AGIS Colloquia Archives, 2010-2016

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.

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.

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.

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].

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.

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.

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.

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.

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.

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.

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.

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.

AGIS Colloquium