Here is a selection of some of the mathematics I have written over the years. See also ArXiv and MathSciNet.

• ** German Diplom Thesis **

My Diplom thesis was written in Bonn (Germany) under the
direction of Prof. G. Harder. It is about moduli spaces of vector
bundles with level structures on curves. It has not been published, and it does not exist in
electronic form. Here is a summary.
pdf

• ** Berkeley PhD Thesis **

I wrote my PhD thesis under the direction of Prof. A. Ogus in
Berkeley, California. It is about the Lefschetz trace formula for the
Frobenius on algebraic stacks in general, and the stack of G-bundles
over a curve (G a reductive group scheme) in particular.
Only parts of the thesis have been published. Here is the thesis in its entirety.
pdf

• ** Derived l-adic categories for algebraic stacks **

This is work I did while I was a postdoc at MIT.
I construct l-adic derived categories for algebraic stacks defined
over a finite field. I define pushforward and pullback functors
between these categories. The key notion is that of convergent
complex. In the end I arrive at a proof of the trace formula for the
Frobenius on any algebraic stack. (I conjectured this formula in
1993.) This work eventually appeared in *Mem. AMS 774*.
pdf

• ** Cohomology of Stacks**

These are notes for lectures I gave in 2002 in Trieste at the ICTP.
They cover de Rham cohomomlogy of differentiable stacks as well as de
Rham cohomology with compact supports. I prove Poincaré duality
for proper differentiable Deligne-Mumford stacks.
In the second part of these notes, I cover singular homology and
cohomology of topological stacks. Examples and applications are included.
pdf

• (with Behrang Noohi) ** Uniformization of Deligne-Mumford curves**

We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones,
and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary
Deligne-Mumford curve as a quotient stack. Along the way, we compute the automorphism 2-groups of weighted projective stacks.
We also discuss connections with the theory of F-groups, 2-groups, and Bass-Serre theory of graphs of groups.
Appeared in *Crelle*
pdf

• ** On the de Rham cohomology of differential and algebraic stacks**

We introduce the notion of cofoliation on a stack. A cofoliation
is a change of the differentiable structure which amounts to giving a
full representable smooth epimorphism. Cofoliations are uniquely
determined by their associated Lie algebroids.
Cofoliations on stacks arise from flat connections on groupoids.
Connections on groupoids generalize connections on gerbes and bundles
in a natural way. A flat connection on a groupoid is an integrable
distribution of the morphism space compatible with the groupoid
structure and complementary to both source and target fibres. A
cofoliation of a stack determines the flat groupoid up to etale
equivalence.
We show how a cofoliation on a stack gives
rise to a refinement of the Hodge to De Rham spectral sequence, where
the E1-term consists entirely of vector bundle valued cohomology
groups.
Our theory works for differentiable, holomorphic and
algebraic stacks. This work appeared in the Artin Festschrift.
pdf

• (with Ajneet Dhillon)** On the motive of the stack of bundles**

In connection with my work in Bonn on my Diplom Thesis, Harder explained to me the strange connection between two numbers which are 1:
The homotopy type of the space of connections and the Tamagawa number.
In this paper we make a precise mathematical conjecture which attempts to explain this strange observation.
In fact, we give a conjectural formula for the motive of the stack of G-bundles on a curve in terms of special
values of the motivic zeta function of the curve.
pdf

• (with Y. Manin)** Stacks of stable maps and Gromov-Witten
invariants **

Treats stacks of stable maps, axioms for virtual fundamental classes
and genus zero Gromov-Witten invariants for convex varieties. Special
emphasis on functorial nature of both stacks of stable maps and
Gromov-Witten invariants. Appeared in * Duke Math. Journal. *
pdf

• (with B. Fantechi) ** The intrinsic normal cone **

Shows how to construct virtual fundamental classes, using the
intrinsic normal cone. Appeared in * Invent. Math. *
pdf

• ** Gromov-Witten invariants in algebraic geometry **

We apply the construction of virtual fundamental classes to stacks of
stable maps to get Gromov-Witten invariants in the general
case. Appeared in * Invent. Math. *
pdf

• ** Algebraic Gromov-Witten invariants **

An introduction to the theory developed in the above three papers.
Apeared in the proceedings of the Warwick conference 1996.
pdf

• ** The product formula for Gromov-Witten invariants **

We prove that the Gromov-Witten CohFT of a product is the tensor
product of the Gromov-Witten CohFT's of the factors.
Appeared in *JAG*.
pdf

