# Daniel Halpern-Leistner

## Algebraic Geometry, Derived Categories, Representation Theory, Mathematical Physics

Email: danhl at math dot columbia dot edu

I am currently a Ritt assistant professor at Columbia University. Previously I was an NSF Post Doc at Columbia and a member in mathematics at the Institute for Advanced Study. I completed my PhD at UC Berkeley. In my undergraduate studies at Princeton University, I focused on math and physics.

Here's my CV, last updated August 2016.

My primary research focuses on a cluster of related projects and ideas which I have labeled "beyond geometric invariant theory." Geometric invariant theory (GIT) is a well-studied and successful tool for constructing moduli spaces in algebraic geometry. But it does more than that. It is a framework for understanding the (equivariant) geometry of algebraic varieties with a reductive group action. The "beyond GIT" project attempts to expand geometric invariant theory in two ways: 1) to use the ideas of GIT to understand the structure of derived categories of equivariant coherent sheaves, which in turn leads to new results in classical equivariant topology and geometry, and 2) to expand the methods of GIT to apply to general moduli problems.

## Spring 2017 Teaching

This semester I will be teaching Math UN2500 - Analysis and Optimization. The syllabus is here, but the primary mode of communication will be through the piazza page.

Schedule:

Previous:

## Beyond geometric invariant theory

One of the great challenges of research mathematics is effectively communicating mathematical ideas. I'm experimenting with a concept map describing the "beyond GIT" project (click to interact).

## Talks and expository papers on Beyond GIT

• A colloquium-style slide talk on applications of beyond GIT to the D-equivalence conjecture.
• I have written a proceedings paper for the AMS summer algbraic geometry institute (SLC, 2015). Video of the lecture is available here.
• Video of short member lecture at IAS
• Brief Oberwolfach report.
• Notes from a workshop on new methods in GIT (Berlin, 2015)

## Articles

• The equivariant Verlinde formula on the moduli of Higgs bundles (with an appendix by Constantin Teleman)
• update 8/4/2016

The Verlinde formula expresses the dimension of the space of global sections of certain "determinant" line bundles on the moduli of principal $G$ bundles on a smooth curve $\Sigma$, where $G$ is a semisimple group. We prove an analog of the Verlinde formula on the moduli space of semistable meromorphic $G$-Higgs bundles over a smooth curve for a reductive group $G$ whose fundamental group is free. The formula expresses the graded dimension of the space of sections of a positive line bundle on the moduli space of Higgs bundles as a ﬁnite sum whose terms are indexed by formal solutions of a generalized Bethe ansatz equation on the maximal torus of $G$.

arxiv:1608.01754; last update 8/4/2016

• Theta-stratifications, Theta-reductive stacks, and applications
• update 6/12/2016

This is a proceedings paper for the 2015 AMS summer institute in Salt Lake City. It is an expanded version of my lecture there, and includes a discussion of a few more recent developments, too. I give an overview of the beyond GIT project, from the theory of $\Theta$-stability and $\Theta$-reductive stacks, to some the applications to derived categories. Along the way, I propose a possible generalization of toric geometry, generalizing a toric variety to an arbitrary projective-over-affine compactification of a homogeneous space. I also discuss a version of Kirwan's surjectivity theorem for Borel-Moore homology, and I formulate a conjecture that the Hodge structure on the Borel-Moore homology of a cohomologically proper algebraic-symplectic stack is pure.

arxiv:1608.04797; last update 6/12/2016

• Combinatorial constructions of derived equivalences (with Steven Sam)
• update 2/26/2016

This proves a version of what I like to call the magic windows" theorem for a fairly general class of quotient stacks: those which are quotients of a linear representation $V$ of a reductive group $G$ (where the representation is quasi-symmetric". The magic windows theorem identifies (under some mild hypotheses) a subcategory of the equivariant derived category of coherent sheaves on $V$ with the derived category of coherent sheaves on any GIT quotient of $V$ which is a scheme or more generally a Deligne-Mumford stack. Applications include:

