# School of Mathematics

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## Algebra, Geometry and

Integrable Systems Colloquium

Friday 20th October, 2017

Mall 1, 4:00 PM

**Victor Buchstaber**(

*Russian Academy of Sciences*)

Infinite dimensional Lie algebras with a structure of finitely generated modules

Infinite-dimensional Lie algebras with the structures of finitely generated modules naturally arise in the theory of integrable systems, algebraic geometry, differential topology, singularity theory. Fruitful examples such Lie algebras are polynomial algebras of vector fields on universal bundles of Jacobians of hyperelliptic curves and automorphic Lie algebras. Witt algebra, Kac - Moody algebras and other classical infinite-dimensional Lie algebras have the faithful representations in a number of important examples of such Lie algebras.
The talk is devoted to the problems of general theory and applications of the infinite-dimensional Lie algebras with the structures of finitely generated modules. We discuss the general constructions of such algebras and their morphisms. Important examples will be given and their connections with known infinite-dimensional Lie algebras described.
This work is in collaboration with A V Mikhailov.

Tuesday 10th October, 2017

MALL 1, 3:15 PM

**Nick Manton**(

*University of Cambridge*)

Algebra, Geometry and Skyrmions

Skyrmions arise in a model devised to explain atomic nuclei.
Mathematically, they are spatially localised structures that solve certain PDEs, and quantum mechanics also comes in. However, this talk will stress neither the physics nor the PDEs. Instead it will be mainly about the geometrical structure of the Skyrmions. They resemble three-dimensional clusters of particles on the vertices of a Face-Centred-Cubic lattice, and
those that are energetically favoured often have the shapes of SU(4) weight diagrams. I will present some (possibly) new results about the weight
diagrams, and say something about what this implies for the physics.

Friday 17th March, 2017

Mall 1, 4:00 PM

**Martina Balagovic**(

*Newcastle*)

Degeneration of dynamical difference equations

I will describe how, under Drinfeld’s degeneration of quantum loop algebras to Yangians, the trigonometric dynamical difference equations for the quantum affine algebra degenerate to the trigonometric Casimir differential equations for Yangians. I will explain all necessary prerequisites and results needed to state this claim precisely, and sketch the idea of the proof.

Tuesday 7th March, 2017

EC Stoner 7.73, 3:15 PM

**Sarah Whitehouse**(

*University of Sheffield*)

Derived A-infinity algebras

A-infinity algebras arise when one considers an operation which is associative up to homotopy. As soon as one does this, one is led to a rich structure with an infinite family of operations. These structures have their origins in topology and they have become important in many different areas of mathematics, including algebra, geometry and mathematical physics.
I will explain what they are and briefly survey some of the places they arise. Then I will motivate and discuss a recent generalisation, known as a derived A-infinity algebra. These are important when working over a commutative ground ring rather than a field. Results include some new descriptions of these structures and a hierarchy of different notions of equivalence.

Friday 3rd March, 2017

Mall 1, 4:00 PM

**David Calderbank**(

*Bath*)

Dispersionless integrable PDEs via geometry

A determined nonlinear PDE system on a manifold M is integrable by a dispersionless Lax pair if it arises as the integrability condition for a rank 2 distribution on a rank 1 bundle over M. By establishing that any such Lax pair is characteristic on solutions, we show that when the characteristic variety is a quadric (e.g. for nondegenerate second order scalar PDEs), the PDE is dispersionless integrable if and only if the induced conformal structure is self-dual (for n=4) or Einstein-Weyl (for n=3) on any solution. These results unify and extend work of E. Ferapontov et al.
(Joint work with Boris Kruglikov)

Wednesday 8th February, 2017

Roger Stevens LT12, 3:00 PM

**Bernd Schroers**(

*Heriot-Watt University*)

Integrable vortex tubes in three dimensions

In this talk I will explain how one can combine a rather wide range of ideas and results from the literature - the work of Loss and Yau on zero-modes of a magnetic 3d Dirac operator, a dimensional reduction of the Seiberg-Witten equations, integrable vortex equations in 2d - to obtain linked vortex tubes as exact solutions of a spinorial vortex equation in 3d. There is a new, Lorentzian version of this story which I will sketch as well. This talk is based on joint work with my student Calum Ross.

