London Mathematical Society -- EPSRC Durham Symposium

Geometric and Algebraic Aspects of Integrability

2016-07-25 to 2016-08-04

Schedule of Talks

Jul 25 (Mon)

14:00 - 19:00 Registration at Grey College (in the JCR - Junior Common Room)

19:00 - 20:00 Dinner at Grey College

**School**

Jul 26 (Tue)

08:00 - 08:45 Breakfast at Grey College

09:00 - 10:00 Alexander Veselov: *Integrability: historical overview I*

I will give an overview of the development of the idea of integrability from the work of Euler and Jacobi to the beginning of 20th century and its influence on the classical algebraic geometry and the creation of quantum mechanics.

10:00 - 10:30 Coffee Break

10:30 - 11:20 Alexander Veselov: *Integrability: historical overview II*

11:30 - 12:30 Franco Magri: *Bi-Hamiltonian formalism*

The birth, the meaning, and the use of the concept of bihamiltonian system will be revisited by means of two examples: one in the field of partial differential equations (the KdV equation), and the other in the field of classical mechanics (the Steklov system).

13:00 - 14:00 Lunch at Grey College

14:30 - 15:30 Mark Ablowitz: *Inverse scattering transform: history, background and methods I*

The inverse scattering transform (IST) is a method to solve the Cauchy problem associated with a class of nonlinear evolution equations. The lectures will give some history and background. The associated linear systems and methodology will be outlined for some physically significant equations such as the Korteweg-deVries (KdV), nonlinear Schrodinger (NLS), modified KdV (mKdV), and sine-Gordon equations. The recently discovered symmetry leading to the integrable nonlocal NLS equation and its IST will also be mentioned.

15:40 - 16:30 Mark Ablowitz: *Inverse scattering transform: history, background and methods II*

16:30 - 17:00 Coffee Break

17:00 - 18:00 Alexander Bobenko: *Integrability in discrete differential geometry I*

A presently emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of refinement of the discretization. Current interest in this field derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. The central role in this theory is played by discrete integrable systems. I these lectures we plan to reveal the integrable structure of discrete differential geometry.
The corse is based on the book
Alexander I. Bobenko, Yuri B. Suris,
Discrete Differential Geometry: Integrable Structure,
Graduate Studies in Mathematics, Vol. 98, AMS, 2008
http://page.math.tu-berlin.de/~bobenko/ddg-book.html

18:10 - 19:00 Alexander Bobenko: *Integrability in discrete differential geometry II*

19:30 - 20:30 Dinner at Grey College

Jul 27 (Wed)

08:00 - 08:45 Breakfast at Grey College

09:00 - 10:00 Alexander Mikhailov: *Symmetry approach to integrability I*

The existence of an infinite hierarchy of local symmetries can be
deemed as a constructive definition of integrability for systems of
partial differential
equations. It enables us to formulate the necessary conditions which
are strong enough to tackle the problem of classification of
integrable systems. The derivation of these conditions will is the
main focus of this crash lecture
course. This approach is quite universal, although in the cases of
differential-difference and partial difference equations we meet
rather challenging problems.

10:00 - 10:30 Coffee Break

10:30 - 11:20 Alexander Mikhailov: *Symmetry approach to integrability II*

13:00 - 14:00 Lunch at Grey College

14:30 - 15:30 Richard Ward: *Twistor theory and integrability I*

The idea of a Lax pair associated with integrable systems is used to
motivate twistor theory. This leads to a twistor description of
integrable geometric systems such as the self-dual Einstein and
self-dual Yang-Mills equations, and various reductions of them.

15:40 - 16:30 Richard Ward: *Twistor theory and integrability II*

16:30 - 17:00 Coffee Break

17:00 - 18:00 Robert Bryant: *Integration techniques for exterior differential systems I*

These lectures will describe some of the techniques that have been found
useful for explicitly finding the integrals of exterior differential systems
arising in geometry. These include Darboux’ method and its generalizations,
compatible reductions, integrable extensions and Bäcklund transformations,
conservation laws, and methods connected with analysis of the characteristic variety.
Emphasis will be placed on illustrative examples and computations
using the structure equations of Cartan.
The first lecture will begin with a brief summary of the basics
of exterior differential systems (EDSs) including involutivity, Cartan-Kähler theory,
and the characteristic variety. This will be followed by an introduction to EDS
formulations of some geometric problems via Cartan’s structure equations,
illustrated by examples.
The second lecture will focus mainly on examples, including minimal
submanifolds, pseudo-holomorphic curves, Willmore geometry,
and prescribed curvature problems in Riemannian and Finsler geometry.

