I am an algebraic geometer who uses tools from derived categories to solve problems of interest to algebraists, geometers, and number theorists. Some of my recent work studies the derived geometry of birational maps, investigates the ties between rational points and rational parameterizations and the structure of the derived category, and develops derived approaches to noncommutative algebraic geometry.

My research has received support from the Simons Foundation and National Science Foundation. It has also benefited from a membership at the Institute for Advanced Study. My ORCID number is orcid.org/0000-0001-5819-0159. See also my Google Scholar profile.

For a more information on me, see my CV.

Papers

Titles and abstracts for my works can be found below. All the completed works are available on the ArXiv. However, please note that there may be differences between the final published version, the ArXiv version, and the current version of a paper. Copies of any article listed as in preparation are available by request.

We study smooth Fano toric varieties whose split forms have centrally-symmetric fans. For such varieties, we show that the existence of a full étale exceptional collection is equivalent to rationality of the variety.

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a slight generalization of this conjecture, where the endomorphism algebras of the exceptional objects are allowed to be separable field extensions of the base field. We show this generalization is false by exhibiting a smooth, projective threefold over the the field of rational numbers that possesses a full etale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full etale-exceptional collection by a torsor that is invisible to Brauer invariants. Along the way, we develop some tools for linearizing objects, including a group that controls linearizations.

Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories.

Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.

To any affine scheme with a G_m-action, we provide a Bousfield colocalization on the equivariant derived category of modules by constructing, via homotopical methods, an idempotent integral kernel. This endows the equivariant derived category with a canonical semi-orthogonal decomposition. As a special case, we demonstrate that grade-restriction windows appear as a consequence of this construction, giving a new proof of wall-crossing equivalences which works over an arbitrary base. The construction globalizes to yield interesting integral transforms associated to D-flips.

We develop a generalization of the Q-construction of the first author, Diemer, and the third author for Grassmann flops over an arbitrary field of characteristic zero. This generalization provides a canonical idempotent kernel on the derived category of the associated global quotient stack. This idempotent kernel, after restriction, induces a derived equivalence over any twisted form of a Grassmann flop. Furthermore its image, after restriction to the geometric invariant theory semistable locus, "opens" a canonical "window" in the derived category of the quotient stack. We check this window coincides with the set of representations used by Kapranov to form a full exceptional collection on Grassmannians. Even in the well-studied special case of standard Atiyah flops, the arguments yield a new proof of the derived equivalence.

We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric.

We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along lines of Rickard and Orlov.

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type A. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

We combine the Bondal-Uehara method for producing exceptional collections on toric varieties with a result of the first author and Favero to expand the set of varieties satisfying Orlov's Conjecture on derived dimension.

We study the relationship between derived categories of factorizations on gauged Landau-Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and describe the complementary components. We also verify a question posed by Kawamata: we show that D-equivalence and K-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne-Mumford stacks. This recovers Kawamata's theorem that all projective toric Deligne-Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover Orlov's σ-model/Landau-Ginzburg model correspondence.

We provide descriptions of the derived categories of degree d hypersurface fibrations which generalize a result of Kuznetsov for quadric fibrations and give a relative version of a well-known theorem of Orlov. Using a local generator and Morita theory, we re-interpret the resulting matrix factorization category as a derived-equivalent sheaf of dg-algebras on the base. Then, applying homological perturbation methods, we obtain a sheaf of A∞-algebras which gives a new description of homological projective duals for (relative) d-Veronese embeddings, recovering the sheaf of Clifford algebras obtained by Kuznetsov in the case when d=2.

We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.

We provide a geometric approach to constructing Lefschetz collections and Landau-Ginzburg Homological Projective Duals from a variation of Geometric Invariant Theory quotients. This approach yields homological projective duals for Veronese embeddings in the setting of Landau Ginzburg models. Our results also extend to a relative Homological Projective Duality framework.

Generalizing Eisenbud's matrix factorizations, we define factorization categories. Following work of Positselski, we define their associated derived categories. We construct specific resolutions of factorizations built from a choice of resolutions of their components. We use these resolutions to lift fully-faithfulness statements from derived categories of Abelian categories to derived categories of factorizations and to construct a spectral sequence computing the morphism spaces in the derived categories of factorizations from Ext-groups of their components in the underlying Abelian category.

In the case of toric varieties, we continue the pursuit of Kontsevich's fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating semi-orthogonal decompositions of the B-model on toric varieties to semi-orthogonal decompositions on the A-model on the mirror Landau-Ginzburg models. As evidence, we prove a new case of Homological Mirror Symmetry for a toric surface whose anticanonical bundle is not nef, namely a certain blow-up of P^2 at three infinitesimally near points.

We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we provide a conjectural geometric framework to further understand M. Kontsevich's Homological Mirror Symmetry conjecture. We obtain new cases of a conjecture of Orlov concerning the Rouquier dimension of the bounded derived category of coherent sheaves on a smooth variety. Further, we introduce actions of A-graded commutative rings on triangulated categories and their associated Noether-Lefschetz spectra as a new invariant of triangulated categories. They are intended to encode information about algebraic classes in the cohomology of an algebraic variety. We provide some examples to motivate the connection.