• Explicit combinatorial bases in the K-theory and cohomology of GIT quotients of this kind
• Many new examples of derived equivalences between different Deligne-Mumford GIT quotients of linear representations of this kind
• Fitting these derived equivalences together to form a representation of the fundamental group(oid) of the complexified Kaehler moduli space of the GIT quotient, and
• Equivalences (under some mild hypotheses) between all Deligne-Mumford hyperkaehler quotients of a symplectic representation of a reductive group

arxiv:1601.02030; last update 2/26/2016

• Tannaka duality revisited (with Bhargav Bhatt)
• update 7/05/15

Any map between algebraic stacks $f: \fX \to \fY$ yields a symmetric monoidal functor between derived categories of quasicoherent sheaves $f^\ast : QC(\fY) \to QC(\fX)$. Jacob Lurie showed that when $\fY$ is geometric (meaning quasicompact with affine diagonal), $f$ can be uniquely recovered from $f^\ast$, and the symmetric monoidal functors arising in this way are those satisfying certain hypotheses (continuous, preserving connective objects and flat objects). We generalize this result, showing that for many stacks, it is not necessary to show that $f$ preserves flat objects. This seemingly minor modification allows for a much wider range of applications of this "Tannakian formalism."

arxiv:1507.01925; last update 7/05/2015

• Equivariant Hodge theory and noncommutative geometry (with Daniel Pomerleano)
• update 7/03/15

We develop a version of Hodge theory for a large class of smooth cohomologically proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant $K$-theory of $X$ with respect to a maximal compact subgroup $M \subset G$. The result is a natural pure Hodge structure of weight $n$ on $K^n_M(X^{an})$. We also treat categories of matrix factorizations for equivariant Landau-Ginzburg models.

• Autoequivalences of derived categories and variation of GIT quotient (with Ian Shipman)
• update 3/2013

In geometric invariant theory (see below), the GIT quotient of $X/G$ depends on a choice from a continuous set of parameters. Nevertheless, the parameter space breaks down into "chambers" within which the GIT quotient does not vary, and these chambers are separated by "walls." When the parameters cross a wall, the GIT quotient is modified by a "birational transformation." I have been studying how the geometry and especially the derived geometry of the GIT quotient changes under such a wall crossing. For the special case of these wall crossings known as a "generalized flop," the derived geometry of the GIT quotient does not change at all. These cases are especially interesting -- they can reveal new symmetries of the derived category of the GIT quotient which do not arise in the classical geometry.

• "Autoequivalences of derived categories and variation of GIT quotient," with Ian Shipman (arxiv:1303.5531)
• The derived category of a GIT quotient
• update 8/2012

If an algebraic group $G$ acts on an algebraic variety $X$, such as $\bC^\ast$ acting on affine space by dilation, is there a meaningful notion of a "space of orbits" for that action? Mumford's geometric invariant theory (GIT) answers this question by constructing a well-behaved orbit space for the action of $G$ on an open subset of "semistable points" of $X$. Many of the algebraic varieties we know and love (partial flag varieties, toric varieties,...) can be presented as GIT quotients of affine spaces. Since the 1980's, many beautiful relationships between the geometry and topology of the GIT quotient and the "equivariant" geometry of $X$ have been discovered. My research extends these relationships to the "derived" equivariant geometry of $X$ and the derived geometry of the GIT quotient.

• The final version treats quotient stacks X/G subject to a certain technical hypothesis on the stratification arxiv:1203.0276)
• For newcomers, I'd recommend first looking at the version which treated just the smooth case, arxiv:1203.0276v2. Warning: This version contains some errors in the description of the stratification that I corrected in the final version.
• Notes from an introductory talk
• The Lefschetz hyperplane theorem for Deligne-Mumford stacks
• update 2012

Deligne-Mumford stacks are an abstract method that modern algebraic geometers use to handle spaces whose points have "internal symmetry." Such spaces arise naturally as parameter spaces of objects (such as pointed elliptic curves) which generically have no symmetry, but for which certain points parameterize an object with discrete symmetry. By enlarging your notion of "space," you can deal with these naturally occuring geometric objects, and with the proper formulation, you can even use concrete geometric reasoning such as Morse theory to show that classical theorems still apply in this setting.