18:10 - 19:00 Robert Bryant: *Integration techniques for exterior differential systems II*

19:30 - 20:30 Dinner at Grey College

**Conference**

Jul 28 (Thu)

08:00 - 08:45 Breakfast at Grey College

09:00 - 09:40 Nicholas Manton: *Vortices -- a review*

Vortices are localised solitons in two dimensional space, and can be defined on any Riemann surface. The abelian vortex equations have an
interpretation in terms of curvature and conical singularities, and are integrable in cases where the curvature has an appropriate constant
value. Variants of the equations correspond to negative, zero and positive
curvature. Many explicit solutions have been constructed using holomorphic
maps between surfaces, but this method does not always work globally.

09:40 - 10:20 Prim Plansangkate: *Twistor theory and topology of Ward solitons*

In this talk I shall show that the total energy of time-dependent uniton solutions of the Ward chiral model is proportional to the third homotopy class of the corresponding extended solutions. This explains the classical energy quantisation of moving solitons. I shall also discuss a compactified twistor fibration associated to the model, and use it to show that the second Chern number of the twistor bundle corresponds to the third homotopy class of the extended solution.

10:20 - 10:50 Coffee Break

10:50 - 11:30 David Calderbank: *Integrable background geometries - review and outlook*

I will briefly review integrable background geometries and
their associated gauge theories, which were introduced at the
turn of the century as a framework to unify reductions of
4-dimensional selfduality equations. As time permits, I will
also indicate some recent developments and generalizations.

11:30 - 12:10 Alexey Bolsinov: *Argument shift method and Manakov operators: applications to differential geometry*

We discuss a surprising relationship between Manakov operators, well-known in the theory of integrable systems on Lie algebras, and curvature tensors of projectively equivalent Riemannian and K\"ahler metrics of arbitrary signature. We demonstrate how using this relationship helps to solve a number of natural problems in differential geometry.

13:00 - 14:00 Lunch at Grey College

14:15 - 14:30 LMS: *Welcome from the LMS*

14:30 - 15:10 Peter Olver: *Dispersive Quantization of Linear and Nonlinear Waves*

The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times. The Talbot effect, named
after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these
phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed.

15:10 - 15:50 Felipe Contatto: *Metrisability of Painleve equations and Hamiltonian systems of hydrodynamic type*

Given a particular set of curves on a manifold, we can ask whether they are local geodesics of a pseudo-Riemannian metric. This is called the metrisability problem. I will present its solution for the integral curves of the Painleve equations and conclude that they are geodesics of a metric only for special choices of parameters, those for which the Painleve equations admit a first integral quadratic in first derivatives.
In the second part of the talk I will present the solution to the following problem: given a torsion free affine connection on a surface, when does it admit 0, 1, 2 or 3 Killing 1-forms? The relation between existence of Killing forms and metrisability will be illustrated by the Painleve equations. Then, I will explain how one can interpret the existence of hamiltonian descriptions of hydrodynamic type systems as a particular case of this problem.

15:50 - 16:20 Coffee Break

16:20 - 17:00 Ian Anderson: *What is Darboux integrability?*

In this talk I will briefly review the classical literature on
Darboux integrability; provide a generalization of the classical definition
of Darboux integrability within the context of differential systems; and
discuss recent developments regarding the group theoretical approach to
Darboux integrability. Comparisons with the various notions of integrability
for evolution equations will be made.

17:00 - 17:40 Boris Doubrov: *Cartan's C-class equations: solving differential equations by differentiation*

We prove that the following two conditions on scalar ODEs or systems of ODEs are equivalent: (a) linearization along each solution is trivializable; (b) any differential invariant of a given system under the group of contract transformations (in case of a scalar ODE) or point transformations (in case of systems) is its first integral. The latter condition was introduced by Elie Cartan, who called such differential equations as C-class equations. In particular, they can be either solved explicitly using their differential invariants or reduced to so-called systems of Lie type. The algebra of differential invariants, in its turn, can be computed constructively using the characteristic Cartan connection, which involves only differentiation and algebraic operations.