Thursday 2nd June, 2016

RSLT16, 4:00 PM

**A. Aleman**(

*Lund*)

Harmonic maps and shift invariant subspaces

Thursday 2nd June, 2016

RSLT16, 4:00 PM

**A. Aleman**(

*Lund*)

Harmonic maps and shift invariant subspaces

Tuesday 10th May, 2016

Mall 1, 4:30 PM

**Michael Gekhtman**(

*Notre Dame*)

Higher Pentagram Maps via Cluster Mutations and Networks on Surfaces

The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras, a new and rapidly developing area with many exciting connections to diverse fields of mathematics.
In this talk I will explain that one possible family of higher-dimensional generalizations of the pentagram map is a family of discrete integrable systems intrinsic to a certain class of cluster algebras that are related to weighted directed networks on a torus and a cylinder. After presenting necessary background information on Poisson geometry of cluster algebras, I will show how all ingredients necessary for integrability - Poisson brackets, integrals of motion - can be recovered from combinatorics of a network.
The talk is based on a joint project with M. Shapiro, S. Tabachnikov and A. Vainshtein.

Monday 7th March, 2016

EC Stoner 7.73, 3:15 PM

**Sarah Whitehouse**(

*University of Sheffield*)

Derived A-infinity algebras

A-infinity algebras arise when one considers an operation which is associative up to homotopy. As soon as one does this, one is led to a rich structure with an infinite family of operations. These structures have their origins in topology and they have become important in many different areas of mathematics, including algebra, geometry and mathematical physics.
I will explain what they are and briefly survey some of the places they arise. Then I will motivate and discuss a recent generalisation, known as a derived A-infinity algebra. These are important when working over a commutative ground ring rather than a field. Results include some new descriptions of these structures and a hierarchy of different notions of equivalence.

Monday 24th March, 2014

Roger Stevens LT07, 2:00 PM

**Eugenie Hunsicker**(

*Loughborough University*)

Variations on the Hodge theorem

The famous Hodge theorem relates the space of harmonic differential forms over a smooth compact manifold to the deRham cohomology. It is a landmark in the area of geometric topology, and has had a huge influence on mathematics since its proof. Since the 1970's, various mathematicians have worked on extending this theorem to a broader setting, including singular spaces and non compact manifolds. This talk will discuss the difficulties involved in doing this, as well as some techniques that have been successful. It will also include several examples of general Hodge theorems proved since the 1970's.

Tuesday 2nd July, 2013

MALL 1, 9:30 AM

**Zhenghan Wang, Bernd Schroers, and many others**(

*Microsoft Station Q and many others*)

Quantum Computing: a Quantum Group Approach, mini workshop, 1-2 July 2013

Monday 1st July, 2013

MALL 1, 2:00 PM

**J. Pachos, G. Brennen, G. Palumbo**(

*Leeds, Macquarie, Leeds*)

Quantum Computing: a Quantum Group Approach, mini workshop, 1-2 July 2013

Monday 15th April, 2013

Roger Stevens Lecture Theatre 13, 4:00 PM

**Zoltan Kadar**(

*University of Leeds*)

AGIS Colloquium: New phases of matter by means of modular tensor categories, part II.

In 1992 Turaev and Viro gave the prescription to construct topological invariants of three dimensional manifolds. The procedure is defining a finite state sum of weights associated to every coloured triangulations of a manifold. A weight itself is a products of local weights corresponding to simplices of the triangulation, which form a representation of a quantum group. 12 years later it has been proposed by Levin and Wen that certain types of materials are described by models equivalent to the above, some of which, in turn, are potential candidates for realizing a (topological) quantum computer.
This time we will really get to the above topic starting from the toric code, the simplest model based on the Drinfeld double of Z_2. Then, its generalizations when the group is an arbitrary finite group will be shown to be equivalent to a subset of the Levin-Wen string nets. The general equivalence proof between these and the Turaev-Viro theories was given by Kirillov Jr in 1106.6033. In the talk I plan to show that the ground state projection of the string net on the surface S corresponds to the 3d state sum of the annulus S x [0,1].

Wednesday 20th March, 2013

MALL 1, 4:00 PM

**Barnaby Martin**(

*Middlesex University London*)

"A Random Walk through Infinite-domain CSP

We survey the past, present and (insofar as temporally possible)
future of the area of infinite-domain constraint satisfaction
problems, mainly from the perspective of Bodirsky and his
collaborators. We are especially interested in matters of structure
and complexity as these manifest from a combinatorial,
universal-algebraic and/ or model-theoretic perspective. We give
special attention to recent work in templates that are not
omega-categorical but enjoy other benign model-theoretic properties.

