We will be hosting a Winter School for students and early career researchers at Lancaster University. The event will start at 1pm on Monday 16th December and finish at 1pm on Friday 20th December. 

The school will consist of the following 4 lecture courses:

• Paul Hacking (University of Massachusetts, Amherst) - Calabi-Yau manifolds and mirror symmetry

• Enrica Mazzon (Universität Regensburg) - Introduction to non-Archimedean geometry

• Franco Rota (University of Glasgow) - Surface-like pseudolattices, del Pezzo surfaces, and genus one fibrations 

• Nikolaos Tsakanikas (École Polytechnique Fédérale de Lausanne) - The termination of flips conjecture in birational geometry

Schedule

All talks will take place in Lecture Theatre 1 in the Fylde Building (FYL). A PDF campus map is below, with an interactive map below:

Campus map (PDF) 

Interactive map

Monday 16th December
13:00 - Arrival
14:00 - Tsakanikas 1
15:00 - Tea/coffee break and accommodation key collection
15:30 - Hacking 1

Tuesday 17th December
09:30 - Rota 1
10:30 - Tea/coffee break
11:00 - Mazzon 1
12:00 - Lunch
13:00 - Office hours / discussion session
14:00 - Break
14:30 - Hacking 2
15:30 - Tea/coffee break
16:00 - Tsakanikas 2

Wednesday 18th December
09:30 - Mazzon 2
10:30 - Tea/coffee break
11:00 - Rota 2
12:00 - Lunch
13:00 - Office hours / discussion session
14:00 - Break
14:30 - Tsakanikas 3
15:30 - Tea/coffee break
16:00 - Hacking 3

Thursday 19th December
09:30 - Rota 3
10:30 - Tea/coffee break
11:00 - Mazzon 3
12:00 - Lunch
13:00 - Office hours / discussion session
14:00 - Break
14:30 - Hacking 4
15:30 - Tea/coffee break
16:00 - Tsakanikas 4

Friday 20th December
09:30 - Mazzon 4
10:30 - Tea/coffee break
11:00 - Rota 4
12:00 - Office hours / discussion session
13:00 - Departure

Titles and abstracts

• Paul Hacking (University of Massachusetts, Amherst) - Calabi-Yau manifolds and mirror symmetry

A Kahler manifold is a smooth manifold that is both complex and symplectic in a compatible way, for example a smooth complex projective variety.  A Calabi--Yau manifold is a Kahler manifold with trivial canonical bundle. The mirror symmetry phenomenon posits that Calabi--Yau manifolds come in pairs X and Y such that the complex geometry of X is equivalent to the symplectic geometry of Y and vice versa.  More precisely, the homological mirror symmetry (HMS) conjecture of Kontsevich asserts that the derived category of coherent sheaves on X is equivalent to the Fukaya category of the symplectic manifold Y. The objects of the derived category are generated by holomorphic vector bundles and morphisms are determined by sheaf cohomology. The objects of the Fukaya category are generated by Lagrangian submanifolds and the morphisms are given by Floer cohomology. 

Work of Seidel, Sheridan, and Ganatra-Pardon-Shende provides an approach to proving the HMS conjecture. The main theorem of GPS is roughly a Mayer-Vietoris type result for Fukaya categories of affine varieities (or Stein manifolds). The work of Seidel and Sheridan explains how to use this to compute Fukaya categories of projective varieties via deformation theory. I will give an overview of this approach based on the simplest examples. One goal is to sketch the recent proof of HMS for K3 surfaces by Ailsa Keating and myself. I will attempt to keep prerequisites to a minimum, in particular, I will not assume prior knowledge of the Fukaya category or Floer cohomology. 