17:40 - 17:50 Break

17:50 - 18:30 Alexander Bobenko: *On a discretization of confocal quadrics (followed by the movie `Conform')*

I will describe a novel construction of discrete confocal quadrics (discrete confocal coordinate systems). Our discretization respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (i.e. they allow a conformal parametrization along curvature lines). The construction is based on an integrable discretization of the Euler-Poisson-Darboux equation. This is a joint work with Suris, Schief and Techter.

19:00 - 19:30 Wine Reception (to continue after Dinner)

19:30 - 20:30 Dinner at Grey College

20:30 - 21:30 Wine and Cheese Reception (continued)

Jul 29 (Fri)

08:00 - 08:45 Breakfast at Grey College

09:00 - 09:40 Sara Lombardo: *Automorphic Lie Algebras and Root System Cohomology*

The concept of Automorphic Lie Algebras (ALiAs) arises in the context of reduction groups introduced in the late 1970s by Mikhailov. ALiAs are obtained by imposing a finite symmetry group G on a Lie algebra over a field of rational functions [1], [2]. Since their introduction the goal has been to classify them and the case in which the symmetry group is finite and acts on sl_n by inner automorphisms has been recently classified [3]. Past work shows remarkable uniformity between Lie algebras associated to different reduction groups. For example, if the base Lie algebra consists of traceless 2 × 2 matrices and the representation of G is irreducible then the ALiA is independent of the reduction group [2].
In this talk I will recall the concept of ALiAs, discuss a number of open questions from the algebraic point of view and introduce a cohomology of root systems [4].
The talk is based on joint work with Vincent Knibbeler and Jan Sanders (Vrije Universiteit Amsterdam).
[1]
S. Lombardo, A.V. Mikhailov, Reduction groups and Automorphic Lie Algebras, Comm. Math. Phys. 258(1): 179–2002, 2005
[2]
S. Lombardo, J. A. Sanders, On the Classification of Automorphic Lie Algebras, Comm. Math. Phys. 299, 793-824, 2010
[3]
V. Knibbeler, S. Lombardo, J. Sanders, Higher dimensional Automorphic Lie Algebras, Journal of Foundation of Computational Mathematics, Springer, 2016
[4]
V. Knibbeler, S. Lombardo, J. Sanders, Automorphic Lie Algebras and Cohomology of Root Systems, arXiv:1512.07020

09:40 - 10:20 Maciej Blaszak: *Dispersionless (3+1)-dimensional integrable hierarchies*

Applying r-matrix approach to particular Jacobi algebra instead of Poisson algebra, it is possible to construct integrable hierarchies of (3+1)-dimensional dispersionless systems. The price one has to pay is the loss of Hamiltonian structure of considered systems.

10:20 - 10:50 Coffee Break

10:50 - 11:30 Sergei Tabachnikov: *On the bicycle transformation and the filament equation: results and conjectures*

The simplest model of a bicycle is a segment of fixed length that can move, in n-dimensional Euclidean space, so that the velocity of the rear end is always aligned with the segment (the rear wheel is fixed on the frame). The rear wheel track and a choice of direction uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. The two track are related by the bicycle (Darboux, Backlund) transformation which defines a discrete time dynamical system on the space of curves. I shall discuss the symplectic, and in dimension 3, bi-symplectic, nature of this transformation and, in dimension 3, its relation with the filament equation.
An interesting problem is to describe the curves that are in the bicycle correspondence with themselves (in this case, given the front and rear tracks, one cannot tell which way the bicycle went). In dimension two, such curves yield solutions to Ulam's problem: is the round ball the only body that floats in equilibrium in all positions? I shall discuss F. Wegner's results on this problem and relate them with the planar filament equation.
Open problems and conjectures will be emphasized.

11:30 - 12:10 Claude Viallet: *Algebraic entropy*

The algebraic entropy, first defined as a coarse
characterisation of the complexity of birational maps, has become a
valuable test of integrability of (bi)rational dynamics, and an object
of interest per se. In particular it has been conjectured to always be
the logarithm of an algebraic integer. We will describe three ways of
calculating it exactly: firstly by a heuristic and quick way, then by
the singularity analysis developed for two dimensional systems, an
thirdly by a novel approach, also deeply linked to the singularity
structure of the system, but applicable in all dimensions, including
infinite dimensional systems like the so called 'discrete delay
Painlev\'e equations.

