Madrid

Title: Integrability, Geometry, and Physics: A Unified View

Abstract: What do water waves have to do with mirror symmetry, or even with Kovalevskaya tops? All of them are, or are closely related to, integrable systems,  in particular to a class of algebraically completely integrable systems known as Hitchin systems. In this talk, I will explore how Hitchin systems provide a unifying framework connecting seemingly disparate areas of mathematics and physics. The presentation will be accessible to a general mathematical audience, while offering concrete connections for specialists in geometry, topology, and algebraic geometry.

Waterloo

Title: New examples of compact holomorphic symplectic manifolds

Abstract: A holomorphic symplectic manifold is a complex manifold $X$ together with a closed, non-degenerate holomorphic 2-form $\Omega$. The top power of $\Omega$ gives a trivialisation of the canonical bundle so that $X$ has trivial first Chern class. In the context of K\"ahler geometry, such manifolds play a very important role due to the Bogomolov covering theorem, which states that any compact K\"ahler manifold with vanishing first Chern class has a covering that splits as the product of Calabi–Yau manifolds, complex tori and irreducible holomorphic symplectic manifolds. Among these, the last two are, in fact, compact holomorphic symplectic manifolds. Furthermore, irreducible holomorphic symplectic manifolds correspond to compact hyperkahler manifolds in the K\"ahler setting. In general, finding compact holomorphic symplectic manifolds is very difficult. In this talk, I will present new examples of compact holomorphic symplectic manifolds. These manifolds correspond to moduli spaces of sheaves on Kodaira surfaces and are non-K\"ahler. This is work in progress with Tom Baird and Eric Boulter.

Maryland

Title: "The joint moduli space of Higgs bundles on families of Riemann surfaces"

Abstract: I will discuss a gauge theoretic approach to the construction of the moduli space of Higgs bundles in 2-dimensionswhere the complex structure of the underlying surface also varies. This "joint" moduli space fibers over Teichmueller spacewith fiber the usual moduli space of Higgs bundles.  I will explain why indefinite Hermitian structures arise naturally on the joint moduli space, and I will indicate the existence of pseudo-Kaehler metrics in a number

of cases of Higgs bundles with special holonomy. I will also discuss the relationship between complex tangencies of isomonodromicl eaves and the strict plurisubharmonicity of the energy function. This recovers and extends several recent constructions of various authors. This work is part of a collaboration with Brian Collier and Jeremy Toulisse.

Yale

Title: An introduction to exact WKB and its uses

Abstract: The exact WKB method originated in the study of linear differential equations, e.g. the Schrodinger equation in one variable. Recently it has turned out that exact WKB has connections to various topics in modern geometry, topology and quantum field theory. I will explain what the exact WKB method is, and briefly survey some of its connections to other fields.

North Caoline at Chapell Hill

Title: Lagrangian fibrations by Jacobian and Prym surfaces

Abstract: In this talk we consider compact holomorphic symplectic manifolds (aka hyperkahler manifolds), particular those fibred by Lagrangian submanifolds. The general fibre must be an abelian variety. Focusing on dimension four, we will describe how to construct examples whose fibres are abelian surfaces that are Jacobians of genus two curves or Prym varieties of double covers of curves. We will also describe some classification results for these kinds of fibrations.

Northwestern

Title:  Combinatorial Perspectives in Stringy Geometry

Abstract:  I will give a tour of some places in modern mathematical physics 

where classical combinatorial objects of discrete math arise.  More specifically, I will

explore how counting points (of various moduli spaces) and surfaces (holomorphic curves) in symplectic topology can be related to the enumeration of graph colorings (chromatic polynomial), triangulations (Catalan numbers) and quiver structures (clusters).

Boston College

Title: "An Explicit relationship between the Swapping and Ghost algebras" 

Abstract: I will describe an explicit relationship between the the Swapping and Ghost algebras, which are both Poisson algebras (defined by Labourie and Bridgeman-Labourie resp.) that are useful combinatorial tools in the broad (vast/deep) program of studying Hitchin components of character varieties. [This is joint work with Ming Hong Tee, Boston College]

UIUC

Title: What is the homotopy type of quantum field theory?

Abstract: Spaces of quantum theories are the fundamental objects in modern mathematical physics. Outside of some basic examples, little is known about the geometry and topology of these spaces. In this talk, I will begin with spaces built from quantum mechanics. We will find that twisted equivariant K-theory encodes the homotopy type of the space of (supersymmetric) quantum mechanical systems. Viewing quantum systems as 1-dimensional quantum field theories, a natural generalization suggests a connection between 2-dimensional (supersymmetric) quantum field theories and twisted equivariant elliptic cohomology. This builds on ideas of Segal, Stolz and Teichner, and is intended as an introduction to their program. 

U Chicago

The dynamics of generic symplectomorphisms. 

I will describe some history and work in various collaborations with Artur Avila, Christian Bonatti and Sylvain Crovisier about what one sees generically (in the C^1 topology)  in the dynamics of symplectomorphisms of a closed manifold. 

Notre Dame

Title: The underlying geometry of Conformal Field Theory, the resulting algebra of fields, and how number theory naturally arises

Abstract:  The geometry of propagating strings in Conformal Field Theory (i.e. string theory), naturally gives rise to an algebraic structure called a vertex operator algebra describing the interactions of fields.   The graded traces and pseudo-traces of families of linear operators corresponding to components of the fields (vertex operators) of interacting particles exhibit certain invariance properties with respect to action of the modular group SL(2,Z).  We discuss aspects of modular symmetries that appear in graded trace and pseudo-trace functions for various classes of algebras of fields (vertex algebras) and their modules.