Next Talk

October
15
3pm - 4pm

Tilting sheaves and the Baer-Bondal theorem

Daniel Bernal, The State University of Campinas
The study of derived categories of coherent sheaves has been studied strongly in the last few years. However, sometimes it is difficult to obtain good descriptions on it. One method to studying is via representations of quivers, as they behave nicely and its derived category is rather tame. In this talk we are going to mention some past results which indicate where the derived category of sheaves coincides with the derived category of representations of quivers; which essentially happens when we have a tilting sheaf, or a strong exceptional collection. We expect to prove it and to give a small breath of what is happening behind such equivalence.

Past Talks

October
1
3pm - 4pm

Derived Categories of Cubic Fourfolds

Bogdan Simeonov, LSGNT
The rationality problem of the cubic fourfold is a classical problem which remains unsolved (though there is recent work of Katzarkov-Kontsevich-Pantev-Yu that promises a solution). I will describe two approaches to this problem: one Hodge-theoretic (due to Hassett), and one using derived categories (due to Kuznetsov). Along the way, I will introduce Kuznetsov components, their properties and how they are related to K3 surfaces. If time permits, I will give an overview of a paper by Thomas and Addington, which shows that the two approaches are actually equivalent.
Notes

Past Talks (2023 - 2024)

June
27
3pm - 4pm

Flops of moduli of sheaves

Abhishek Khannur, LSGNT
A stability condition is necessary for constructing well behaved moduli spaces of (complexes of) sheaves on a projective variety. I'll give a brief introduction to stability conditions and mention how varying this stability condition changes the moduli space as we cross a wall. In particular, I will describe, hopefully with a concrete example, a local model for flops of moduli of Bridgeland semistable (complexes of) sheaves on a K3 surface in terms of GIT or King stability for quiver varieties. If time permits, I'll say a thing or two about wall crossing formulas for the derived category of coherent sheaves of the moduli spaces.
June
11
3pm - 4pm

DAnG it! An Introduction to Derived Analytic Geometry

Rhiannon Savage, University of Oxford
Derived algebraic geometry (DAG) can be viewed as homotopical algebraic geometry relative, in the sense of Toën and Vezzosi, to the category of simplicial commutative rings. In this talk we will explore a new model for derived analytic geometry (DAnG), proposed by Ben-Bassat, Kelly, and Kremnizer, as homotopical algebraic geometry relative to the category of simplicial commutative complete bornological rings. I will also briefly mention some work I have done looking at representability of derived analytic stacks in this framework. This will mainly be an ideas/vibes talk and no knowledge of analytic geometry or infinity categories will be assumed!
May
28
3pm - 4pm

Some things I know about derived intersections

Oscar Finegan, Cardiff University
Intersection theory (in the sense of Fulton) sets up intersections of objects considered only up to rational equivalence. This leads to nice numerics in intersections according to a principle of continuity. However, it's somewhat weird and unintuitive. A self-intersection of a line in the plane is clearly distinguished from the transverse intersection of two lines. Derived Algebraic Geometry produces a more rigid framework in which these degenerate cases are interesting to study. I'll talk about this a little bit, then actually talk about the homological methods I've used over the course of my doctorate (maybe even including a proof of a result 😲).
Notes
Recording
May
14
3pm - 4pm

Categorical Torelli theorems for Fano threefolds

Augustinas Jacovskis, University of Luxembourg
Let \(X\) be a smooth Fano threefold. Due to a theorem of Bondal-Orlov from the 90s, \(X\) is determined up to isomorphism by its derived category \(D^b(X)\). Since \(X\) is Fano, \(D^b(X)\) admits a semiorthogonal decomposition into a number of exceptional objects, and the orthogonal complement to them. It is a natural question to ask whether \(X\) is determined up to isomorphism by less data than the whole of \(D^b(X)\). A natural candidate for "less data" is the aforementioned orthogonal complement to the exceptional objects. This subcategory of \(D^b(X)\) is known as the Kuznetsov component \(Ku(X)\), and when \(Ku(X)\) determines \(X\) up to isomorphism a categorical Torelli theorem is said to hold.