13:00 - 14:00 Lunch at Grey College

14:00 - 14:15 Conference Photograph - please gather at Grey College Main Entrance

17:00 - 19:00 Meeting of the Editorial Board of 'Journal of Integrable Systems': Old Library, Grey College

19:00 - 20:00 Dinner at Grey College

Jul 30 (Sat)

08:00 - 08:45 Breakfast at Grey College

09:00 - 09:40 Paolo Lorenzoni: *Bi-flat F-manifolds, Painlevé transcendents and complex reflection groups*

We study F-manifolds equipped with a pair of flat connections (and a pair of F-products), that are required to be compatible in a suitable sense. In the first part of the talk we show that bi-flat F-manifolds in dimension three are locally parameterized by solutions of the full Painlevé IV,V and VI equations, according to the Jordan
normal form of the operator of multiplication by the Euler vector field. In the second part of the talk we discuss some examples related to complex reflection groups.
Based on joint works with Alessandro Arsie.

09:40 - 10:20 Lionel Mason: *Tau functions and anomalous dimensions*

This talk concerns integrable structures in N=4 super Yang-Mills theory. I will discuss a twistor correspondence that applies on the one hand to solutions to the self-dual Yang-Mills equations that are stationary and axisymmetric that is based on a parametrised family of Joukowski (or Zhukovski) transformations. This can be used to obtain tau functions that give rise to certain near BPS anomalous dimensions. In particular we see that such anomalous dimensions satisfy integrable differential equations. This is joint work with Andrea Ferrari.

10:20 - 10:50 Coffee Break

10:50 - 11:30 Youjin Zhang: *Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs*

For an arbitrary semisimple Frobenius manifold we construct an integrable hierarchy of Hamiltonian partial differential
equations. In the particular case of quantum cohomology the tau-function of a solution to the Hodge hierarchy generates the
intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic
classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a
deformation of the Korteweg-de Vries hierarchy depending on an infinite number of parameters. We conjecture that this hierarchy is a universal
object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.
This is a joint work with Boris and Di Yang

11:30 - 12:10 Todor Milanov: *Eynard--Orantin recursion for simple singularities of type A*

In my joint work with B. Bakalov we have proposed a certain vertex algebra approach to the higher genus reconstruction associated with a semi-simple Frobenius manifold. Our construction gives a differential operator constraint for each state in the so called W-algebra. The problem is that in general the W-algebra is difficult to compute, so our construction has only limited applications. In this talk, I would like to discuss a case in which the Eynard--Ornatin recursion can be used to determine states in the W-algebra.

13:00 - 14:00 Lunch at Grey College

14:30 - 15:10 Claus Hertling: *Geometry of the movable poles of real solutions of Painleve III(0,0,4,-4)*

In 1986 Its and Novokshenov studied the asymptotics and the movable poles of real solutions on the real positive line of the Painleve III equation of type (0,0,4,-4). They had results on the behaviour near 0 and near infinity. I will combine this with the global geometry of the moduli spaces for initial data and monodromy data. This will lead to facts on the movable poles (and movable zeros) on the whole positive real line. Behind this is an interpretation of the corresponding isomonodromic connections as TERP-structures (or noncommutative Hodge structures) and results on them by T. Mochizuki, Sabbah and myself. The results are joint work with M.A. Guest.

15:10 - 15:50 Martin Guest: *Convexity for a certain space of solutions to the Hitchin equations*

In previous work with Alexander Its and Chang-Shou Lin we have solved a special case of the 2D Toda equations, which we call the tt*-Toda equations. This special case has various interpretations: harmonic maps of a surface, the topological-antitopological fusion equations of Cecotti-Vafa and Dubrovin, the Hitchin equations (Higgs bundles). We shall review this example, and describe/explain a convexity property of the solutions, which arises from the third point of view, i.e. moduli spaces of flat connections. This is joint work with Nan-Kuo Ho.

15:50 - 16:20 Coffee Break

16:20 - 17:00 Philip Boalch: *Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams *

