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Séminaire ALPE

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

organized by Tristan Bozec (IMAG), Ricardo Campos (IMT) and Joost Nuiten (IMT)

June 2023

March 2023

June 2022

March 2022

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

June 2023

- Date: 6-7 June 2023.

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

- Schedule

  • 6 June, 14h30-15h30, Alex Takeda (IHES)
       coffe break

  • 6 June, 16h00-17h00, Alex Vitanov (Montpellier)
       dinner

  • 7 June, 9h30-10h30, Victor Carmona (Seville)
       coffe break

  • 7 June, 11h00-12h00, Yuqing Shi (Utrecht)
       lunch

  • 7 June, 14h15-15h15, Anna Barbieri (Verona)

- Program

Anna Barbieri: Moduli spaces of Bridgeland stability conditions and of multi-differentials

The space of Bridgeland stability conditions is a complex manifold attached to a triangulated category. It parametrises t-structures of the category and it is morally interpreted as the mathematical counterpart of a moduli space of conformal field theories. In some interesting cases, it is isomorphic to a moduli space of quadratic differentials on a Riemann surface. I will review this correspondence, which is due to Bridgeland-Smith in the simple zeroes case, and was extended in [BMQS] to (almost) arbitrary critical profiles, and discuss applications, in particular a tentative construction of a compactification of the stability manifold. This is based on joint works (partially in progress) with M.Moeller, Y.Qiu, and J.So.

Victor Carmona: Enveloping operads; one construction to govern them all

Enveloping algebras have been the focus of numerous studies and a possible reason for that is their appearance in different fields of mathematics and mathematical physics. In many of those cases, the universal enveloping algebra construction is restricted to Lie-algebras, but enveloping algebras are a general device that can be considered for a quite broad notion of algebra. Furthermore, enveloping algebras are just the unary part of a more general construction, enveloping operads. While this operadic enhancement has been of crucial importance in work of Berger-Moerdijk, Fresse, Harper, Muro, Pavlov-Scholbach, White-Yau, etc, it has not been given the attention that it deserves. It is curious that a quick search of “universal enveloping algebra” in Mathscinet gives back around 3200 matches, whereas searching “enveloping operad” just gives 42 matches.  

In this talk, we are going to discuss all these enveloping notions and present current work on applications of enveloping operads that will hopefully convince the audience about the potential of these objects.

Yuqing Shi: Higher enveloping algebras in monochromatic layers

The universal enveloping algebra functor assigns to a Lie algebra a unital associative algebra, characterised by the property that it sends free Lie algebras to free associative algebras. A spectral Lie algebra in an algebra over the so-called spectral Lie operad in the infinity category of spectra, generalising the classical notion of Lie algebras. Ben Knudsen generalises the universal enveloping algebra construction to a functor from the infinity category of spectral Lie algebras to the infinity category of augmented spectral E_n-algebras for any natural number n, known as the higher enveloping algebra functor. Recall that an E_1-algebra is an associative algebra up to coherent homotopies. A monochromatic layer of height h is the localisation of the infinity category of spectra, where the so-called v_h-periodic equivalences are inverted, generalising the rational localisation of spectra. In my talk, I will first introduce the construction of higher enveloping algebras, using a different approach than that of Knudsen. Then I will discuss a relationship between spectral Lie algebras and E_n algebras in the monochromatic layer of height h, which is exhibited by these higher enveloping algebra functors. This is a joint work with Gijs Heuts.

 

Alex Takeda: Pre-Calabi-Yau structures, string topology and the categorical formal neighborhood of infinity

Pre-Calabi-Yau structures are noncommutative versions of Poisson structures appearing in homological mirror symmetry, and can be used to describe certain types of TQFT operations on Hochschild homology. In this talk, I will describe how to use these structures to give a certain algebraic model for the string topology of non-simply connected manifolds; this method uses Efimov's notion of the "categorical formal punctured neighborhood of infinity", a noncommutative version of perfect complexes near the divisor at infinity in a compactification. This talk is based on a current joint project with Manuel Rivera and Zhengfang Wang.

Alex Vitanov: Ring structure on the Chen-Ruan and Hochschild cohomology of a quantized symplectic quotient orbifold

I explain the ring structure on the Chen-Ruan cohomology and on the Hochschild cohomology of a quantized symplectic global quotient orbifold and discuss an ongoing work with Damien Calaque related to a conjecture by Viktor Ginzburg and Dmitry Kaledin, which claims the existence of a ring isomorphism between both cohomology rings.

March 2023

- Date: 29-30 March 2023. 