## Working papers

These form the backbone of the Beyond GIT project. While they are currently complete papers, I have decided to edit and expand them. I will keep the most up-to-date versions here.

• $\Theta$-stratifications, derived categories, and moduli of complexes on a K3 surface
• update 2/09/17

A continuation of the derived Kirwan surjectivity project. It turns out that a version of the result is true in much greater generality if one works in the context of derived algebraic geometry. This leads to new structure theorems for equivariant derived categories even for classical quotient stacks, when the stack is singular and fails to meet the technical hypotheses used in "The derived category of a GIT quotient." The key observation is that even for classical stacks, the stratification obtains a non-trivial derived structure coming from a particular modular interpretation. The final paper will include the main structure theorem for the derived category of a stack with a $\Theta$-stratification and applications to the the D-equivalence conjecture for moduli spaces of sheaves on a K3 surface.

• "Theta stratifications and derived categories" -- will prove the results announced in the paper Theta-stratifications, Theta-reductive stacks, and applications" (in progress)

Preliminary documents:

• "The D-equivalence conjecture for moduli spaces of sheaves on a K3 surface" (draft; last update 2/08/17) -- this will ultimately be a section of the paper above. It is a sketch of the proof of one of the main applications, that the D-equivalence conjecture holds for Calabi-Yau varieties which are birational to a modli space of sheaves on a K3.
• "An appendix to 'Theta stratifications and derived categories'" (draft; last update 2/04/15) -- proves a version of the main theorem for global quotient stacks, which is enough for many applications
• Towards a general theory of instability in moduli theory
• update 10/21/2014

A common kind of problem in algebraic geometry is to find a space, called a moduli space, parameterizing isomorphism classes of some kind of algebro-geometric objects -- let's call them widgets. Many attempts to form a moduli space for widgets proceed by finding a scheme, $X$, which parameterizes a family of widgets, and an algebraic group, $G$, acting on $X$ such that points in the same orbit under $G$ parameterize isomorphic widgets. Then hopefully one can apply geometric invariant theory to find an open subset in X which has a good quotient under the action of $G$, and whose $G$-orbits classify semistable'' widgets. However, there are many situations where a stability condition can be specified on widgets without referring to any GIT problem. We discuss a framework for defining a notion of semistability for an arbitrary moduli problem, and we introduce a structure on the unstable locus, which we call a Theta-stratification, which generalizes classical stratifications of the unstable locus in GIT as well as of the moduli of vector bundles on a curve. We identify a class of moduli problems, which we call reductive, for which the GIT story carries over nicely into this more general framework -- for these stacks the existence of a $\Theta$-stratification on the unstable locus can be reduced to checking a relatively simple hypothesis.

• "On the structure of instability in moduli theory," (draft)
• Mapping stacks and the notion of properness in algebraic geometry (with Anatoly Preygel)
• update 2/1/2014

One essential feature of a scheme $X$ which is flat and proper over a base $S$ is that for any scheme $Y$ which is of finite type over $S$, there is an algebraic space $Map(X,Y)$ classifying maps from $X$ to $Y$. There are extensions of this statement when $X$ is a proper stack over $S$ and $Y$ has some reasonable hypotheses. While studying the notion of instability in algebraic geometry, I noticed that the quotient stack $\bC / \bC^\ast$ has this property as well. This lead to long investigation with Anatoly Preygel into just what properties of a stack X guarantee the existence of algebraic mapping stacks into "geometric" target stacks. We reformulated the notion of "properness" for algebraic stacks in terms of properties of the derived category of those stacks, in such a way that these categorical properties guarantee the existence of algebraic mapping stacks. Our notion generalizes the classical definition of proper Artin stacks, but in addition there are many global quotient stacks which are definitely not proper in the classical sense but are "proper" in our sense. We were able to construct a very large class of examples by introducing a notion of "projective morphism" of stacks, a property which can be readily verified in examples. Along the way, we prove some surprising new descent properties for derived categories of coherent sheaves in derived algebraic geometry.