In 1987 Hitchin discovered a new family of algebraic integrable systems, solvable by spectral curve methods. One novelty was that the base curve was of arbitrary genus. Later on it was understood how to extend Hitchin's viewpoint, allowing poles in the Higgs fields, and thus incorporating many of the known classical integrable systems, which occur as meromorphic Hitchin systems when the base curve has genus zero. However, in a different 1987 paper, Hitchin also proved that the total space of his integrable system admits a hyperkahler metric and (combined with work of Donaldson, Corlette and Simpson) this shows that the differentiable manifold underlying the total space of the integrable system has a simple description as a character variety
Hom(\pi_1(\Sigma), G)/G
of representations of the fundamental group of the base curve \Sigma into the structure group G (typically G=GL_n(C)). This misses the main cases of interest classically, but it turns out there is an extension. In work with Biquard from 2004 Hitchin's hyperkahler story was extended to the meromorphic case, upgrading the speakers holomorphic symplectic approach from 1999. Using the irregular Riemann-- Hilbert correspondence the total space of such integrable systems then has a simple explicit description in terms of monodromy and Stokes data, generalising the character varieties. The construction of such ``wild character varieties'', as algebraic symplectic varieties, was recently completed in work with D. Yamakawa. In this talk I will describe some simple examples of wild character varieties including some cases of complex dimension 2, familiar in the theory of Painleve equations, although their structure as new examples of complete hyperkahler manifolds (gravitational instantons) is perhaps less well-known. This abstract viewpoint on integrable system also leads to a theory of Dynkin diagrams for them, which we hope can be used in their classification.

17:00 - 17:40 Guido Carlet: *Deformations of Poisson and bi-Hamiltonian structures on formal loop spaces*

Dispersive Poisson brackets and Poisson pencils on formal loop spaces play a important role in the description of integrable hierarchies, especially in the setting of hierarchies of topological type. We will first review the general framework and motivation for the study of such objects, including main results like Getzler’s theorem and the notion of central invariants by Liu, Dubrovin and Zhang.
Then we will discuss our recent work, which include the proof, using spectral sequences techniques, of the triviality of the bi-Hamiltonian cohomology of semisimple Poisson brackets of DN type. That in turn implies the existence of arbitrary order dispersive deformations, starting from any choice of central invariants, thus solving a conjecture of Dubrovin et al. Finally we will briefly describe our recent results on the generalisation to the multivariable setting and outline some open problems. Based on joint works with H. Posthuma, S. Shadrin and with M. Casati, S. Shadrin.

17:50 - 18:30 Sergey Cherkis: *From the Bow Integrable System to the Kaehler Potential on the Moduli Spaces of G-monopoles and instantons*

We study monopoles in three-space and instantons on a multi-Taub-NUT space with some gauge group G. The index bundle of the associated family of Dirac operators defines a bow representation as well as an integrable system associated with it. Solutions of this integrable system are in one-to-one correspondence with the gauge equivalence classes of these monopoles or instantons. This view leads to a natural Kaehler potential on their moduli space.
The first part of this talk is based on the recent work with Mark Stern and Andres Larrain-Hubach

19:00 - 20:00 Dinner at Grey College

Jul 31 (Sun)

08:00 - 08:45 Breakfast at Grey College (please collect a packed lunch at Breakfast - there will be no cooked lunches today)

09:00 - 18:00 Excursion to Whitby and Robin Hood's Bay

Electronic Itinerary included in delegate information pack

19:00 - 20:00 Dinner at Grey College

Aug 01 (Mon)

08:00 - 08:45 Breakfast at Grey College

09:00 - 09:40 Tamara Grava: *Small dispersion limit in the KP equation*

We study the Kadomtsev-Petviashvili equation in the limit of small dispersion and we show that the critical behaviour of the solutions is universal and it is described by a Painleve’ transcendent.

09:40 - 10:20 Christian Klein: *Numerical study of $2+1$ dimensional nonlinear dispersive PDEs *

We present several numerical studies of solutions to PDEs from the
family of nonlinear Schrödinger and Korteweg-de Vries equations. We study
the formation of dispersive shocks and of potential blow-ups in the
solutions. Conjectures for the asymptotic description of break-up and
blow-up of the solutions is presented.

10:20 - 10:50 Coffee Break

10:50 - 11:30 Raffaele Vitolo: *Bi-Hamiltonian structures of KdV type*

Combining an old idea of Olver and Rosenau with the classification of second and third order homogeneous Hamiltonian operators (Ferapontov, Pavlov, V. 2014-2016) we classify compatible trios of two-component homogeneous Hamiltonian operators. The trios yield pairs of compatible bi-Hamiltonian operators whose structure is a direct generalization of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give rise to multi-parametric families of bi-Hamiltonian systems. We recover known examples and we find apparently new integrable systems.
(Joint work with P. Lorenzoni, A. Savoldi.)

