**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 2021/22 are, Erroxe Etxabarri-Alberdi (Loughborough), Oscar Finegan (Cardiff), and Samuel Johnston (Cambridge), as well as 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 any 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.

The Calf will maintain a hybrid approach to delivering the seminar, with talks delivered in person and with the option for those who cannot make the seminar in person to join 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:**

**Cambridge, Wednesday 8th December 2021,**CMS, Pavilion E, MR13

13:00-17:00

Federico Barbacovi (UCL): SemiOrthogonal Decompositions, gluing, and spherical twists

Abstract: Since their introduction, SemiOrthogonal Decompositions have played an important role in the study of derived categories. The reason is that they allow to “chop up” the category is smaller (generally easier) pieces. We will start by defining a SOD and working through some examples. Then, we will see how to glue two categories in such a way to get a SOD, and finally, if time permits, I will explain how this all fits into the picture of gluing spherical functors. To make things explicit, we will use the example of the category of modules over an algebra.

Felix Küng (University of Liverpool): Twisted Hodge diamonds give rise to non-Fourier-Mukai functors

Abstract: We apply computations of twisted Hodge diamonds to construct an infinite number of non-Fourier-Mukai functors. To do this we first recall the construction by Rizzardo, Van den Bergh, and Neeman of the passing through by a non-geometric deformation of a variety along a Hochschild cocycle of suitably large degree. We then use twisted Hodge diamonds to control the dimensions of the Hochschild cohomology of hypersurfaces in projective space and prove that there are a large number of Hochschild cohomology classes that allow this type of construction. In particular we can use these calculations to construct non-Fourier-Mukai functors for abitrary degree hypersurfaces in arbitrary high dimensions.

Xiong Yirui (University of Sheffield): What is a t-structure and how do we construct one

Abstract: Derived categories of (coherent sheaves of) algebraic varieties encode much useful information about the geometry of the varieties, such as the cohomology of sheaves, higher push-forwards and pull-backs. t-structure ("t" for truncation) is a powerful tool to study the internal structure of the derived category, which enables us to find interesting abelian categories therein. In my talk, I will first give a quick review of the definitions of derived categories and t-structures. Then I will explain our approach to construct new t-structures other than the standard one. Finally I will explain my work on constructing a family of such t-structures in the derived category of contangent bundle on a projective plane, based on the work of the Rudakov school.

**Recent Past 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.

This page is maintained by Chris Seaman.