Workshop
Alps 2024
Sainte-Foy-Tarentaise
(8-12 January)
Workshop
Alps 2024
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| Abstract
: BV formalism is a theory that arise in theoretical physics
to quantize field theories. In this talk I will try to explain
its link with derived geometry and how can we see it as a kind
of derived symplectic reduction. For any smooth manifold with
an action functional, Felder and Kazhdan have solved the
classical master equation building an object they call a "BV
variety". I will propose to see their construction, firstly as
a construction of a symplectic L_∞-algebroid on the
(classical) critical locus of the action and secondly as a
symplectic (coisotropic) reduction. I will then try to explain
why the construction is universal among reductions. References:
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| Abstract:
The
interaction of Hodge structures with rational homotopy
theory is a powerful tool to provide restrictions on the
homotopy types of Kähler manifolds and of complex algebraic
varieties. An example is the well-known result of Deligne,
Griffiths, Morgan and Sullivan, stating that compact Kähler
manifolds are formal. In the simply connected case, it
implies, for instance, that the rational homotopy groups of
such manifolds are a formal consequence of the cohomology.
Despite this fact, the mixed Hodge structure on their
rational homotopy groups is not, in general, a formal
consequence of the Hodge structures on cohomology. To
understand this phenomenon, we will introduce a stronger
notion of formality which arises from studying homotopy
theory in a category encoding the Hodge structures. We will
also introduce obstructions to this strong formality,
generalizing the classical ones, and study when are Kähler
manifolds formal in this stronger sense. References:
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| Abstract:
For complex manifolds, there exists a refined notion of
weak equivalence related to both Dolbeault and
anti-Dolbeault cohomology. This class of weak equivalences
naturally defines a stronger formality notion. In
particular, the ddbar-Lemma does not imply formality in
this new sense. The goal of this talk is to introduce a
novel operadic framework designed to understand this
homotopical situation. I will present pluripotential
A-infinity algebras as well as a homotopy transfer
theorem, and will discuss a potential Koszul duality
theory in the pluripotential setting. |
| Abstract: The Lambrechts-Stanley model is a proposed model for the rational homotopy type of ordered configuration spaces of manifolds. In the particular case of a formal manifold, this model is simply the E_2-page of the Leray-Serre spectral sequence for the inclusion of the configuration space in the product. It was proved Kriz that this is indeed a model for smooth projective algebraic varieties over the complex numbers by constructing an explicit quasi-isomorphism with another model built from the Fulton-MacPherson compactification. It was later proved by Idrissi that the real Lambrechts-Stanley model is indeed a model for simply connected closed manifolds. I will explain some ongoing work with Joana Cirici in which we revisit Kriz's proof using the tool of E_1-formality. Our method lets us also prove that the Lambrechts-Stanley model works for certain algebraic varieties in p-adic homotopy theory in the "tame range" (i.e. in degrees low enough that the Steenrod operations don't create problems). |
| Abstract:
I will explain recent developments about the formality or
non-formality of the Swiss-Cheese operad and its potential
uses. In particular, I will present joint work with R. V.
Vieira about the non-formality of Voronov's version of the
Swiss-Cheese operad, as well as our current work in
progress about a conjectural reinterpretation in terms of
symmetric Massey products. References:
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| Abstract: I will present two ongoing projects on a connection between E_n-algebras and some discrete models in statistical physics and quantum field theory (one project is with Victor Carmona, and the other one with Damien Lejay). |
| Abstract:
Upon hearing the last words from the speaker, the
chairman slowly got up and started a round of applause.
As the claps diminished, an uncomfortable silence filled
the room. He tried to look, in vain, for any raised
hands. Awkwardly, he asked “are there any questions?”,
knowing beforehand that there would be none. He looked
at the speaker in pure anguish, unable to find a single
question that could fit with the theme of talk.
Suddenly, he remembered the good old trick, the one that
had saved him in so many occasions. And confidently, he
asked: “But do you think this could work in positive
characteristic?” – The goal of the talk will be to
answer this question.” |
| Abstract: Une structure d'algèbre différentielle graduée A (e.g. une algèbre associative, une algèbre de Lie, une opérade, etc.) est formelle si elle est reliée à son homologie H(A) par un zig-zag de quasi-isomorphismes préservant le type de structure algébrique. Les classes de Kaledin ont été introduites comme une théorie de l'obstruction caractérisant entièrement la formalité des algèbres associatives sur un corps de caractéristique nulle. Dans cet exposé, je présenterai une généralisation des classes de Kaledin à n'importe quel anneau de coefficients, mais également à d'autres structures algébriques (encodées par des opérades, éventuellement colorées, ou par des propérades). Je démontrerai de nouveaux critères de formalité basés sur ces classes et en donnerai des applications. |
| Abstract: I will try to explain how one can encode more of the geometry of manifolds via new operadic structures on the spaces of differential forms of manifolds. There will notably be two new "dual" differential graded operads: one for Poisson manifolds and one for manifolds equipped with a non-degenerate metric (Riemannian, Minkovsky). I will try to treat their respective homotopy property (resolutions, homotopy transfer theorem, etc.) and applications. |
| Abstract:
I will explain how almost Hermitian and Hermitian metrics
give rise to homotopy BV algebras satisfying the Hodge to de
Rham degeneration condition. These results extend in some
sense the BV algebra introduced by Koszul for Poisson
manifolds. I will compute part of the induced homotopy
hypercommutative structures on various geometric examples
and discuss possible conjectures connected to homological
mirror symmetry and to formality. This is a work in progress
together with Scott Wilson. |