• "Mapping stacks and categorical notions of properness," with Anatoly Preygel (draft; Feb 1)

My PhD thesis included chapters that became the papers "The derived category of a GIT quotient," "Autoequivalences of derived categories and variation of GIT quotient," and "On the structure of instability in moduli theory."

As an undergraduate I thought about algebraic approaches to information theory. Here is a primer on my undergraduate work (Last update 4/30/08)

Cohomologically proper algebraic stacks:

The notion of a proper (i.e. "compact") algebraic variety is essential to algebraic geometry. In the context of stacks, we argue that the standard notion of a proper algebraic stack is too restrictive, and we provide an alternative notion of cohomological properness: the stack should satisfy the 'Grothendieck existence' theorem "universally" (i.e. after arbitrary base change to a complete Noetherian ring, or more generally a stack which is complete along a closed substack). The Grothendieck existence theorem is usually regarded as a theorem, but taking it as the definition of properness leads to a notion which is well-behaved and generalizes the more geometric definition of properness in Champs algébriques in useful ways.

Relevant papers:
• "Mapping stacks and categorical notions of properness" - develops the general theory
• "Equivariant Hodge theory and noncommutative geometry" - inspired by the analogy between cohomologically proper stacks and proper schemes

Equivariant Hodge theory:

For any dg-category, there is a spectral sequence starting with the Hochschild homology and converging to the periodic cyclic homology. When the category is $Perf(X)$ for a scheme, this can be identified with the classical Hodge-to-deRham spectral sequence, and it degenerates when $X$ is smooth and proper. The non-commutative HdR sequence degenerates for many cohomologically proper smooth stacks and fails to do so for many smooth stacks which are not cohomologically proper. This provides further evidence that cohomologically proper is a good generalization of the notion of properness to Artin stacks.

Another interesting aspect of this degeneration is that it allows one to construct a canonical pure Hodge structure on the topological $K$-theory of many smooth and cohomologically proper stacks. It raises the question of whether there is a canonical mixed Hodge structure on the topological K-theory of an arbitrary stack, and whether there is a motivic framework which is well-suited for stacks.

Relevant paper:
• "Equivariant Hodge theory and noncommutative geometry"

Mapping stacks:

If $\fX$ and $\fY$ are stacks, then one can form a mapping stack, $Map(\fX,\fY)$, which by definition is the moduli functor parameterizing families of maps from $\fX$ to $\fY$. When $\fX$ and $\fY$ are algebraic spaces (locally finitely presented over a Noetherian base) and $\fX$ is proper and flat, then it is a classical result that $Map(\fX,\fY)$ is in fact represented by an algebraic space. When $\fX$ is an algebraic stack, this mapping stack will again be an algebraic stack, as long as $\fY$ is geometric and $\fX$ is cohomologically proper. The special case where $\fX = \Theta$ is central to the theory of $\Theta$-stratifications.

Relevant papers:
• "Mapping stacks and categorical notions of properness" - develops the general theory
• "Tanaka duality revisited" - one of the key inputs

Theta stratifications:

If $X$ is a symplectic manifold admitting a Hamiltonian action by a compact group, $K$, then one has a $K$-equivariant stratification of $X$ by the gradient descent flow of the norm-squared of the moment map. When $X$ is a projective variety, then this stratification has an alternative, purely algebraic, description in terms of the Hilbert-Mumford numerical criterion in GIT, and one can think of this as a stratification of the algebraic stack $\fX = X/K_{\bC}$. Theta stratifications provide a generalization of this to stacks which are not global quotients stacks, such as the non finite-type stacks appearing in many moduli problems. The theory provides a framework for studying "stability" of algebro geometric objects generalizing GIT and many other commonly studied notions of stability.

Key idea: the strata have canonical modular interpretations -- they parameterize maps $f : \bC/\bC^\ast \to \fX$ which exhibit "optimally destabilizing" data for the unstable point $f(1) \in \fX$.

Relevant papers:
• "On the structure of instability in moduli theory" -- develops the machinery and foundational theorems regarding $\Theta$-stratifications. Also discusses methods of constructing $\Theta$-stratifications.