In my talk I will overview two methods (Bridgeland moduli-theoretic and K-theoretic) of proving categorical Torelli theorems for certain classes of prime Fano threefolds. I will in particular discuss joint work with Xun Lin, Zhiyu Liu, and Shizhuo Zhang; and Hannah Dell and Franco Rota.
Notes
April
30
3pm - 4pm

Preprojective algebras via Serre functors and HRS-tilting

Jon Cheah, University of Hong Kong
The preprojective algebra of a quiver is the quotient of the path algebra of the doubled quiver by certain commutator relations. This brief description hides many questions as to why such a construction is natural/useful: Why these particular commutators? What does preprojective mean? I will give a gentle introduction to preprojective algebras and use examples to demonstrate the classical definitions. Using results which relate the Coxeter functors and AR-translate to Serre functors on the derived category, I will show how the defining relations arise naturally from an explicit computation of the Serre functor. This derived construction gives a dg-algebra whose zeroth cohomology gives the classical preprojective algebra. If time permits, I will discuss the Serre functor as a composition of derived reflection functors from the point of view of HRS-tilting.
Notes
March
19
3pm - 4pm

Bayer-Macrì nef divisor on Bridgeland moduli spaces: Q&A

Uttaran Dutta, University of South Carolina
Last week I talked about Bayer-Macrì nef divisor, which is a general way of showing projectivity of a Bridgeland Moduli space. But since there is no GIT construction available for Bridgeland Moduli spaces, existence of universal family is not at all easy to prove. There are other techniques such as using F-M transform and the "Gieseker wall", using exceptional collections and their "quiver region", etc. I can discuss these things if people are interested.
March
12
3pm - 4pm

Bayer-Macrì nef divisor on Bridgeland moduli spaces

Uttaran Dutta, University of South Carolina
There is a notion of flat families of Bridgeland-semistable objects parametrized by an algebraic space, and correspondingly, there is a moduli stack parametrizing flat families of Bridgeland-semistable objects of a fixed character. There are many open questions about the geometry of these moduli spaces. One of the major questions is: when is there a projective coarse moduli space parameterizing S-equivalence classes of Bridgeland-semistable objects of a fixed character?

Bayer and Macrì defined a nef divisor on such moduli spaces and gave a description of the contracting curves. This led to a general way of showing the projectivity of these moduli spaces. The construction of the nef divisor is related to the classical determinant line bundle construction on moduli of sheaves. I shall start by giving an overview of both the theories and how they are related. In the end, if time permits, I shall add a few comments on the ampleness of the Bayer-Macrì divisor on K3 surfaces.
Notes
March
5
3pm - 4pm

Toric GIT quotients and some homological mirror symmetry: Q&A

Michela Barbieri, University College London
A Q&A session following on from the previous talk!
February
27
3pm - 4pm

Toric GIT quotients and some homological mirror symmetry

Michela Barbieri, University College London
Anything about mirror symmetry refers to some mysterious relationships between complex and symplectic geometry. Roughly speaking, homological mirror symmetry says that if you have some complex geometry X, there is a mirror symplectic geometry Y such that D^b(Coh X) is equivalent to the Fukaya category of Y, denoted Fuk(Y). In fact, starting from a complex geometry (think an algebraic variety) there isn't just one mirror. There's a family of mirrors living over a parameter space, which is sometimes referred to as the Stringy Kähler Moduli Space (SKMS). The fundamental group of the SKMS acts naturally on Fuk(Y) via monodromy, and by mirror symmetry, we expect to see this action carry over. The way it carries over is believed to be through spherical functors. My goal will be to explain some details of this story in the context of Calabi-Yau toric geometric invariant theory. I'll start off explaining some things about GIT, mainly focusing on examples, and see where we get to!
Notes
February
20
3pm - 4pm

Derived autoequivalences on algebraic flops: Q&A

Caroline Namanya, Makerere University and University of Glasgow
A Q&A session following on from the previous talk!
February
13
3pm - 4pm

Derived autoequivalences on algebraic flops

Caroline Namanya, Makerere University and University of Glasgow
In this talk, i will give a construction of derived autoequivalences associated to an algebraic flopping contraction \(X\to X_{\text{con}}\) where \(X\) is quasi-projective with only mild singularities. These functors are constructed naturally using bimodule cones, and are locally two-sided tilting complexes by using the local-global properties.
Notes
January
30
3pm - 4pm

