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Nouveau séminaire ALPE

(Au-delà de la Ligne de Partage des Eaux)

organized by Tristan Bozec (IMAG) and Joost Nuiten (IMT)

Next meeting

- Date: 29-30 March 2023. 

- Place: IMAG (direction), campus Triolet, building 9, room 109.

- Speakers (tentative list): Gabriel Angelini-Knoll (Paris Nord), Bérénice Delcroix-Oger (Montpellier), Elena Dimitriadis Bermejo (Toulouse), Lander Hermans (Antwerp), Renata Picciotto (Angers), Špela Špenko* (Bruxelles)

More information coming soon.

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Previous meetings :

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- Date: 27-28 June 2022.

- Place: IMT (direction), campus Université Paul Sabatier, salle Pellos (building 1R2)

 

- Schedule


- Program

Eloise Hamilton: 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 C^n and curve singularities.

Ludovic Monier: 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.

Guglielmo Nocera: 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 RN.

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 RN. 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).

 

Morgan Opie: Classification and construction of rank 3 vector bundles on CP^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 CP^5.  

The previous interesting case is that of rank 2 bundles on CP^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 CP^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.

Maria Yakerson: 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.

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- Date: 16-17 March 2022. 

- Place: IMAG (direction), campus Triolet, building 9, room 109.

 

- Schedule

- Program

Joost Nuiten: Lie algebroids as curved Lie algebras.

Differential graded Lie algebras play an important role in deformation theory as algebraic objects classifying the infinitesimal neighbourhoods of moduli spaces around a basepoint. An informal principle asserts that geometric objects without a fixed basepoint should admit a similar description in terms of curved Lie algebras, which have a `differential' whose square is controlled by a curvature element. In this talk, I will discuss the relation between two algebraic models for the formal neighbourhood of a moduli space around a variety, rather than around a single point: in terms of dg-Lie algebroids and in terms of curved Lie algebras over the de Rham complex. In particular, I will describe an embedding of the homotopy category of dg-Lie algebroids into the homotopy category of such curved Lie algebras. Joint work with Damien Calaque and Ricardo Campos.

Ben Davison: The decomposition theorem for moduli stacks of objects in 2-Calabi-Yau categories.

Given (a certain kind of) 2CY structure on a category C, its derived moduli stack of objects acquires a 0-shifted symplectic structure.  Via a general formality result for such categories, if the underlying classical stack has a good moduli space, the stack may be etale-locally modelled as the stack of representations of a preprojective algebra.  Furthermore, it is possible to show that the derived direct image of the dualizing complex along the morphism to the good moduli space satisfies the famous BBDG decomposition theorem, and is furthermore pure, when considered as a mixed Hodge module.  I will explain all this, as well as applications to Kac polynomials and (time permitting) nonabelian Hodge theory for stacks.

Nields Feld: Quadratic refinements in motivic homotopy theory and non-commutative geometry.

In the nineties, Voevodsky proposed a radical unification of algebraic and topological methods. The amalgam of algebraic geometry and homotopy theory that he and Fabien Morel developed is known as motivic homotopy theory. Roughly speaking, motivic homotopy theory imports methods from simplicial homotopy theory and stable homotopy theory into algebraic geometry and uses the affine line to parameterize homotopies. Voevodsky developed this theory with a specific objective in mind: prove the Milnor conjecture. He succeeded in this goal and won the Fields Medal for his efforts in 2002.
In this talk, we will start by recalling some facts in motivic homotopy theory, and we then present some results in motivic enumerative geometry (Euler characteristic, base change theorem, trace formula, ramifications). Finally, we will try to generalize these results by using ideas coming from non-commutative geometry.

Ángel González-Prieto: Quantization of parabolic character varieties and interference phenomena.

The algebraic structure of moduli spaces of representations of surface groups (aka character varieties) has been widely studied in the past decades, partially due to their close relation with the moduli spaces of Higgs bundles and flat connections. Nevertheless, very little is known about the geometry of character varieties when we allow poles in the Higgs field, the so-called parabolic setting. In this framework, new singularities arise in the moduli space that prevent the classical methods to work.

In this talk, we will introduce a new hope. We will construct a TQFT that encodes the Grothendieck motives of parabolic character varieties and we will apply it to obtain explicit expressions of these motives, even with highly non-generic parabolic data. This framework also provides a new interpretation of the singularities: at the side of the TQFT they arise as an interference phenomenon that leads to drastic changes in the geometry.

Corina Keller: Generalized character varieties and quantization via factorization homology.

 

Factorization homology is a local-to-global invariant which "integrates" disk algebras in symmetric monoidal higher categories over manifolds. In this talk I will focus on a particular instance of factorization homology on surfaces where the input algebraic data is a braided monoidal category. If one takes the representation category of a quantum group as an input, it was shown by Ben-Zvi, Brochier and Jordan (BZBJ) that categorical factorization homology provides a quantization of the Fock-Rosly Poisson structure on the classical G-character variety. I will discuss two applications of the factorization homology approach for quantizing (generalized) character varieties. First, I will explain how to compute categorical factorization homology on surfaces with principal D-bundles decorations, for D a finite group. The main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We will see that in this case factorization homology gives rise to a quantization of Out(G)-twisted character varieties (This is based on joint work with Lukas Müller). In a second part we will consider surfaces that are decorated with marked points. It was shown by BZBJ that the algebraic data governing marked points are braided module categories and I will discuss an example related to the theory of dynamical quantum groups.

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The previous meetings were organized by the ERC project NEDAG: see here and here

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