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

- Schedule

- Program

 

Gabriel Angelini-Knoll: Deformations of equivariant homotopy theory


Deformations of stable homotopy first arose in the setting of motivic homotopy theory over the complex numbers. In this context, they have been used to make significant progress on computations of the stable homotopy groups of spheres by work Isaksen—Wang—Xu. There is an analogous relationship between real motivic homotopy theory and C_2-equivariant homotopy theory by work of Burkland—Hahn—Senger, but this relies on the existence of the real bordism spectrum, which is not available for other groups of equivariance. In joint work with M. Behrens, E. Belmont, and H. J. Kong, we provide a deformation of the Borel equivariant homotopy category that applies to any finite group. From another point of view, we construct a replacement the C_2-effective slice filtration for an arbitrary finite group after Borel completion. Even in the case of C_2, our filtration has some benefits in the sense that differentials in our spectral sequence can be determined from the attaching maps of projective space and Adams spectral sequence differentials.

Bérénice Delcroix-Oger: From pre-Lie to post-Lie operads through hypertrees

Hypertrees are some generalizations of trees where edges are not just pairs but sets of vertices. They can be ordered by refinement.The resulting posets of  hypertrees were introduced by McCullough and Miller in 1996 in terms of labeled bipartite trees and described in terms of hypertrees by McCammond and Meier in 2004. Following a conjecture of Chapoton, it was shown in 2013 that the action of the symmetric group on its homology is the same as the one on the underlying vector spaces of pre-Lie operad.

This framework is really reminiscent of the one of partition posets. The posets of partitions of a fixed finite set, ordered by refinement, have been widely studied since the 1980s. Stanley and Hanlon computed the action of the symmetric group on their homology and Joyal pointed out the link with Lie operad. It was finally proven at the beginning of the 2000's by Fresse that this correspondence follows from the theory of Koszul operads, since the chain complex of the poset is exactly the leveled bar construction of the Comm operad.

In a joint work with Clément Dupont (IMAG, Montpellier), we investigated the link between hypertree posets and pre-Lie operad. Surprisingly, we uncover an operad structure on nested sets of hypertree posets which is isomorphic to post-Lie operad and a left-post-Lie-module structure on the pre-Lie operad explaining the previous results. 

Elena Dimitriadis Bermejo: tba

Lander Hermans: Deforming prestacks: a calculus of rectangles

In his foundational work Gerstenhaber furnishes the guiding example for algebraic deformation theory: for an associative algebra A he defined a dgLie bracket on its Hochschild complex and showed that it controls the deformations of A through the Maurer-Cartan equation. Algebraic geometry motivates the natural question whether a similar story exists for presheaves of associative algebras, e.g. when applied to the structure sheaf of a scheme. In this talk I will explain how the Gerstenhaber-Schack complex fulfils this role, yet also motivates to generalize from presheaves to prestacks (i.e. pseudofunctors) as the most suitable objects to start with. By means of elementary drawings of rectangles, I will explain both the new differential on the GS-complex, established by Dinh Van-Lowen, and the new L-infinity-structure we recently obtained, thus completing the story: the higher Lie brackets on the GS complex control the deformations of prestacks through the generalized MC equation.

Renata Picciotto: tba

Špela Špenko: HMS symmetries of toric boundary divisors

Let X be a complex manifold. By homological mirror symmetry one expects an action of the fundamental group of the "moduli space of Kähler structures" of X on the derived category of X. If X is a crepant resolution of a Gorenstein affine toric variety we obtain an action on the derived category of the toric boundary divisor of X which leads to an action on the Grothendieck group of X. This is a joint work with Michel Van den Bergh.

June 2022

- Date: 27-28 June 2022.

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

 

- Schedule

  • 27 June, 14h00-15h00, Maria Yakerson (Zurich) - slides
       coffe break

  • 27 June, 15h30-16h30, Ludovic Monier (Toulouse) - slides
       dinner

  • 28 June, 9h30-10h30, Guglielmo Nocera (Pise)
       coffe break

  • 28 June, 11h00-12h00, Eloise Hamilton (Cambridge)
       lunch

  • 28 June, 14h00-15h00, Morgan Opie (UCLA)


- 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.

March 2022

- Date: 16-17 March 2022. 

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

 

- Schedule

  • 16 March, 14h30-15h30: Joost Nuiten (Toulouse)
      coffee break

  • 16 March, 16h00-17h00: Ángel González-Prieto (Madrid) - slides
       dinner

  • 17 March, 10h00-11h00: Corina Keller (Montpellier)
       coffee break

  • 17 March, 11h30-12h30: Niels Feld (Toulouse)
     
    lunch

  • 17 March, 14h30-15h30 : Ben Davison (Edinburgh) - slides

- 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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