Geometry and Mathematical Physics Seminar

The Geometry and Mathematical Physics Seminar usually takes place on Wednesdays at 13.30 during term time in Watson Building.

Autumn 2019

Strong positivity for quantum cluster algebras

  • Travis Mandel, University of Edinburgh
  • Tuesday 19 November 2019, 14:30
  • Poynting Small Lecture Theatre (S06)

 

I will describe recent joint work with Ben Davison in which we construct “quantum theta bases” for skew-symmetric quantum cluster algebras. These bases satisfy many nice properties; e.g., the structure constants for the multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, thus proving the strong positivity conjecture from quantum cluster theory. Our approach combines structures coming from mirror symmetry (i.e., scattering diagrams and broken lines, as used by Gross-Hacking-Keel-Kontsevich in the classical limit) with results from the DT-theory of quiver representations.

BPS invariants in geometry and physics

  • Sven Meinhardt, University of Sheffield
  • Tuesday 12 November 2019, 14:30
  • Watson (Mathematics) Room 310

Note the unusual room

BPS indices were first introduced in physics as a count of ‚short‘ irreducible representations of a supersymmetric extension of the well-known 4-dimensional Lorentz group occurring in special relativity. String theorists suggested to compute these numbers by counting D-branes living in the hidden 6 dimensions of 10-dimensional spacetime. This idea has been made more precise by many mathematicians in the last 20 years and is nowadays known as Donaldson-Thomas theory. In my talks I will sketch the main ideas of Donaldson-Thomas theory and report on recent results obtained in collaboration with Ben Davison. Starting with the problem of counting D-branes, we will introduce BPS invariants and the Hall algebra leading to a categorification of BPS invariants. If time permits, the relation to physics and cluster transformations will be explained.

Bridgeland Stability on Threefolds

  • Benjamin Schmidt, Leibniz Universität Hannover
  • Tuesday 05 November 2019, 14:30
  • Poynting Small Lecture Theatre (S06)

In 2002 Bridgeland introduced the notion of a stability condition in a triangulated category to provide a mathematical framework for work by Douglas in homological mirror symmetry. A fundamental Theorem says that all stability conditions on a given category form a complex manifold. However, in general it is still not known whether it is non-empty. The construction is easy for the bounded derived category of coherent sheaves on a smooth projective surface, but for threefolds examples are scarce. I will give an introduction to the topic, and explain recent progress.

Hodge structures of K3 type in Fano varieties

  • Enrico Fatighenti, Loughborough University
  • Tuesday 29 October 2019, 14:30
  • Poynting Small Lecture Theatre (S06)

Subvarieties of Grassmannians (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type (FK3), for their deep links with hyperkähler geometry. In this talk we will present some examples of recently discovered FK3 varieties, and a general procedure that allows us to spread a (Hodge) K3 structure as a component of the Hodge structure of different varieties. This is in collaboration with Giovanni Mongardi and Marcello Bernardara—Laurent Manivel.

Topological Recursion, Hurwitz theory, and moduli spaces of curves

  • Danilo Lewanski, Max Planck Institute for Mathematics in Bonn
  • Tuesday 22 October 2019, 14:30
  • Poynting Small Lecture Theatre (S06)

Note room and time change

Topological recursion (TR) is a technique developed by Chekhov, Eynard and Orantin about ten/fifteen years ago, which computes invariants recursively from the given input data of a spectral curve, even in the case the spectral curve is not provided by a matrix model. Examples of these invariants include Mirzakhani’s volumes of moduli spaces of hyperbolic surfaces, Gromov-Witten invariants, Hurwitz numbers of several kinds, Tutte’s enumeration of maps, asymptotics of coloured Jones polynomials of knots, and more. In particular, Hurwitz theory provides a good set of enumerative geometric problems whose numbers are (in some cases still conjecturally) generated via TR for different explicit spectral curves. Interestingly enough, these numbers are always linked to the cohomology of the moduli spaces of curves, and at the same time with integrable hierarchies of type 2D Toda or KP, contributing to the understanding of the interaction between Geometry and Mathematical Physics.

Quivers and supersymmetric QFTs

  • Cyril Closset, University of Oxford
  • Tuesday 15 October 2019, 14:30
  • NUFF-G07

Time and room changed

Quivers appears in string theory as a convenient way of summarising the spectrum of open-string BPS states at “orbifold” points. Similarly, in QFT , quiver methods are useful either to characterize the field theory itself, or to characterize its spectrum of BPS states. After a general introduction, I will focus on quivers that appear in the context of five-dimensional supersymmetric field theories, including new results about the Coulomb branch of 5d SCF Ts.

Asymptotics and Resurgence in String and Gauge Theories

  • Inês Aniceto, University of Southampton
  • Tuesday 08 October 2019, 17:00
  • Watson Building (Mathematics, R15 on map) Lecture Room A

The perturbative expansions of many physical quantities are divergent, and defined only as asymptotic series. It is well known that this divergence reflects the existence of nonperturbative, exponentially damped contributions, such as instanton effects, which are not captured by a perturbative analysis. This connection between perturbative and non-perturbative contributions of a given physical observable can be systematically studied using the theory of resurgence, allowing us to construct a full non-perturbative solution from perturbative asymptotic data. In this talk I will start by reviewing the essential role of resurgence theory in the description of the analytic solution behind an asymptotic series, and its relation to the so-called Stokes phenomena, phase transitions and ambiguity cancelations. I will then exemplify with some recent applications of resurgence in the context of string and gauge theories.

There is a complete list of the talks in the Geometry and Mathematical Physics seminar on talks@bham.