$\Theta$-reductive stack:

Given an algebraic stack representing a certain moduli problem, one can ask what data is required to define a Theta-stratification. For $\Theta$-reductive stacks, all one needs is a class in $H^2(\fX;\bQ)$ and $H^4(\fX;\bQ)$ satisfying a "boundedness" hypothesis.

Simplest example: $X/G$, where $X$ is affine and $G$ is a reductive group

Non-example: $X/G$ where $X$ is projective and $G$ is reductive

Interesting examples: Moduli of objects in the heart of a t-structure on the derived category of coherent sheaves of a projective variety

Formal definition: the map of "evaluation at 1" $Map(\bC/\bC^\ast,\fX) \to \fX$ should be proper on connected components

Our research suggests that a good starting point for analyzing a moduli problem is to find a $\Theta$-reductive enlargement of that moduli problem, then apply the theory of $\Theta$-stability.

Relevant papers:
• "On the structure of instability in moduli theory" -- developes the theory of $\Theta$-reductive stacks and $\Theta$-stability.

Generalized buildings:

A key construction in the theory of Theta-stability assigns to any point in an algebraic stack $p \in \fX$, a topological space $\sD(\fX,p)$, called the degeneration space. A map $\bC/\bC^\ast \to \fX$ along with an isomorphism $f(1) \simeq p$ determines a point of $\sD(\fX,p)$, and points of this form are dense in $\sD(\fX,p)$.

When $\fX = BG$ for a semisimple group $G$, then $\sD(\fX,p)$ is homeromorphic to the spherical building of $G$, and when $\fX = X/T$ for a normal toric variety $X$, the degeneration space of a generic point of $X$ is homeomorphic to $(|\Sigma| - \{0\}) / \bR^\times_{\geq 0}$, where $|\Sigma|$ denotes the support of the fan defining $X$. Thus these degeneration spaces can be thought of as "generalized buildings," and they connect the theory of buildings in representation theory with toric geometry.

Relevant papers:
• "On the structure of instability in moduli theory" -- defines the degeneration space associated to a point in an arbitrary stack

Extensions of Kirwan surjectivity:

Kirwan surjectivity states that for a GIT quotient of a smooth variety, $X^{ss} / G \subset X/G$, the restriction map on cohomology $H^\ast(X/G) \to H^\ast(X^{ss}/G)$ is surjective. The main structure theorem for derived categories leads to two extensions of this theorem:

1. It provides a "categorification" of this result to a statement about derived categories of coherent sheaves. The restriction functor $D^b Coh (X/G) \to D^b Coh(X^{ss}/G)$ is always surjective on the level of objects, so that's not the right categorification. But it turns out there is a subcategory $G_w \subset D^b Coh(X/G)$ such that the restriction functor gives an equivalence $G_w \simeq D^b Coh(X^{ss}/G)$. This implies that the restriction functor admits a section $D^b Coh(X^{ss}/G) \to D^b Coh(X/G)$, and so for any invariant which can be extracted from the derived category, the restriction functor from $X/G$ to $X^{ss}/G$ is surjective. For instance, you get Kirwan surjectivity for higher algebraic K-theory.

2. It turns out that this categorical form of Kirwan surjectivity continues to hold (under certain hypotheses) for the category $D^bCoh$ when $X$ is quasi-smooth; either a local complete intersection or a space with quasi-smooth derived structure. Combined with the de-categorification results extracting topological invariants from derived categories, this leads to a version of Kirwan surjectivity for equivariant Borel-Moore homology (again assuming certain hypotheses on the first homology of the cotangent complex of $X$)

Relevant papers:
• "The derived category of a GIT quotient" -- the smooth case, and some singular cases (without using derived algebraic geometry)
• "Remarks on derived categories and $\Theta$-stratifications" -- works out the generalized structure theorem in the quasi-smooth case
• "Equivariant Hodge theory and noncommutative geometry" -- proves that one can extract Borel Moore K-theory from $D^bCoh$

Topological invariants of derived categories:

A result of Feigin and Tsygan holds that the cohomology of (the analytification of) an affine variety, $X$, over $\bC$ is isomorphic to the periodic-cyclic homology of the coordinate ring. This agrees with the periodic-cyclic homology of the category of perfect complexes on $X$, so some topological information can be extracted directly from the derived category. For equivariant categories, Thomason showed that the topological equivariant K-theory modulo a prime power can be recovered from the derived category of equivariant coherent sheaves. It turns out that the equivariant K-theory itself can be recovered from the derived category of equivariant coherent sheaves, at least when $X$ is smooth. This result allows one to de-categorify categorical Kirwan surjectivity and recover classical Kirwan surjectivity. When $X$ is singular, one can recover the equivariant Borel-Moore $K$-theory from the derived category of coherent sheaves this leads to new versions of Kirwan surjectivity in Borel-Moore homology.

Relevant papers:
• "Equivariant Hodge theory and noncommutative geometry"

Virtual non-abelian localization theorem:

When $X$ is a compact manifold and $T$ is a torus acting on $X$, localization theorems in equivariant cohomology provide a method for reducing the integrals of equivariant cohomology classes on $X$ to integrals over the fixed locus $X^T$ (which could be just a sum over a finite set of points). There is another, closely related, flavor of localization theorems for stacks with a $\Theta$-stratification. The integral $\int_X \omega$ is replaced by the K-theoretic integral $\chi(X/G,F) := \Sigma (-1)^p \dim R^p \Gamma(X,F)^G,$ where $F \in Perf(X/G)$, and the localization formula expresses $\chi(X/G)$ as a sum of $\chi(X^{ss}/G,F)$ and "correction terms" coming from each stratum. In some cases the correction terms vanish, leading to an identification between the integral over $X$ and over $X^{ss}$, and in other cases $X^{ss} = \emptyset$, leading to an formula for $\chi(X/G,F)$ in terms of the fixed locus as in the cohomological version. Thus the $K$-theoretic localization theorem is a little more flexible and has the advantage of working for non-abelian $G$, and the cohomological localization formula can be recovered for classes of the form $ch(F)$.

For smooth global quotient stacks, the non-abelian localization theorem was developed by Teleman and Woodward. In fact the non-abelian localization formula is intrinsic to any stack with $\Theta$-stratification, and does not require a local quotient description. Furthermore, using a little bit of derived algebraic geometry, we extend the non-abelian localization theorem to stacks which are quasi-smooth. If $\fX^{cl} \hookrightarrow \fX$ is the underlying classical stack of a quasi-smooth stack $\fX$, and $F \in Perf(\fX)$, then the virtual non-abelian localization theorem is a formula for $\chi(\fX^{cl},F \otimes \mathcal{O}_{\fX}^{vir})$, where $\mathcal{O}_{\fX}^{vir} \in Perf(\fX^{cl})$ is the "virtual" structure sheaf. (In practice most classical stacks with a perfect obstruction theory are the underlying classical stack of a quasi-smooth derived stack.)

Relevant papers:
• "Remarks on Theta-stratifications and derived categories"

Structure theorems for equivariant derived categories:

The main structure theorem for stacks with a $\Theta$-stratification concerns the derived category of coherent sheaves. A semi-orthogonal decomposition of a pre-triangulated dg-category consists of the data of a collection of pre-triangulated (i.e. stable) subcategories which are semiorthogonal to one another (Homs only go in one direction with respect to some total ordering of the categories), and such that every object has a functorial filtration whose associated graded pieces lie in these subcategories. The main structure theorem states that when $\fX$ is a (derived) stack with a $\Theta$-stratification, then $D^- Coh(\fX)$ admits a semiorthogonal decomposition where one piece, $G^w \subset D^-Coh(\fX)$, is identified with $D^-Coh(\fX^{ss})$ via the restriction functor. As a consequence, objects in $D^-Coh(\fX^{ss})$ can be lifted functorially to $D^-Coh(\fX)$. The other pieces of the semiorthogonal decomposition consist of objects which are set-theoretically supported on the unstable locus in $\fX$. There are certain situations in which this structure theorem is even stronger:

