**Calf Seminar**

About the Calf

About the Calf

Calf is the junior COW, an algebraic geometry seminar group primarily aimed at PhD students.

The organisers for academic year 2020/21 are Calla Tschanz (Bath), Geoffrey Mboya (Oxford) and Joe Prebble (Loughborough), and we have an extensive network of local organisers at different universities. Calf announcements are made using the COW mailing list. If you would like to get involved in the organisation, or suggest your institution as the next venue, please contact either of the people named above.

The COW seminar has some funding for travel expenses, and information on reimbursement can be found on the main COW webpage.

Due to the current situation with COVID-19, we are organising a series of online seminars via Zoom. The link to join will be in an email announcement made via the COW mailing list. If you do not have the link and wish to attend, please email one of the organisers.

**Upcoming Meeting:**

Imperial, Friday 15th October 2021, Huxley Building, rooms 642, 658, 6M42

Imperial, Friday 15th October 2021

13:00 - 18:00

Michael Hallam (Oxford): Stability of fibrations through geodesic analysis.

Abstract: A celebrated result in geometry is the Kobayashi--Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite--Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.

Alyosha Latyntsev (Oxford): BPS states, vertex algebras and torus localisation.

Abstract: To understand the ideas coming from string theory, mathematicians have noticed the importance of moduli spaces (of objects in a category). By this method the physics notions of 1. BPS states/D-branes, 2. a conformal field theory, 3. \Pi stability, ... have been turned into 1. cohomological Hall algebras, 2. vertex/chiral algebras, 3. Bridgeland stability conditions, ... , all of which are extremely rich mathematical objects.

In this talk, I will

a. explain what a cohomological Hall algebra and vertex algebra is (and why you should care),

b. sketch their physics analogues (no physics knowledge required!),

c. show how they are connected.

The main tool used for c. is a new version of the torus localisation which works for singular/derived spaces. Loosely, torus localisation methods "turn geometry into combinatorics", and are an important tool in Gromov Witten theory, toric geometry, and Donaldson Thomas theory. Before indicating how this works, I will

d. give a quick algebraic-geometric proof of torus localisation for smooth spaces (and explain how to use it).

Liam Stigant (Imperial): Minimal Models in Mixed characteristic.

Abstract: The Minimal Model Program (MMP) has been a hugely successful endeavour in characteristic 0, particularly in low dimensions where it is essentially complete. These ideas have been adapted to a wide range of situations and extensions of the original problem. Most notably for this talk is the recent work of Bhatt et al which establishes the bulk of the MMP program for a large class of three-dimensional schemes in mixed characteristic - specifically klt threefold pairs which are projective and surjective over an excellent normal ring, which admits a dualising complex and has residue fields of characteristic p > 5.

In this talk I will present a conceptual overview of the MMP and its aims, before focusing on the mixed characteristic setting and some of the specific difficulties that arise here. I will also discuss some of the ways in which the study of threefolds in mixed characteristic is easier than for varieties of the same dimension over a field, in particular giving sketches of some relatively simple proofs of termination and abundance for threefolds. These results are traditionally very difficult even in dimension 3 over a field, in fact they remain unknown in full generality for positive characteristic threefolds. The proof of abundance is part of some joint work with Fabio Bernasconi and Iacopo Brivio.

**Recent Past Meeting:**

Online Calf Seminar, Thursday 29th April 2021

Online Calf Seminar, Thursday 29th April 2021

**15:00**Fredrik Vaeng Rotnes (Imperial)

**Title**: Foliations and Birational Geometry

**Abstract**: In a single breath, a foliation is a partitioning of a manifold into submersed submanifolds all of the same codimension, called the leaves of the foliation.

In this talk we'll explore holomorphic foliations from a birational point of view. The main case is surface foliations, which are defined simply by holomorphic differential equations on the plane.

But instead of going about solving these equations to find particular solutions, our interest is in a qualitative understanding of the geometry of a foliation.

There'll hopefully be many examples to display the different geometric behaviours, like the geometry of the leaves or the holonomies around singular points.

In this way we can try to abstract out features shared among groups of foliations, for the purpose of a classification that in the surface case very much resembles the classical Kodaira-Enriques classification.

The goal towards the end will be to explain one major helpful tool in this direction: The development of a Minimal Model Program foliations.

Contact one of the organisers to get

**the recorded talk.**

calf_120221_kvg.pdf |

This page is maintained by Chris Seaman.