Abstract. Given a complex algebraic variety X, a classical problem in algebraic geometry is to determine the real forms of X, where a real form is an algebraic variety Y which becomes isomorphic to X over the complex numbers. We are interested in a variant of this problem, in the case where X is a G-variety, i.e., an algebraic variety equipped with the action of a complex algebraic group G, and Y is an F-variety defined over the real numbers, for some real algebraic group F. Here we require that over the complex numbers, the algebraic groups F and G should become isomorphic, and X and Y should become G-equivariantly isomorphic. After defining these notions and illustrating them with examples, I will consider the special case where G = SL2 and X is an almost homogeneous SL2-threefold. This means that X contains an open dense SL2-orbit. These threefolds are well known in algebraic geometry; for example, they come up in the classification of Fano varieties and in the study of algebraic subgroups of the Cremona group. I will explain how we can classify their equivariant real forms. I will also discuss the connection between equivariant and non-equivariant real forms ("non-equivariant" means that we disregard the SL2-action).

Abstract. Lattice cohomology is an invariant of complex surface singularities introduced by Nemethi and inspired by Ozsvath and Szabo's work on Heegaard Floer homology. I will give an introduction to lattice cohomology and survey some recent work aimed at expanding the theory to higher dimensions.

Abstract. Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters needed to define it. We will explain how valuations can be used to prove lower bounds on the essential dimension of Galois cohomology classes of a field (with an emphasis on the Brauer group). If time permits, analogous results for Witt classes of Hermitian forms generalizing a theorem of Chernousov and Serre will be discussed. This is joint work with Zinovy Reichstein.

Abstract. A Kashiwara crystal is a combinatorial gadget associated to a representation of a reductive algebraic group that enables us to understand the structure of the representation in purely combinatorial terms. We will describe a type-independent combinatorial construction of crystals of the form $B_w(n\lambda)$, using the heap associated to a fully commutative element $w$ in the Weyl group. Then we will discuss how we can use the heap to also define a module for the preprojective algebra of the underlying Dynkin quiver. Using the work of Savage and Tingley, we also realize the crystal $B_w(n\lambda)$ via irreducible components of the quiver Grassmannians of n copies of this module, and we describe an explicit crystal isomorphism between the two models. This is joint work with Anne Dranowski, Joel Kamnitzer and Calder Morton-Ferguson.

Abstract. Given a family of smooth curves over a punctured disk, to extend this family to a family over (a ramified cover of) the whole disk, we have to allow the central fiber to be a stable nodal curve. If we want to extend families of curves together with some “additional data” (e.g. a vector bundle over the family, a fibration in elliptic curves over the family, G-torsors, etc), then the problem becomes more complicated, and in general stable nodal curves are not enough (for example, in the case of curves with a line bundle, or admissible covers). In this talk I will present a result that says that, if we allow the central fiber to be a nodal twisted curve whose coarse space is quasi stable (i.e. it might contain rational components with two marked points), then we can always extend the original family of curves plus “additional data” (e.g. vector bundle, fibration, G-torsor,…) to a family over the whole disk, as long as the “additional data” is given by maps to quotient stacks admitting a projective good moduli space. Moreover, the algorithm for extending this family is explicit. This is joint work with Andrea Di Lorenzo.

Abstract. In this talk, I will first describe how the classical Dieudonne module of finite flat group schemes and p-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then I will explain how to classify finite flat group schemes over a fairly general base by using classifying stacks and the formalism of prismatic F-gauges due to Drinfeld and Bhatt--Lurie.

Abstract. In this expository talk, let X be a smooth complex projective variety and V a holomorphic vector field on X such that 0 < |zero(V)| < \infty. Then the cohomology algebra H*(X) over C can be recovered from local data near the zero scheme Z of V. More precisely, when the upper triangular subgroup B of SL(2,C) acts on X with exactly one fixed point and V arises from the unipotent subgroup of B, then H*(X) is exactly the coordinate ring A(Z). Here the grading is given by a natural action of the diagonal torus T in B on A(Z). Moreover, the T-equivariant cohomology H*_T(X) is obtained from the union of all B-stable curves in X x P^1 for the natural action of B on P^1. In a different vein, one can ask about singular subvarieties Y of X to which V is tangent. Here, the natural result would be that H*(Y) is isomorphic to A(Y\cap Z), where Y\cap Z is the schematic intersection of Y and Z. In fact, there is a general result in this direction which we will describe if time permits. A lovely illustration of this is given by the flag variety F(n) of GL(n,C) where, using a certain B-action, it has been shown that for any B-stable Schubert variety Y in F(n), H*(Y) is isomorphic to A(Y\cap Z).