11:30 - 12:10 Boris Konopelchenko: *Jordan hydrodynamic type systems. Confluence, integrability, parabolicity and regularization. *

Hydrodynamic type systems of the Jordan blocks matrix form associated with the confluent Lauricella functions are considered. It is shown that such nondiagonalizable systems can be obtained from the generic diagonal hydrodynamic
type systems via the confluence process similar to that for the Gauss hypergeometric function. These systems are integrable and govern dynamics of critical points for confluent Lauricella type functions. Such systems appear in the
change of type transitions for the quasi-linear systems of PDEs of mixed type. Gradient catastrophe and its regularization for pure parabolic systems are considered. Differential reductions of such systems are discussed too.

13:00 - 14:00 Lunch at Grey College

14:30 - 15:10 Paolo Maria Santini: *The inverse spectral transform for integrable dispersionless PDEs: Cauchy problem, longtime behavior, wave breaking and applications to physics.*

We present the main features of the inverse spectral transform (IST) of integrable dispersionless PDEs in multidimensions, stressing the main differences from the classical IST of soliton PDEs. We use such a novel IST to solve the Cauchy problem, construct the longtime behavior of solutions, and investigate analytically wave breaking phenomena, when they take place. This theory, applicable to several distinguished examples, like the dispersionless Kadomtsev - Petviashvili, the 2D dispersionless Toda, and the heavenly equations, has been developed in collaboration with S. V. Manakov, and this talk is dedicated to his memory. If time permits, we also discuss some rigorous aspects of this theory, recently understood in collaboration with P. G. Grinevich and D. Wu, and applications to two Cauchy problems for nonlinear wave phenomena in Nature.

15:10 - 15:50 Leonid Bogdanov: *SDYM equations on the self-dual background*

We introduce the technique combining the features of integration schemes for SDYM equations and multidimensional dispersionless integrable equations to
get SDYM equations on the conformally self-dual background. Generating
differential form is defined, the dressing scheme is developed. Some special
cases and reductions are considered.

15:50 - 16:20 Coffee Break

16:20 - 17:00 Wojciech Krynski: *GL(2,R)-geometries and integrable systems*

I will consider GL(2,R)-structures in even dimensions and present a general construction of almost-complex manifolds associated to the structures. The integrability of the structures is expressed in terms of the torsion and curvature of the GL(2,R)-structures. As a application I will present an integrable system whose solutions are in a one-to-one correspondence with torsion-free GL(2,R)-structures in dimension 4.

17:00 - 17:40 Boris Kruglikov: *Integrability in Grassmann and other Geometries*

We will discuss the question whether it is possible to read off
dispersionless integrability via geometry of the equation or its solution
space. We will demonstrate the affirmative answer in several particular
important cases, related to projective, Lagrangian and Grassmann geometries.
This will be done via relations to Einstein-Weyl geometry in 3D,
self-duality in 4D, and GL2-structures.
Based on joint work with E.Ferapontov, B.Doubrov, and V.Novikov.

17:40 - 17:50 Break

17:50 - 18:30 Michael Atiyah: *Why does the 6-sphere not have a complex structure?*

This question is difficult because the 6-sphere has an almost complex structure which would not be kahler. I will explain, using old ideas, that a complex structure would be "even" while the almost complex structure is "odd", yielding a contradiction.

19:00 - 20:00 Dinner at Grey College

Aug 02 (Tue)

08:00 - 08:45 Breakfast at Grey College

09:00 - 09:40 Adam Doliwa: *Yang-Baxter maps in non-commuting variables, and their geometric origin*

Starting from the non-commutative discrete KP system we construct, by applying periodic constraints, a family of solutions of the functional
Yang-Baxter equation. In the commutative case these maps
were studied in relatation to geometric crystals. Their counterpart on the level of integrable discrete systems is provided by recently introduced non-commutative Gelfand-Dikii systems.

09:40 - 10:20 Andy Hone: *Somos sequences in algebra, geometry and number theory*

Somos sequences are integer sequences that are generated by recurrence relations that are quadratic (or bilinear) in the dependent variables. They naturally arise in the theory of integrable systems, as finite-dimensional reductions of the discrete KP or BKP equation. In the case of three-term Somos (or Gale-Robinson) recurrences, they also come equipped with the structure of a cluster algebra. Here we review current progress in understanding these sequences from algebraic, geometric and arithmetic viewpoints.

10:20 - 10:50 Coffee Break

10:50 - 11:30 Yuri Suris: *A construction of a large family of integrable symplectic birational maps*

We give a construction of completely integrable (2m)-dimensional 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 он 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.