Monday 11th March, 2013

Roger Stevens Lecture Theatre 13, 4:00 PM

**Zoltan Kadar**(

*University of Leeds*)

AGIS Colloquium: New phases of matter by means of modular tensor categories.

In 1992 Turaev and Viro gave the prescription to construct topological invariants of three dimensional manifolds. The procedure is defining a finite state sum of weights associated to every coloured triangulations of a manifold. A weight itself is a products of local weights correspondig to simplices of the triangulation, which form a representation of a quantum group. 12 years later it has been proposed by Levin and Wen that certain types of materials are described by these models, some of which, in turn, are potential candidates for realizing a (topological) quantum computer.
An introduction to the above concepts will be given and key physical quantities will be shown to correspond to specific Turaev-Viro state sums.

Wednesday 22nd February, 2012

RSLT 24, 3:00 PM

**Tom Coates**(

*Imperial College*)

Gromov--Witten Invariants and Modular Forms

The Gromov--Witten invariants of a space X count, roughly
speaking, the number of holomorphic curves that meet various cycles in
X. They have important applications in algebraic geometry, symplectic
topology, and theoretical physics. Let X be the canonical line bundle
over the projective plane P^2. I will describe joint work with Iritani
in which we show that generating functions for Gromov--Witten
invariants of X are modular forms for the group Gamma_0(3). There are
tantalizing hints of a connection with integrable systems.

Friday 3rd February, 2012

Mall 1, 4:00 PM

**Franco Vivaldi**(

*Queen Mary, London University*)

Integrable vs. non-integrable when the space is discrete

In dynamical systems with discrete space, the distinction
between regular and irregular motions must be re-considered.
I articulate the main questions using examples, with emphasis
on arithmetical phenomena.

Wednesday 8th June, 2011

MALL 1, 3:00 PM

**Domingo Toledo**(

*University of Utah*)

Periods of Cubic Surfaces

Cubic surfaces in complex projective space can be classified by a map to
the quotient of the unit ball in C^4 by an arithmetic group. This map is called the
period map. The arithmetic group reflects and refines classical the classical
relations between cubic surfaces and the Weyl group of E_6. We will explain the
construction of this map, explain some of its geometry, and then talk about what is
known and what we would like to know about values of this map. This is joint work with
Allcock and Carlson.

Wednesday 23rd March, 2011

RSLT 18, 3:00 PM

**Richard Szabo**(

*Heriot-Watt*)

Instantons on noncommutative toric varieties

We construct new noncommutative deformations of toric varieties by
combining methods from toric geometry, isospectral deformation theory and
noncommutative geometry in braided monoidal categories. We apply these
techniques to the construction of a certain class of noncommutative
instantons and discuss the interrelationships between their description in
terms of deformed ADHM data, torsion-free modules and a noncommutative
twistor correspondence.

Wednesday 26th January, 2011

RSLT 10, 3:00 PM

**Fran Burstall**(

*University of Bath*)

Geometry and dynamics of isothermic submanifolds

This talk will have three parts: in the first, I will
describe the beautiful classical theory of isothermic surfaces in the
3-sphere due to Christoffel, Darboux, Bianchi and others. Then I will
indicate how the 3-sphere may be replaced by any symmetric R-space (a
conjugacy class of real parabolic subalgebras with abelian
nilradicals) with essentially no loss of integrable structure.
Finally, I shall show how dynamics of the simplest examples (curves in
the real projective space) provide a geometric interpretation of the
KdV equation, its relation to the mKdV equation via the Miura
transform and the Baecklund transformations of KdV discovered by
Walhquist-Estabrook.

Wednesday 20th October, 2010

Roger Stevens LT 11 (10.11), 3:00 PM

**Roger Bielawski**(

*University of Leeds*)

Natural geometry of hyperbolic monopoles

I shall present a solution to a long standing problem: what is the natural geometry of moduli spaces of hyperbolic monopoles. The answer, it turns out, is a completely new type of geometry, generalising hyperkaehler and hypercomplex geometries. Remarkably, even the underlying linear algebraic structure appears not to have been studied before. In this talk, I'll present the geometry "from bottom up", starting with the linear-algebraic version and then describing integrability conditions required for manifolds with such geometry. No knowledge of monopoles is necessary. This work is joint with L. Schwachhoefer.

Wednesday 24th February, 2010

RSLT16, 3:00 PM

**Tamas Hausel**(

*Oxford*)

Mirror Symmetry, Langlands duality and the Hitchin system

AGIS Colloquium

### Colloquia

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