Matthew Ballard

Title: Kernels from compactifications
Abstract: I will talk about how compactifying group actions leads to fully-faithful functors.


Daniel Bergh

Title: Destackification and some applications
Abstract: We give an algorithm which given a smooth tame Artin stack produces a sequence of stacky blow-ups in smooth centres such that the coarse space of the modified stack becomes smooth. Since this removes the "stackiness" from the stack in a controlled way, we call the process destackification.

I will briefly describe what destackification is and discuss some possible applications. In particular, I will mention a result on geometricity for derived categories of stacks from recent work together with Lunts and Schnürer. The work on destackification is joint with Rydh.


Daniel Edidin

Title: Canonical reduction of stabilizers of Artin Stacks
Abstract: We consider a class of birational transformations of Artin stacks with good moduli spaces called Reichstein transformations. Using recent étale slice results of Alper, Hall and Rydh we show that if X is a stable (meaning it contains an open DM substack which is saturated with respect to the good moduli space morphism) smooth Artin stack with good moduli space M then there is a canonical sequence of Reichstein transformations which produces a smooth, tame stack X' and projective birational morphism M' --> M where M' is the coarse moduli space of X'. This generalizes earlier work in GIT of Kirwan and Reichtstein, and the author and Yogesh More for toric stacks. It is in part joint work with David Rydh.


Barbara Fantechi

Title: Jun Li's degeneration formula via twisted targets
Abstract: Jun Li's original proof of the degeneration formula is made technically difficult since the moduli stack of stable maps he uses isn't open in the moduli of prestable maps, hence the relevant obstruction theory is rather difficult to work with. In this joint work with Dan Abramovich we propose a different approach, inspired by the notion of twisted stable curves; we construct a different moduli stack, prove that is proper over the one of Jun Li, and show that it is open, thus has a natural obstruction theory.


Yi Hu

Title: Modular Blowup and Applications
Abstract: We propose a method to modularly blow up the moduli of stable maps. We will explain how to carry it out for genus one and genus two (joint with Jun Li).


Ariyan Javanpeykar

Title: Finiteness results for smooth hypersurfaces over finitely generated domains over Z
Abstract: The Lang-Vojta conjecture states that a smooth quasi-projective variety is Brody hyperbolic if and only if it is arithmetically hyperbolic. In this talk we will first explain the statement of the Lang-Vojta conjecture and then, assuming the Lang-Vojta conjecture, prove that the set of smooth hypersurfaces over a finitely generated domain over Z of fixed dimension and fixed degree is finite. We will explain that this conditional finiteness result is a consequence of the fact that the complex algebraic stack of smooth hypersurfaces admits a representable immersive period map, under suitable assumptions on the degree and dimension. Finally, to illustrate how far one can get without assuming the Lang-Vojta conjecture, we will prove (unconditionally) that, for all finitely generated domains A over Z, the set of smooth sextic surfaces in P^3_A is finite. This is joint work with Daniel Loughran.


Masoud Kamgarpour

Title: The Hitchin map for the moduli stack of $\mathcal{G}$-torsors
Abstract: TBA


Amelendu Krishna

Title: Riemann-Roch theorem for quotient Deligne-Mumfords stacks.
Abstract: The Grothendieck Riemann-Roch theorem proves the covariant functoriality of the Riemann-Roch map from K-theory of coherent sheaves on a scheme to its Chow groups. This was a revolutionary work in algebraic geometry and was generalized by Bloch to higher K-theory. In this talk, we shall explain how the Riemann-Roch map makes for quotient Deligne-Mumford stacks and what one can say about it.


Daniel Litt

Title: Non-Abelian Lefschetz Hyperplane Theorems
Abstract: How are the properties of a variety X related to those of an ample divisor D in X? Classical Lefschetz hyperplane theorems answer this question by comparing the cohomology or homotopy type of X to that of D. I'll describe new results, encapsulating some of these older Lefschetz theorems, which compare F(X) to F(D) where F is any functor which is representable in a suitable sense. For example, F can be \pi_1, or a moduli functor. While the main results are in characteristic zero, the method of proof passes through positive characteristic.


