Derived
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Graduate
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Sketching some stuff about:

• How atomic SODs relate to Hodge atoms from [KKPY].
• Group actions on SODs and rationality over non-closed fields.

Atomic SODs, (bi)rationality, and the MMP.

I will give an overview of some results and conjectures on semiorthogonal decompositions for the derived categories of some interesting spaces arising from an intersection of two quadrics. I'll explain how we can use variation of GIT to compute the "main" component of these semiorthogonal decompositions.

Given a singular variety Y and a resolution of singularities f:X→Y, it is interesting to understand how this resolution behaves at the level of derived categories. Bondal and Orlov conjectured that, if Y has only rational singularities, then the induced functor between derived categories should be a localization. This conjecture has been proven in several cases using different approaches. In this talk, we will go from t-structure arguments to the more abstract and technical approach of Efimov (arXiv:1308.0135), in order to provide an overview of the problem.

The McKay correspondence establishes a derived equivalence between the classical minimal resolution and a noncommutative crepant resolution of a Kleinian surface singularity. Based on joint work with Ruth Wye, I will explain how the McKay correspondence extends to a larger class of noncommutative crepant resolutions of the singularity, and how their Hilbert schemes of points are related through variation of GIT quotients (VGIT). Time permitting, I will also sketch some recent ideas from work in progress with Austin Hubbard on how to relate the resolutions themselves via VGIT by taking into account the variation of monoidal structures on their mutual derived category.

This talk will be to some degree complementary to the one given by the speaker in the DANCE seminar on May 11th. You can find the recording and notes of that talk on the DANCE website.

The study of Bridgeland stability conditions has been an exciting area of research in the past two decades. A celebrated result of Bridgeland ensures that the space of stability conditions forms a complex manifold. This space is, in general, non-compact, and many possible notions of "degenerate" stability conditions have been worked out throughout the years.

We will give a gentle introduction to this framework by studying the stability manifold of the derived category of coherent systems, building upon the seminal work of Feyzbakhsh and Novik. Given a general genus four curve, we will prove the existence of a degeneration to a weak stability condition. We use this to construct stability conditions in the Kuznetsov component of a nodal cubic threefold, relying on a description by Alexeev and Kuznetsov. Finally, we will discuss various future directions and applications.

Homological projective duality is a homological extension of classical projective duality, introduced by Alexander Kuznetsov. When algebraic varieties X and Y in dual projective spaces are homologically projectively dual, then their hyperplane sections admit semiorthogonal decompositions with equivalent non-trivial components. The main statement will be explained but without proof.

Then, we discuss classical- and homological projective duality for Hirzebruch surfaces. Finally, we discuss the duality between Gr(2, 6) and a non-commutative resolution of Pf(4, 6), and compare the derived categories of their dual hyperplane sections. In particular, we see that the non-trivial component in the semiorthogonal decomposition for a Pfaffian cubic fourfold is equivalent to the derived category of a K3 surface.

In many occasions in algebraic geometry, one can relate the G-equivariant geometry of ℂⁿ to the geometry of the minimal resolution of ℂⁿ/G. I will discuss the case of surface cyclic quotient singularities, where G is a finite cyclic subgroup of GL(2,ℂ). It was proved by Ishii and Ueda arXiv:1104.2381 (categorifying earlier work of Wunram) that the derived category of a surface 𝓧 with cyclic quotient singularities admits a semi orthogonal decomposition with two components: one being a copy of the derived category of the minimal resolution Y of the coarse space X of 𝓧, and the other being a residual component indexed by what are called non-special representations. I will explain this result and, if time permits, describe an interpretation of it using mirror symmetry (as in my paper arXiv:2602.04866).

Phantom categories, or admissible subcategories on which all additive invariants vanish, were once considered pathological phenomena unlikely to occur on simple enough varieties. So when Krah constructed a phantom on a blowup of ℙ², it came as a surprise and disproved several conjectures! Since then, there has been work on extending this construction to other rational surfaces. In this talk, I will explain these constructions and how they are proved using Kuznetsov's machinery of heights. I will also describe some tools used to study these phantoms, like Hochschild cohomology and a spectral sequence for computing Hom's between objects.