Some references:

1. Notes from a graduate course on mirror symmetry, by Denis Auroux (2009), here 

(the most relevant part being the second half, starting from "Intro to HMS")

2. A beginner's introduction to Fukaya categories, by Denis Auroux. here 

3. Introduction to the work of Ganatra-Pardon-Shende, notes from a lecture by Nick Sheridan (2022). here

Examples:

4. Categorical Mirror Symmetry: The Elliptic Curve, by A. Polishchuk and E. Zaslow (1998), here

5. Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves (2005), by D. Auroux, L. Katzarkov, D. Orlov, here

6. Homological mirror symmetry for the quartic surface, by Paul Seidel (2013), here

7. Homological mirror symmetry for log Calabi--Yau surfaces, by P. Hacking and A. Keating (2020), here

WARNING: While these examples may be instructive, the hope is that future calculations will be easier and more conceptual (not relying on matching generators on both sides), using GPS and deformation theory.

• Enrica Mazzon (Universität Regensburg) - Introduction to non-Archimedean geometry

This mini-course provides an introduction to the fundamentals of non-archimedean geometry. Starting with the definition and basic properties of non-archimedean fields, we will explore the theory of Berkovich spaces, emphasizing their construction and key features. The course will conclude with some applications of non-archimedean geometry in algebraic geometry, focusing on the study of degenerations of varieties.   

• Franco Rota (University of Glasgow) - Surface-like pseudolattices, del Pezzo surfaces, and genus one fibrations 

A pseudolattice abstracts the properties of the (numerical) Grothendieck group of a triangulated category. When the category is the derived category of a smooth algebraic surface, the associated pseudolattice reflects much of the geometry involved.

As a start, we will review some foundational results of surface theory and translate them in the pseudolattice language. Then I will survey the theory of pseudolattices and some of their applications, especially to obstructing the existence of full exceptional collections.

Keeping with the principle of translating from categorical to numerical language, we will introduce spherical homomorphisms of pseudolattices, which abstract spherical functors, as introduced by A. Harder and A. Thompson. In the same paper, the authors point to a – different, but related – source of surface-like pseudolattices: the Fukaya-Seidel category of genus 1 Lefschetz fibrations. The two sources of pseudolattices are related by mirror symmetry: time permitting we will illustrate this in detail in the case of a low degree Del Pezzo surface.

• Nikolaos Tsakanikas (École Polytechnique Fédérale de Lausanne) - The termination of flips conjecture in birational geometry

The main objective of this mini course is to provide a gentle introduction to the Minimal Model Program (MMP), which plays a central role in the birational classification theory of higher-dimensional algebraic varieties. I will first discuss the 1- and 2-dimensional cases, which serve as a motivation for the general picture, and I will then present the general form of the MMP in arbitrary dimension, together with its fundamental predictions. Moreover, I will recall the basic notions of positivity of divisors (e.g., nef and big) and I will also define the standard classes of singularities of varieties or pairs (e.g., klt and log canonical).

Afterwards, the mini course will focus on one of the main open problems of the MMP in dimension at least four, the so-called termination of flips conjecture. I will review the known cases of this conjecture, presenting sometimes the main ideas or even the strategy of the proofs. If time permits, I will also discuss in some detail the termination of flops for irreducible holomorphic symplectic and Enriques manifolds.

Accommodation and Travel

We can provide accommodation for participants and cover reasonable travel costs for students and postdocs based in the UK. Regrettably, due to limited funding we cannot support other participants' travel.

Please indicate on the registration form if you would like to apply for accommodation and/or travel funding, and include an estimate of how much travel funding you expect to require. Should we receive more applications than we have funding to cover, accommodation and travel funding will be allocated on a first-come, first-served basis.

To allow us to fund as many participants as possible, we ask that participants purchase "advance" or "off-peak" train tickets where practical. For those under the age of 30, we also recommend looking into getting a railcard, which can offer substantial savings on the cost of train travel around the UK.

Registration

Registration for this event has now closed.

Funding

This event is taking place as part of the UK Algebraic Geometry Network. The network is funded by a Network Support for the Mathematical Sciences grant from the Isaac Newton Institute, via EPSRC grant EP/V521929/1. This event is also partially funded by the EPSRC Programme Grant Enhancing Representation Theory, Noncommutative Algebra and Geometry and the Heilbronn Institute for Mathematical Research.