• ** Localization and Gromov-Witten invariants **

These are notes of lectures I gave in the summer of 1997 in Cetraro,
Italy, at the CIME session on Quantum cohomology. The first lecture
is an introduction to stacks. The second lecture is about equivariant
intersection theory. Finally, in the third lecture I prove the
localization formula for Gromov-Witten invariants of projective space.
My approach is different from the approach of Graber-Pandharipande
(Invent. Math. 1999). It avoids equivariant virtual fundamental
classes. These notes appeared in *LNM 1776*.
pdf

• (with A. O'Halloran)** On the Cohomology of Stable Map
Spaces **

We describe an approach to calculating the
cohomology rings of stable map spaces of genus zero curves without
marks to projective space. Our method uses an equivariant vector
field on the big Bialynicki-Birula cell and is based on work of
Akildiz and Carrell. Our main result is a complete determination
of the stable cohomology as the dimension of the target goes to
infinity of the space of degree 3 maps in terms of generators and
relations. Appeared in *Invent. Math.*
pdf

• ** Differential Graded Schemes I**
* Perfect Resolving Algebras *

I introduce perfect resolving algebras and study their fundamental
properties. These algebras are basic for any theory of
differential graded schemes, as they give rise to affine differential
graded schemes. I also introduce étale morphisms. The
purpose for studying these, is that they will be used
to glue differential graded schemes from affine ones with respect to
an étale topology. This has not been published yet.
pdf

• ** Differential Graded Schemes II**
* The 2-category of differential graded schemes *

I construct a 2-category of differential graded schemes. The local
affine models in this theory are differential graded algebras, which
are graded commutative with unit over a field of characteristic zero,
are concentrated in non-positive degrees and have perfect cotangent
complex. Quasi-isomorphic differential graded algebras give rise to
2-isomorphic differential graded schemes and a differential graded
algebra can be recovered up to quasi-isomorphism from the differential
graded scheme it defines. Differential graded schemes can be glued
with respect to an étale topology and fibered products of
differential graded schemes correspond on the algebra level to derived
tensor products. While this theory is without internal contradictions,
it does not give rise to "correct" derived geometry: 2-categories are not sufficient for that purpose.
Many results in this preprint will hopefully still be useful in derived geometry,
for example, a theory based on simplicial presheaves.
pdf

• ** Donaldson-Thomas Type Invariants via Microlocal Geometry **

This is the paper where I introduce the *Behrend function*.
I prove that Donaldson-Thomas type invariants are equal to
weighted Euler characteristics of their moduli spaces. In particular, such
invariants depend only on the scheme structure of the moduli space, not
the symmetric obstruction theory used to define them. I also introduce
new invariants generalizing Donaldson-Thomas type invariants to mod-
uli problems with open moduli space. These are useful for computing
Donaldson-Thomas type invariants over stratifications. Appeared in * Annals of Mathematics*.
pdf

• (with B. Fantechi) ** Donaldson-Thomas Type Invariants via Microlocal Geometry **

We evaluate the Behrend function for a
singularity which is an isolated point of a C*-action and
which admits a symmetric obstruction theory compatible with the
C*-action. The answer is (-1)^d, where d is the dimension
of the Zariski tangent space.
We use this result to prove that for any threefold, proper or not, the
weighted Euler characteristic of the Hilbert scheme of n points on
the threefold is, up to sign, equal to the usual Euler
characteristic. For the case of a projective Calabi-Yau threefold,
we deduce that the Donaldson-Thomas invariant of the Hilbert scheme
of n points is, up to sign, equal to the Euler characteristic. This
proves a conjecture of Maulik-Nekrasov-Okounkov-Pandharipande.
pdf

• (with J. Bryan) ** Super-rigid Donaldson-Thomas Invariants **

We solve the part of the Donaldson-Thomas theory of Calabi-Yau
threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-
Thomas correspondence of [MNOP] for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for
the quintic threefold in degrees one and two.
pdf

• (with B. Fantechi) ** Gerstenhaber and Batalin-Vilkovisky
structures on Lagrangian intersections **

Let M and N be Lagrangian submanifolds of a complex symplectic
manifold S . We construct a Gerstenhaber algebra structure on
Tor^S(O_M,O_N) and a compatible Batalin-Vilkovisky module structure on
Ext_S(O_M,O_N). This gives rise to a de Rham type cohomology theory for Lagrangian intersections.
It is an attempt to categorify Lagrangian intersection numbers. This work has appeared in the Manin Festschrift.
pdf

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Last updated March 10, 2010.