Brill-Noether theory is a way to understand the behaviour of smooth curves: it initially studied line bundles and their sections, but after works of Mukai, Lazarsfeld and others, (semi)stable bundles are also of interest. After Bridgeland introduced stability conditions, Bayer showcased a use for curves on K3 surfaces, by demonstrating the applications of wall-crossing. In this talk, I will talk about Lazarsfeld-Mukai bundles, give a brief introduction to tilt stability on K3 surfaces, and discuss works of Bayer and Feyzbakhsh.

Since the introduction of Bridgeland stability conditions, a very interesting problem has been that of constructing (proper) moduli spaces of semistable objects. In this talk, I will explain why the classical GIT approach fails in this case and how the machinery of good moduli spaces and intrinsic GIT can be employed to overcome the issue, drawing connections between the two approaches when possible.

I will survey the current state of research, namely that of stability conditions over noncommutative varieties. Lastly, I will outline ongoing work on how the properties necessary for the existence of good moduli spaces are preserved within connected components of the stability manifold.

A celebrated result of Orlov equates the derived category of a Calabi-Yau hypersurface (or the Kuznetsov component of a Fano hypersurface) with a category of "graded matrix factorizations" for the defining equation. Physically, this was predicted by Witten from viewing both sides as different limit points of a physical theory (a GLSM), which mathematically manifests as a variation of GIT quotients (a flop). In this talk I will introduce graded matrix factorizations, and explain the main ideas of the proof, shamelessly nabbed from my advisor's paper on it.

The construction of Bridgeland stability conditions on explicit triangulated categories remains an intriguing and challenging problem. So far, most known examples occur on varieties of dimension less than four. In this talk, after a quick review of the Bayer–Macrì–Toda construction on threefolds, we will focus on Toda’s series of works on Gepner-type stability conditions, motivated by graded matrix factorizations and mirror symmetry. We will also discuss some recent progress toward the construction of Gepner-type stability conditions on the quintic threefold.

Bridgeland stability conditions have gain popularity recently in algebraic geometry as an important tool in the study of smooth projective varieties and their moduli spaces. In this talk, we give an introduction to this important construction, show how we can build such stabilities for Fano 3-folds and computes some stable objects for Picard rank 1. We also explain the difficulties for going beyond dimension 3 and what has been done recently to try to achieve this goal.

Bridgeland stability is notoriously hard to construct on varieties of dimension gt; 2. One promising workaround is to focus on the Kuznetsov component of Fano varieties - a residual piece of the derived category that often captures the essential geometric information. In this talk, I will explain the construction of stability conditions on Kuznetsov components using the machinery of quadric fibrations introduced by Bayer--Lahoz--Macrì--Stellari (arXiv:1703.10839). I will illustrate how this technique applies to cubic 3-folds and 4-folds, and poke at the curious cubic 5-fold creatures I have been exploring recently. If time permits, I will discuss some applications to moduli spaces.

We introduce the moduli spaces of representations of quivers. Following the resemblance to moduli spaces of vector bundles on algebraic curves, we propose a probing method for their derived categories: by embedding several copies of the derived category of the quiver itself, we obtain partial semiorthogonal decompositions and, in some cases, full strongly exceptional collections.

This work is supported by the Luxembourgish National Research Fund (AFR-17953441)

In this seminar I will review some conjectures by Halper-Leistner on the non-commutative MMP. One of the central aspects of this program is to show how quantum cohomology induces, via Bridgeland stability conditions, a sort of “canonical” semi-orthogonal decomposition of the derived category. Halpern-Leistner studied the NCMMP for curves. Time permitting I will show how quantum cohomology of projective spaces P^n induces semiorthogonal decompositions.

A smooth projective variety is said to be a Fano visitor if its bounded derived category D^b(X) can be embedded into the derived category of a smooth Fano manifold. In 2011, Bondal conjectured that every smooth projective variety is a Fano visitor. In this talk, I will provide a proof of this conjecture for 75% of K3 surfaces, using the work of Bayer and Macrì, which describes the birational geometry of moduli spaces of sheaves on K3 surfaces through Bridgeland stability conditions, and the study of fixed loci of antisymplectic involutions on hyperkähler manifolds by Saccà, Macrì, O'Grady, and Flapan.

The minimal resolution (S) of an ADE singularity is known to admit a tilting bundle, giving an equivalence of categories between the bounded derived category of coherent sheaves on S, and modules over a certain algebra, based on the path algebra of the McKay quiver. In this talk, I will explain how we can amend this construction for partial resolutions, giving a 'cornered' quiver and algebra, and how we put together this proof of a partial McKay correspondence.