1. When $\fX$ is smooth, this semiorthogonal decomposition induces a decomposition on $Perf(\fX)$, and
2. When $\fX$ is quasi-smooth but the coherent sheaf $H^1(T_{\fX})$ satisfies a certain weight condition, then this induces a semiorthogonal decomposition of $D^bCoh(\fX)$. The condition holds automatically for algebraic-symplectic stacks.
Relevant papers:
• "The derived category of a GIT quotient" -- works out the case when $\fX = X/G$ is a global quotient stack which is either smooth or is singular (and classical) and satisfies certain technical hypotheses.
• "Remarks on $\Theta$-stratifications and derived categories" -- works out the derived case for global quotient stacks. The full version (to appear) is similar (although the proof is conceptually cleaner).

Monodromy representations on derived categories:

Let $V$ be a quasi-symmetric linear representation of a reductive group $G$. The magic windows theorem gives more than derived equivalences between different GIT quotients of $V$. There are different choices of magic windows that one can use to construct derived equivalences between the different GIT quotients, one can use these different equivalences to construct an action of the fundamental groupoid of the complexified Kaehler moduli space." This space admits an explicit combinatorial description as the complement of a certain hyperplane arrangement in a torus, based on the character of $V$. Under homological mirror symmetry, this action of the fundamental groupoid on the derived category is conjectured to be mirror to an action by symplectic parallel transport on the mirror family of varieties. Thus we refer to it as a monodromy representation, even though its relationship with monodromy is still conjectural, for the moment.
• "Combinatorial constructions of derived equivalences"--prove the magic windows theorem in the smooth case and establishes the action of the fundamental groupoid of the Kaehler moduli space

Equivariant Verlinde formula on the moduli of Higgs bundles:

The Verlinde formula expresses the dimension of spaces of "generalized Theta-functions," defined as sections of a certain line bundle on the moduli of semistable $G$-bundles on a smooth curve. One consequence of the virtual localization formula is a proof of a version of this formula for the moduli of semistable Higgs bundles. Building on previous work of Teleman and Teleman-Woodward, one can compute the dimension of the space of sections of this line bundle (of any given weight under the $\bG_m$-action which scales the Higgs field) on the stack of all Higgs bundles, and our methods allow one to identify this with the corresponding space of sections of this line bundle on the space of semistable Higgs bundles.

Relevant paper:
• "The equivariant Verlinde formula on the moduli of Higgs bundles"

Categorical representations of Yangians:

This is still work in progress with Davesh Maulik and Andrei Okounkov. We are using the magic windows theorem to categorify some earlier work of theirs on the quantum cohomology and quantum K-theory of Nakajima quiver varieties.

Derived equivalences and variation of GIT quotient:

The main application of the structure theorem is to proving cases of the K-equivalence implies D-equivalence" conjecture for K-equivalences arising from variation of GIT quotient: as the stability parameter for the GIT quotient changes, $X^{ss}_- \leftrightarrow X^{ss}_+$, the stratification changes and thus the subcategories $G^\pm_w \subset D^bCoh(X/G)$ differ. In nice situations, the category corresponding to one GIT quotient contains the category corresponding to the other GIT quotient, $G_w^- \subset G_w^+$. Using the full structure theorem, one can identify the semiorthogonal complement of $G^-_w$ in $G^+_w$ explicitly.

This model for how to construct derived equivalences is due to Ed Segal and to Hori, Herbst, and Page, who discovered the phenomenon in some basic examples. Ballard,Favero, and Katzarkov and I independently extended these methods to construct derived equivalences for all VGIT wall crossings which are "balanced" and satisfy a "Calabi-Yau" condition. The current state of the art on this method is the "magic windows theorem," which identifies subcategories of $D^b(V/G)$ which are identified with every GIT quotient for which $V^{ss} = V^s$, when $V$ is a quasi-symmetric" linear representation of a reductive group $G$ satisfying a mild technical condition.

Relevant papers:
• "The derived category of a GIT quotient" -- the first version, gets derived equivalences for balanced wall crossings (in the smooth case)
• "Combinatorial constructions of derived equivalences" -- establishes the magic windows theorem and many new cases of the derived equivalence conjecture.