Séminaire ALPE

27 juin 2022, 14:00 → 28 juin 2022, 15:00 Europe/Paris

IMT 1R2 207 (Salle Pellos)

Séminaire en commun entre l'IMAG de Montpellier et l'IMT de Toulouse autour de la géométrie et topologie algébriques.

Détails pratiques sur la page web du séminaire.

lundi 27 juin

1 On the cohomology of Quot schemes of infinite affine space

Hilbert schemes of smooth surfaces and, more generally, their Quot schemes are well-studied objects, however not much is known for higher dimensional varieties. In this talk, we will speak about the topology of Quot schemes of affine spaces. In particular, we will compute the homotopy type of certain Quot schemes of the infinite affine space, as predicted by Rahul Pandharipande. This is joint work in progress with Joachim Jelisiejew and Denis Nardin.

Orateur: Maria Yakerson (ETH Zürich)

2 Graded loop spaces and de Rham-Witt algebra

This talk will focus on the study of a variation of the graded loop space construction for mixed graded derived schemes endowed with a Frobenius lift. We developed a theory of derived Frobenius lifts on a derived stack which are homotopy theoretic analogues of δ-structures for commutative rings. This graded loop space construction is the first step towards a definition of the de Rham-Witt complex for derived schemes. In this context, a loop is given by an action of the "crystalline circle", which is a formal analogue of the topological circle, endowed with its natural endomorphism given by multiplication by p. In this language, a derived Dieudonné complex can be seen as a graded module endowed with an action of the crystalline circle.

Orateur: Ludovic Monier (Ecole Normale Supérieure)

mardi 28 juin

3 Whitney stratifications and conically smooth structures

A classical problem in geometry is the following: can we stratify non smooth spaces (e.g. algebraic varieties, quotients of smooth manifolds by group actions, ...) into smooth strata, in such a way that good “equisingularity conditions” for strata are matched? The notion of Whitney stratification arises to give an answer to this question, and indeed algebraic varieties, analytic varieties, semialgebraic sets and semianalytic sets admit a Whitney stratification. The notion of Whitney stratification is not intrinsic, in that it depends on an embedding of the space in $\mathbb{R}^N$.

On the other hand, the notion of conically smooth structure was introduced by Ayala, Francis and Tanaka in 2017. We will explain how this latter notion is an “intrinsic” version of specifying a Whitney stratification, i.e. it does not depend on the choice of an embedding into $\mathbb{R}^N$. In particular, we show that a Whitney stratified space always admits a canonical conically smooth structure. If time permits, we will provide an application of this result: namely, the affine Grassmannian associated to a reductive group, which is a fundamental object in the Geometric Langlands Program, is a conically smooth space. We will also illustrate other basic features of the notion of conically smooth structure.

This is joint work with Marco Volpe (University of Regensburg).

Orateur: Guglielmo Nocera (Scuola Normale Superiore Pisa)

4 An overview of Non-Reductive Geometric Invariant Theory

Geometric Invariant Theory (GIT) is a powerful theory for constructing and studying the geometry of moduli spaces in algebraic geometry. In this talk I will give an overview of a recent generalisation of GIT called Non-Reductive GIT, and explain how it can be used to construct and study the geometry of new moduli spaces. These include moduli spaces of unstable objects (for example unstable Higgs/vector bundles), hypersurfaces in weighted projective space, k-jets of curves in $\mathbb{C}^n$ and curve singularities.

Orateur: Eloise Hamilton (Cambridge University)

5 Classification and construction of rank 3 vector bundles on $\mathbb{C}P^5$

Finding complete invariants for (unstable) complex vector bundles on complex projective spaces is a surprisingly subtle problem -- even for low-rank bundles in the topological category. In this talk, I will give a solution to this problem for rank 3 bundles on $\mathbb{C}P^5$.

The previous interesting case is that of rank 2 bundles on $\mathbb{C}P^3$, solved in the 70s by Atiyah--Rees via a KO-theory valued invariant of rank 2 bundles. I show that rank 3 bundles on $\mathbb{C}P^5$ with the same Chern classes are distinguished by an invariant of rank 3 bundles with values in the generalized cohomology theory of 3-local topological modular forms. I will explain how the Atiyah--Rees invariant and my invariant are analogous, at least from the point of view of chromatic homotopy theory. I will also discuss a method for constructing vector bundles, which can be viewed as a topological analogue of an algebraic construction due to Horrocks.

At present, my theorems all are for topological vector bundles; however, there are algebraic analogues for many of the questions of interest. As time allows, I will discuss future algebro-geometric directions for this project.

Orateur: Morgan Opie (UCLA)