11:30 - 12:10 Vladimir Novikov: *Discrete integrable equations in 3D*

We propose a novel approach to the classification of integrable differential/difference as well as fully discrete equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limit are `inherited’ by the full dispersive equation. The only constraint of the method is that the initial ansatz possesses a non-degenerate dispersion less limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify 3D integrable Hirota type equations within various particularly interesting subclasses.

13:00 - 14:00 Lunch at Grey College

14:30 - 15:10 Evgeny Sklyanin: *Quantisation of Kadomtsev-Petviashvili equation*

A quantisation of the KP equation on a cylinder is proposed
that is equivalent to an infinite system of one-dimensional bosons
carrying masses m=1,2,... The Hamiltonian is Galilei-invariant
and includes the cubic split/merge (m1,m2)<->(m1+m2) terms
for all combinations of particles with masses m1, m2 and m1+m2,
with a special choice of coupling constants.
The Bethe eigenfunctions for the model are constructed.
The consistency of the coordinate Bethe ansatz, and therefore,
the quantum integrability of the model is verified for the sectors up to the total mass M=8.

15:10 - 15:50 Oleg Chalykh: *Hypergeometric adelic Grassmannian and bispectrality*

We will describe differential-difference bispectral families that can be constructed starting from the hypergeometric equation. This can be viewed as a generalisation of Wilson's adelic Grassmannian and its trigonometric analogue studied by Haine, Horozov and Iliev. The constructed bispectral operators generalise various previously known examples related to the exceptional Jacobi polynomials. This is joint work with A Khalid (Leeds).

15:50 - 16:20 Coffee Break

16:20 - 17:00 Peter Clarkson: *Orthogonal Polynomials and Integrable Systems*

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 Painlev\'e equations. The determinants arise as partition functions in random matrix models and the recurrence coefficients satisfy a discrete Painlev\'e equation. The semi-classical orthogonal polynomials discussed will include a generalization of the Freud weight and an Airy weight.

17:00 - 17:40 Allan Fordy: *$Z_N$ graded discrete Lax pairs and discrete integrable systems*

We introduce a class of Z_N graded discrete Lax pairs, with N x N matrices, linear in the spectral parameter and classify the associated discrete integrable systems. Several well known examples belong to our scheme for N=2 (discrete MKdV equation, Hirota's discrete sine-Gordon equation, discrete potential KdV, Schwarzian KdV equation), so many of our systems may be regarded as generalisations of these. Even at N=3, several new integrable systems arise.
We also present continuous isospectral deformations these Lax pairs, giving compatible differential-difference systems, which play the role of continuous symmetries of the discrete systems. Master symmetries and their respective hierarchies can be constructed.
We present two nonlocal symmetries of our discrete systems, which give rise to the two-dimensional Toda lattice, with our nonlocal symmetries being the B\"acklund transformations and our discrete system being the nonlinear superposition formula (for the generic case).
This talk is joint work with Pavlos Xenitidis and based on the paper: arXiv:1411.6059 [nlin.SI].

19:00 - 19:30 Conference Dinner Wine Reception

19:30 - 20:30 Conference Dinner

Aug 03 (Wed)

08:00 - 08:45 Breakfast at Grey College

09:00 - 09:40 Yuri Berest: *Double affine Hecke algebras and character varieties of knots*

Let G be a complex reductive algebraic group. In this talk, we will
discuss a general conjecture that there is a natural action of the
double affine Hecke algebra of type G on a (quantized) character
variety of the complement of a knot in S^3. We will explain a motivation behind this conjecture and give an explicit construction,
which is inspired by the theory of quantum integrable systems.
The main implication is the existence of 3-variable polynomial knot
invariants that specialize to the famous Jones polynomial and its
colored versions introduced by Witten, Reshetikhin and Turaev.
Time permitting, we show that, in the classical limit (q=1), our conjecture follows from (and essentially reduces to) a known
conjecture in knot theory due to G. Brumfiel and H. Hilden (1990).
(The talk is based on joint work with P. Samuelson.)

09:40 - 10:20 Robert Marsh: *Braid groups and quiver mutation*

Joint work with Joseph Grant.
We give a presentation of an Artin braid group of type ADE for any quiver which is mutation-equivalent to a Dynkin quiver of the same type. We show how these presentations can be understood topologically in types A and D using a disk and a disk with a cone point of order two respectively.