Martin Olsson

Title: TBA
Abstract: TBA


Brian Osserman

Title: Relative dimension of stacks
Abstract: Infinitesimal deformation theory is a common tool for showing that objects can be deformed in families. However, in some cases, such as the theory of limit linear series and its higher-rank generalization, it works better to use more naive dimension-based arguments. For morphisms of varieties over a field, there is no subtlety in carrying this out, but for schemes or stacks, the pathologies of dimension theory demand a more careful approach. With this motivation, we describe a notion of a morphism having "at least" a given relative dimension. This condition is quite well behaved, generalizes transparently to algebraic stacks, and arises naturally in many settings.


David Rydh

Title: Weak factorization of Deligne-Mumford stacks
Abstract: The weak factorization theorem relates two birational smooth varieties in characteristic zero via a sequence of blow-ups and blow-downs in smooth centers. I will present a variant of the weak factorization theorem for Deligne-Mumford stacks in characteristic zero where blow-ups are replaced with stacky blow-ups.

As for weak factorization for schemes, the main idea is to use cobordisms (Morelli, Włodarczyk) which are analogous to cobordisms in Morse theory. I will outline the proof for Deligne-Mumford stacks which actually is simpler than for schemes.

Time permitting, I will also mention functorial flatification and étalification following ideas of Hironaka. This shows that the weak factorization algorithm is functorial with respect to smooth morphisms. As a corollary, we recover the weak factorization theorem for simplicial toric varieties (the weak Oda conjecture, proven by Włodarczyk).


Matthew Satriano

Title: Is every variety with quotient singularities a global quotient by a finite group?
Abstract: By taking a stacky perspective on this question posed by Fulton, we obtain a necessary and sufficient criterion for when a variety is a quotient by a finite abelian group. As a consequence, we show that the answer to Fulton's question is yes for toric varieties. Along the way, we obtain a completely general "bottom-up" construction of Deligne-Mumford stacks as root stack and canonical stack constructions over the coarse space. We discuss what this bottom-up construction means in terms of Fulton's problem. This is based on joint work with Anton Geraschenko.


Ronan Terpereau

Title: Young person's guide to invariant Hilbert schemes
Abstract: The aim of this talk is to introduce the so-called invariant Hilbert schemes which are moduli spaces parameterizing certain affine schemes equipped with a reductive algebraic group action whose coordinate ring is isomorphic to a fixed linear representation. We will see that these moduli spaces have two main applications: classification of affine schemes with a "nice" group action and resolution of quotient singularities. Also I will give many examples and describe the main tools used to describe the geometry of the invariant Hilbert schemes.


Fabio Tonini

Title: Representations of the Nori fundamental gerbe
Abstract: The Nori fundamental group scheme of a scheme X with a rational point x is a profinite group scheme that "controls" torsors over X under finite group schemes with a trivialization on x. Interpreting a group scheme as a trivial gerbe, one can more generally associate a profinite gerbe, called the Nori fundamental gerbe, without the need of a rational point. In both cases Tannaka's duality assures those objects (affine group schemes of affine gerbes) are completely determined by their category of representations. In the talk I will give a description of the category of representations of the Nori fundamental gerbe by considering vector bundles with extra structure.


Angelo Vistoli

Title: Fundamental Gerbes
Abstract: Let $X$ be a connected and geometrically reduced variety over a field k, with a fixed rational point $x_0$ in $X(k)$. Nori defined a profinite group scheme $N(X,x_0)$, usually called Nori's fundamental group scheme, with the property that homomorphisms $N(X,x_0)$ to a fixed finite group scheme $G$ correspond to $G$-torsors $P \to X$, with a fixed rational point in the inverse image of $x_0 \in P$. If $k$ is algebraically closed of characteristic 0 this coincides with Grothendieck's fundamental group, but is in general very different.

Nori's main theorem is that if $X$ is complete, the category of finite-dimensional representations of $N(X,x_0)$ is equivalent to an abelian subcategory of the category of vector bundles on $X$, the category of essentially finite bundles.

Furthermore, Nori defined a similar fundamental groups with unipotent group schemes, whose category of finite-dimensional representations is equivalent to the category of vector bundles admitting a filtration with trivial quotients.

In my talk I will discuss my joint work with Niels Borne, in which we remove the dependence on the base point, substituting Nori's fundamental group with a profinite gerbe, and simplify and generalize the proof of the correspondence between essentially finite bundles and representations of this gerbe. Furthermore I will explain a systematic approach for finding fundamental gerbes for several classes of group schemes, going well beyond the finite or unipotent cases.


Jonathan Wise

Title: TBA
Abstract: TBA