Higher coherences/infinity categories pop up naturally in Algebraic geometry

Omer Bojan, Tel Aviv University
In this talk I hope to convince you these are intuitive objects using repetitive topological slogans, examples, and deceitful charm. Plan:
  1. Infinity sheaves (Stacks, moduli spaces...)
  2. Stable categories
  3. Coherent group actions, and more
January
23
3pm - 4pm

Fourier-Mukai transforms: Q&A

Luke Naylor, University of Edinburgh
A Q&A session following on from the previous talk!
Notes
January
16
3pm - 4pm

Fourier-Mukai transforms

Luke Naylor, University of Edinburgh
Shigeru Mukai first introduced a "Fourier functor" in the 80s, now known as the Fourier-Mukai transform, which has since been used extensively for derived equivalences in algebraic geometry. In this talk, I will cover some material around the following topics:
  • Where does the "Fourier" come from?
  • Some homological algebra around the definition and early propositions
  • Some roles it plays in algebraic geometry
Notes
November
28
3pm - 4pm

Demystifying Bridgeland stability conditions: Q&A

Hannah Dell, University of Edinburgh
A Q&A session following on from the previous talk!
Notes from Q&A
November
21
3pm - 4pm

Demystifying Bridgeland stability conditions

Hannah Dell, University of Edinburgh
Bridgeland stability conditions have proved to be a useful tool in algebraic geometry and beyond. However, if one looks at the definition, the connection to geometry can seem quite mysterious. In this talk, I will introduce them and present some examples and applications.
Notes
November
7
3pm - 4pm

…a big fan of your heart

Parth Shimpi, University of Glasgow
Just like dogs, the derived categories of finite dimensional algebras have hearts. Unlike dogs, a single category can have many hearts. Fortunately the zoo of hearts is not completely unclassifiable— in this talk I will describe various ways to classify hearts, first homologically by detecting various shadows in the module category itself, and then combinatorially much like how fans classify toric varieties.
Notes
October
31
3pm - 4pm

Fourier-Mukai transforms on Jacobians and Pryms: Q&A

Robert Hanson, Instituto Superior Técnico
A Q&A session following on from the previous talk!
October
24
3pm - 4pm

Fourier-Mukai transforms on Jacobians and Pryms

Robert Hanson, Instituto Superior Técnico
Shigeru Mukai has constructed certain derived equivalences between an abelian variety \(A\) and its dual \(A^{\vee}\). These equivalences are called Fourier-Mukai transforms, named so for their similarity to the classical Fourier transform on \(L^2\) spaces. Given a smooth curve \(C\), one gets a derived autoequivalence on the Jacobian \(A = A^{\vee} = Jac(C)\), and given a finite morphism of smooth curves \(\beta : C \to X\), one gets a derived equivalence between the Prym variety \(A = Prym(\beta)\) and the GIT quotient \(A^{\vee} = Prym(\beta) / \Gamma\), where \(\Gamma\) is a group of torsion line bundles on \(X\). Motivated by mirror symmetry and geometric Langlands, this talk explores extensions of these derived equivalences to cases where \(C\) is singular and we compactify the Jacobians and Pryms. This is based on joint work with Emilio Franco and Joao Ruano in our preprint arXiv 2206.12349.
Notes
October
17
3pm - 4pm

A tour of the McKay correspondence: Q&A

Austin Hubbard, University of Bath
A Q&A session following on from the previous talk!
October
10
3pm - 4pm

A tour of the McKay correspondence

Austin Hubbard, University of Bath
The classical McKay correspondence is a great example of a deep geometric connection realised as an equivalence of derived categories. We will discuss the classical case and various generalisations.
Notes
October
03
3pm - 4pm

Derived Categories of Nodal Varieties: Q&A

Aporva Varshney, UCL
A Q&A session following on from the previous talk!
September
26
3pm - 4pm

Derived Categories of Nodal Varieties

Aporva Varshney, UCL
It can be daunting to leave the warm, cosy duvet of smooth varieties, no matter how much the minimal model program may tempt us. In this talk, we will take small steps by looking at nodal varieties using recent results of Kuznetsov-Shinder. We will see how, at the level of the derived category, nodal singularities can sometimes be "absorbed" by producing a semiorthogonal decomposition into a "singular" component and a "smooth" component. I will then sketch a proof of a very cool result which shows that for curves and threefolds admitting this kind of absorption, the singular component in the decomposition vanishes when smoothing the variety.
Notes
Website made by Aporva Varshney.