10:20 - 10:50 Coffee Break

10:50 - 11:30 Dmitry Korotkin: *Periods of meromorphic quadratic differentials and Goldman bracket*

We study symplectic properties of monodromy map of second order
linear equation with meromorphic potential with simple poles and zeros on
a Riemann surface. The periods of the square root of the quadratic
differential on the canonical cover of the Riemann surface provide
a natural system of Darboux coordinates on the cotangent bundle
$T^*\Mcal_{g,n}$. Furthermore, the canonical symplectic structure on
$T^*\Mcal_{g,n}$ induces the Goldman Poisson structure on the
corresponding character variety under the monodromy map of second order
differential equation on a Riemann surface. Being combined with results of
S.Kawai this result shows the symplectic equivalence of embeddings of the
cotangent bundle into the space of projective structures given by the Bers
(quasi-fuchsian) and the Bergman projective connections.
The case of holomorphic quadratic differentials was considered in a recent
joint work with M.Bertola and C.Norton.

11:30 - 12:10 David Brander: *Geometric Cauchy problems for surfaces associated to harmonic maps*

I will describe how to solve geometric Cauchy problems for surfaces associated to Riemannian and Lorentzian harmonic maps using special potentials in the DPW method and its generalizations.

13:00 - 14:00 Lunch at Grey College

14:30 - 15:10 Maria Przybylska: *Integrability obstructions of certain homogeneous Hamiltonian systems in 2D curved spaces*

Recently differential Galois integrability conditions have appeared to be very effective for various Hamiltonian systems. The necessary conditions for the integrability of a Hamiltonian system in the Liouville sense are given in terms of properties of the differential Galois group of variational equations obtained by linearisation of equations of motion in a neighbourhood of a particular solution. The fundamental Morales-Ramis theorem of this approach says that if a Hamiltonian system is meromorphically integrable in the Liouville sense in a neighbourhood of a phase curve corresponding to a particular solution, then the identity component of thedifferential Galois group variational equations along this phase curve is Abelian.
In the case of natural Hamiltonian systems on $2n$ dimensional Euclidean phase space with the standard kinetic energy and with homogeneous potentials of integer degrees generically exist certain particular solutions. Moreover differential Galois groups of variational equations along these solutions can be calculated effectively.
Successful integrability analysis of Hamiltonian systems with homogeneous potentials in flat Euclidean spaces motivated us to look for systems in curved spaces with similar properties. We propose two classes of Hamiltonians that can be considered as such analogues of homogeneous Hamiltonian systems in flat spaces. Moreover we present differential Galois integrability conditions for more general case when potentials are homogeneous of rational degrees functions of variables i.e. they are algebraic functions. Obtained integrability conditions are applied to certain classes of potentials and integrable cases are identified.

15:10 - 15:50 Simon Ruijsenaars: *Poeschl-Teller made relativistic *

We begin this seminar by reviewing one-dimensional potential scattering, and discuss the repulsive and attractive Poeschl-Teller potentials g(g-1)/sinh^2(x) and -g(g-1)/cosh^2(x) as explicit examples. After relating this to the hyperbolic integrable N-particle Calogero-Moser systems, we sketch the relativistic version of the latter, and generalizations of the Poeschl-Teller results to this relativistic setting. We present an overview rather than details, linking in particular to factorized scattering and the sine-Gordon quantum field theory.

15:50 - 16:20 Coffee Break

16:20 - 17:00 Misha Feigin: *V-systems*

V-systems are special configurations of vectors defined in terms of linear algebra. They were introduced by Veselov in the end of 90th in relation with Frobenius manifolds. V-systems generalize Coxeter root systems. The class of V-systems is closed under natural operations of taking subsystems and restrictions. A number of examples of V-systems is known but classification remains open. After explaining these properties of V-systems I am going to discuss more recent study of the special subclass of harmonic V-systems. The corresponding arrangements of hyperplanes are shown to be free in the sense of Saito, and the basic logarithmic fields have explicit potentials for classical families of V-systems. The talk is based on joint works with A.P. Veselov.

17:00 - 17:40 Vladimir Roubtsov: *Quantization of Painleve monodromy varieties and generalized Sklyanin algebras*

We discuss two ways of coalescent Painlevé monodromy varieties quantization. Both are depend upon two different choices of flat coordinates on these varieties. We show that there is a flat deformation which generalizes the quantum coordinate rings of the monodromy varieties and includes (generalized) Sklyanin algebras with three generators.

19:00 - 20:00 Dinner at Grey College

Aug 04 (Thu)

08:00 - 08:45 Breakfast at Grey College

10:00 - 10:00 Please check out of your room by 10.00am today