We provide a factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we call the extended Hochschild cohomology algebra of the category of equivariant factorizations. In some cases of geometric interest, extended Hochschild cohomology contains Hochschild cohomology as a subalgebra and Hochschild homology as a homogeneous component. We use our factorization model for the internal Hom to calculate the extended Hochschild cohomology for equivariant factorizations on affine space.

Combining the computation of extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and extends Griffiths' classical description of the primitive cohomology of a smooth, complex projective hypersurface in terms of homogeneous pieces of the Jacobian algebra. In the process, the primitive cohomology is identified with the fixed subspace of the cohomological endomorphism associated to an interesting endofunctor of the bounded derived category of coherent sheaves on the hypersurface. We also demonstrate how to understand the whole Jacobian algebra as morphisms between kernels of endofunctors of the derived category.

Finally, we present a bootstrap method for producing algebraic cycles in categories of equivariant factorizations. As proof of concept, we show how this reproves the Hodge conjecture for all self-products of a particular K3 surface closely related to the Fermat cubic fourfold.

The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator's generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one.

We extend Orlov's result on representability of equivalences to schemes projective over a field. We also investigate the quasi-projective case.

We give a new upper bound for the generation time of a tilting object and use it to verify, in some new cases, a conjecture of Orlov on the Rouquier dimension of the derived category of coherent sheaves on a smooth variety.

We investigate sheaves supported on the zero section of the total space of a locally-free sheaf E on a smooth, projective variety X when the top exterior power of E is isomorphic to the canonical bundle of X. We rephrase this construction using the language of A-infinity algebra and provide a simple characterisation of the case E is simply the canonical bundle itself.

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties.

In this paper, we introduce the interested reader to homological mirror symmetry. After recalling a little background knowledge, we tackle the simplest cases of homological mirror symmetry: curves of genus zero and one. We close by outlining the current state of the field and mentioning what homo- logical mirror symmetry has to say about other aspects of mirror symmetry.

0. Derived categories of sheaves on quasi-projectives schemes. Thesis. 2008.

Books

1. Superschool on derived categories and D-branes. Edmonton, Canada, July 17-23, 2016. Lectures from the PIMS Superschool. Edited with Charles Doran, David Favero and Eric Sharpe. Springer Proceedings in Mathematics & Statistics, 240. Springer, Cham, 2018.

Selected lectures

A conjecture of Orlov asks whether all varieties with full exceptional collections are rational. We will discuss two interpretations of this question over non-closed fields. The strict one, where End(E)=k is the definition of exceptional for X over k, holds for toric varieties over a general field. The looser one, End(E)/k is a separable extension of fields, does not guarantee the existence of a k-point much less rationality. The work discussed is joint with Alexander Duncan, Alicia Lamarche, and Patrick McFaddin.

The general expectation, attributed to Orlov, is that a smooth projective variety with a full exceptional collection must be rational over its base field. We will discuss this question over non-closed fields. We will show that there exists smooth projective geometrically rational 3-folds which possess full etale-exceptional collections (where End(E) is a finite separable extension of the base field) but not any points over the base field. In the other direction, we will show the expectation holds for smooth projective toric varieties over any base field: if a smooth projective (neutral) toric variety over k possesses a full k-exceptional collection then it is in fact k-rational. This is joint work with A. Duncan, A. Lamarche, and P. McFaddin.

A healthy body of evidence says that birational geometry and derived categories are intimately bound. Even so, many basic questions are still open. One of the most central questions is the conjecture of Bondal and Orlov (later extended by Kawamata) that says two smooth projective varieties related by a flop are actually derived equivalent. The first step in resolving this question is understanding how to produce functors from rational maps. In work with Diemer and Favero, we provided a method to construct an integral kernel associated to any D-flip of normal varieties with Q-Cartier D. Conjecturally, this can be used to answer Bondal and Orlov's question. In this talk, we will discuss the construction and natural extensions of it. In particular, we will highlight work with Chidambaram, Favero, McFaddin, and Vandermolen relating to the what has been termed a Grassmann flop.

There is a framework that underlies a vast number (all?) semi-orthogonal decompositions in algebraic geometry: wall-crossing in moduli with stability. This builds on previous work on VGIT. We will discuss this in the smooth setting and illustrate with examples.

Here is a basic question. Take your favorite finite set A of n \times n matrices over \C. Call this an alphabet if every matrix can be written as a linear combination of products (words) in A. How long is the longest word? How about if we take the maximum over all A? The Orlov spectrum of a triangulated category captures exactly this data when we use cones for products. It is notoriously difficult to compute thanks to a failure of additivity but existing results offer a testament to its appeal. In this talk, I will introduce the Orlov spectrum, discuss some examples and conjectures. Themes to be touched on include: rationality, Hall algebras, and braid groups. Some results are joint with David Favero and Ludmil Katzarkov.

I will outline how wall-crossing in moduli problems can be used to understand the derived category via windows. Then, in the specific case of wall crossing in GIT, I will provide kernels for these windows coming from compactifications - of the group and the action groupoid. This talk includes joint work with Diemer, Favero, Katzarkov, and Kontsevich.