We will present several semiorthogonal decompositions and identities in the Grothendieck ring of categories arising from the geometry of cubic hypersurfaces and complete intersections of two quadrics, some of which will be conjectural. Along the way, we will present some interesting birationality constructions, and time permitting, some work in progress in proving some of these decompositions/identities. 

We compute several types of dimension for the bounded derived categories of coherent sheaves of orbifold curves. This completes the calculation of these dimensions for derived categories of noncommutative curves in the sense of Reiten-van den Bergh. Along the way we construct stability conditions for orbifold curves. We also obtain a characterisation of orbifold curves with hereditary tilting bundle in terms of diagonal dimension.

A conjecture of Orlov predicts that for two derived equivalent varieties $X$ and $Y$ over the complex numbers, there is an isomorphism between their cohomology that preserves the grading and Hodge structure. A few years ago, Taelman showed that the conjecture holds for hyperkähler varieties. In this talk, I will show how this generalizes to more general varieties carrying a symplectic form, under some assumptions. The main technical tool is the so-called LLV algebra; I will introduce it and explain its relevance to the problem.

The derived autoequivalence group of a variety has really interesting applications due to its connections to mirror symmetry and stability conditions. For a smooth projective variety X with an ample or anti-ample canonical bundle, Bondal and Orlov showed that the variety only admits certain "trivial autoequivalences". When the canonical bundle is trivial, however, then the autoequivalence group is much richer. As a relative analog of this fact, birational morphisms \(X \to Y\) with trivial relative canonical bundle, known as a crepant contractions, could be expected to induce new derived symmetries. In this talk, I will construct autoequivalences induced by these contractions and characterise them as twists around spherical objects and spherical functors. This talk will mostly be examples based. We will consider some examples in which these autoequivalences exist and some examples in which they don't. We will hopefully extract some "hidden smoothness" and "spherical" criteria which allow us to construct these symmetries.

Matrix factorizations give a deformation of the derived category of coherent sheaves on a variety, originally found in the study of hypersurface singularities, but more recently cropping up in homological mirror symmetry with superpotentials. Sticking to algebraic geometry in relation to derived categories, I will introduce and try to give some geometric intuition for matrix factorizations. A crucial result allowing us to compare these categories is Knörrer periodicity, which I will give some motivation for and describe an application to singular varieties.

The Coherent-Constructible Correspondence provides a categorical equivalence between the complexes of coherent sheaves on toric varieties and complexes of certain constructible sheaves on the mirror torus. In this talk, I will explain how to use this equivalence together with the stratification of the mirror torus introduced by Bondal to generalize Hilbert's Syzygy Theorem to any smooth, projective toric variety. This talk is based on my joint work with David Favero (https://arxiv.org/pdf/2411.17873).

The Derived Torelli Theorem by Mukai ('87) and Orlov ('96) connects the derived categories, Hodge theory, and geometry of K3 surfaces. Each of these three aspects is important in their own right, which makes the Derived Torelli Theorem a rich source for the study of the geometric implications of derived equivalence. In this talk, I will present some of the main ideas of Mukai and Orlov, and broadly explain how to prove the Derived Torelli Theorem. This means diving into the theory of moduli spaces of sheaves on K3 surfaces, Brauer groups, and of course Hodge theory. I will also give some explicit examples of derived equivalent elliptic K3 surfaces that will show all three aspects in action.

Working with triangulated categories in general, it is important to determine whether a given exact functor is fully faithful or not. In algebraic geometry, one often encounters exact functors of Fourier-Mukai type relating derived categories of smooth projective varieties. In my talk, I will focus on a typical geometric situation and present a criterion allowing to verify fully faithfulness of certain Fourier-Mukai transforms (built from projections and closed embeddings). The goal is to explain the role of normal bundles in such computations and to provide as many examples as possible.

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.

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.

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.

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!

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 😲).

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.

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.

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.

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.

A Q&A session following on from the previous talk!

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!

A Q&A session following on from the previous talk!

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.

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

A Q&A session following on from the previous talk!

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

A Q&A session following on from the previous talk!

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.

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.

A Q&A session following on from the previous talk!

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.

A Q&A session following on from the previous talk!

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.

A Q&A session following on from the previous talk!

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.