*Please note that the room of the seminar will change throughout the term. *

*Vaibhav Gadre (University of Glasgow). ***Cusp excursions of random Teichmulller geodesics..** Friday 3rd May, 1:30-2:30pm. Huxley 308.

** Abstract: ** Dynamics on moduli spaces of Riemann surfaces is often studied by considering analogies with homogeneous dynamics. The talk will begin with a short survey of moduli spaces from this point of view. It will then focus on random geodesics on moduli spaces as a specific context for considering these analogies.

*Pierrick Bousseau (ETH Zurich). ***A relative holomorphic anomaly equation.** Friday 10th May, 1:30-2:30pm. Huxley 308.

** Abstract: ** The holomorphic anomaly equation is some recursive relation conjecturally satisfied by higher genus Gromov-Witten invariants of Calabi-Yau 3-folds. I will start reviewing the case of the local projective plane, for which this result is known. The main point of the talk will be to present a new conjectural holomorphic anomaly equation for some family of relative Gromov-Witten invariants related to the projective plane, and a proof in genus one. This is work in progress, joint with Longting Wu.

*Nikolaos Tziolas (University of Cyprus). *** Vector fields and moduli of canonically polarized surfaces in positive characteristic.** Friday 17th May, 1:30-2:30pm. Huxley 139.

** Abstract: ** The purpose of this talk is to present some results about the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, equivalently non reduced automorphism scheme, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces.

*Ed Segal (UCL). ***Mirrors to punctured surfaces.** Friday 24th May, 1:30-2:30pm. Huxley 144.

** Abstract: ** There are various known ways to construct a category which is mirror to the Fukaya category of a punctured surface. I’ll discuss some work-in-progress (with Yanki Lekili) where we find algebro-geometric proofs that all the constructions give equivalent categories.

No seminar on Friday 31st May due to this event.

*Andreas Krug (University of Marburg). ***Stability of Tautological Bundles on Symmetric Products of Curves.** Friday 7th June, 1:30-2:30pm. Huxley 144.

** Abstract: ** Given a vector bundle E over smooth variety X, there is a natural way to associate a vector bundle, called tautological bundle, on the Hilbert scheme of points on X. In this talk, we will discuss stability of tautological bundles in the case that X is a curve.

* Vera Vértesi (University of Strasbourg). ***Additivity of the minimal genus for tight contact structures.** Friday 14th June, 1:30-2:30pm. Huxley 144.

** Abstract: ** A corollary of Haken’s Lemma from 1968 gives that the minimal genus of Heegaard splittings of 3-manifolds is additive under connected sum. In this talk I generalise this result for contact 3-manifolds. As a consequence of Giroux’s 2002 paper about open book decompositions, contact 3-manifolds have contact Heegaard splittings i.e. contact 3-manifolds decompose as the union of two contact handlebodies. After the basics of contact geometry I describe the main idea of the proof, that the minimal genus for contact Heegaard splittings of tight contact structures is additive under connected sum.

*Hamid Ahmadinezhad (Loughborough University). ***Two decades on birational rigidity of del Pezzo fibration.** Friday 21st June, 1:30-2:30pm. Huxley 144.

** Abstract: ** Birational rigidity of Mori fibre spaces is a central problem in post MMP birational classification. In this talk, I will focus on 3-fold Mori fibre spaces with fibre dimension two, the so-called del Pezzo fibrations. I will discuss the importance of the problem and give an overview of the central results, with some recent progress.