## About meEmail: 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:**

- Fall 2016: Topics course - taming moduli problems in algebraic geometry
- Spring 2016: Calc III - Multivariable differential calculus.
- In Fall 2015 I developed a concept map (similar to the one above) in tandem with Johan de Jong's graduate seminar on the Tate conjecture for surfaces.

## 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*

- Notes from lectures on Theta-stability and existence of good moduli spaces, with applications to Donaldson invariants (Thun, Switzerland, 2017). Lecture 1, Lecture 2
- 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

This is a short note which discusses a construction involving equivariant derived categories which allows one to state a categorical version of the classical Atiyah-Bott localization formula. The classical theorem states that for a smooth manifold with a $\bC^\ast$-action, the fundamental class in equivariant cohomology can be decomposed as a finite sum with each term corresponding to a connected component of the fixed locus, but in order to do this one must formally invert certain elements in the equivariant cohomology of the point. We show that if $X$ is a smooth scheme, then the structure sheaf of $X$ admits a filtration whose associated graded pieces correspond to connected components of the fixed locus, but in order to do this one must work in a slightly larger category of equivariant complexes, which we define.

"A categorification of the Atiyah-Bott localization formula; last update 5/21/2016

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

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

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

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

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.

- arxiv:1507.01924; last update 7/03/15

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)

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

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.

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 $\Theta$-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)*

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

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)