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.