Mardi 26 Janvier 14h - Tristan Bozec (IMAG, Montpellier) : Lieux critiques relatifs et espaces de modules de carquois.

Dans cet exposé je donnerai un procédé pour construire des sous-variétés lagrangiennes de variétés carquois. Je m'inspirerai d'outils de géométrie symplectique pour définir de telles sous-variétés, a priori nouvelles, par exemple dans le schéma de Hilbert de points sur le plan. La construction généralise les algèbres différentielles graduées de Ginzburg (analogue 'dérivé' des algèbres préprojectives), et on verra que le pendant algébrique des variétés lagrangiennes consiste en des structures dites Calabi-Yau. Le travail reporté a été réalisé avec Damien Calaque et Sarah Scherotzke.

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Mardi 16 Février 13h - Matthew Morrow (IMJ, Paris) : K-theory of smooth algebras over perfectoids.

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In Niziol’s approach to p-adic Hodge theory via algebraic K-theory, the following lemma plays a crucial role as it implicitly links syntomic and etale cohomology: given a smooth algebra R over the ring of integers V of the algebraic closure of a p-adic field, then R and R[1/p] have the same p-adic K-theory. I will present some generalisations and variants of this result in which V is replaced by an arbitrary perfectoid ring. Joint work with Ben Antieau and Akhil Mathew.

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Mardi 2 Mars 14h - Benjamin Antieau (Northwestern) : Homological lifting, Gauss—Manin connections, and derived commutative rings.

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 I will speak about ongoing work seeking to answer the following question. Suppose that k is a commutative ring and that R is a smooth commutative k-algebra. If M is an R-module with integrable connection, when does M admit a `homological lift’? A homological lift is a module over HKR-filtered Hochschild homology in filtered complexes with filtered S^1-action which reduces to M after taking associated gradeds and `shearing down'. I will explain the theory of mixed homology and remixing which allows one to find such lifts in case k -> R has small fiber dimension. This generalizes work of Ben-Zvi—Nadler, Preygel, and Toën—Vezzosi in characteristic zero. Then, I will discuss the Gauss—Manin connection on Hochschild homology as another source of positive answers.

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Mardi 9 Mars 14h - Joachim Kock (UAB, Barcelona): Petri nets and processes.

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Petri nets are a very useful formalism for modelling processes, with applications in many different fields of science and engineering such as chemistry, epidemiology, computer science, and business and production modelling. Their operational semantics come in two main flavours: geometric (in terms of posets, graphs, and such), and algebraic (in terms of monoids, monoidal categories, etc.). People have struggled for many years to reconcile the two viewpoints, the problem being an issue with symmetries. In this talk I will explain how the problem can be overcome with the help of some elementary homotopy viewpoints. The new formalism for Petri nets is based on polynomial-style finite-set configurations and etale maps. The processes of a Petri net P are etale maps G -> P from graphs. The main result I want to arrive at is that P-processes form a symmetric monoidal Segal space, which is the free prop-in-groupoids on P, but most of the time will be spent just explaining Petri nets, markings, firings, and the token game.

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Mardi 16 Mars 14h - Elden Elmanto (Harvard) : On Real algebraic cycles and equivariant homotopy theory.

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The capitalized Real of the title refers to Atiyah's "Real K-theory" which amalgamates the spectra KO and KU, bringing equivariant homotopy theory into focus within geometry. I will explain the beginnings of a similar program in algebraic geometry over the real numbers, starting with a reconstruction of the category of C_2 spectra purely in algebro geometric terms and the construction of a cycle class map valued in Bredon cohomology with coefficients in the Burnside ring (with Shah) and a study of zero cycles in this context (with Asok).

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Mardi 30 Mars 14h - Felix Wierstra (Amsterdam) : Models for non-simply-connected homotopy theory.

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In algebraic topology one often tries to model spaces by algebraic objects. This has been done in many interesting cases, but there are almost always strong assumptions on the fundamental group of the spaces, it can be nilpotent at best. In this talk I explain how some of these results can be extended to non-simply-connected spaces. Recently, Manuel Rivera and Mahmoud Zeinalian have shown that it is possible to extract the fundamental group of a space from its singular chains. This remarkable discovery makes it possible to model non-simply-connected spaces. In this talk, I will explain how the simplicial coalgebra of chains determines a space one prime at a time. In particular, I will show that the fiberwise Bousfield localizations of two connected spaces are weakly equivalent if and only if the cocommutative coalgebras of chains are equivalent in some sense called cobar-quasi-isomorphism. I'll further explain that when the spaces have universal covers which are of finite type, then the spaces are weakly homotopy equivalent if and only if the simplicial chains with integer coefficients are Omega-quasi-isomorphic. This is joint work with Manuel Rivera and Mahmoud Zeinalian.

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Mardi 6 Avril 14h - Joana Cirici (Barcelona) : Galois actions, purity and formality with torsion coefficients.

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The notion of formality first arose in rational homotopy theory, but it makes sense in any context in which chain complexes interact with multiplicative structures and has multiple applications beyond its original purpose. The idea that purity implies formality goes back to Deligne, Griffiths, Morgan and Sullivan, who used the Hodge decomposition to show that compact Kähler manifolds are formal over the rationals. Following the ideas behind Deligne's philosophy of weights, I will explain how to use Galois actions on étale cohomology to study formality with torsion coefficients for algebraic structures associated to certain schemes defined over a finite field. As an application, I will review results for configuration spaces on the complex space and for the operad of little disks. This is joint work with Geoffroy Horel.

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Mardi 20 Avril 16h - Arpon Raksit (Stanford) : Thoughts on the topological filtered circle

Moulinos–Robalo–Toën and I introduced (from two different perspectives) something called "the filtered circle": this is an object in derived algebraic geometry that refines and gives rise to the HKR filtration on Hochschild homology and related filtrations on the various forms of cyclic homology. It is natural to wonder what this idea can buy you in the "topological" setting, and this talk will be a report on ongoing investigation into this matter. I do not know how to recover the BMS filtrations, but I'll discuss a variant thereof (with two main differences: (1) the variant makes sense for a larger class of inputs; (2) the BMS filtrations are definitely significant, while I don't know whether the variant is significant or not.)

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Mardi 11 Mai 14h - Adeel Khan (IHES) : Microlocal analysis on singular spaces

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I will describe a way to use derived geometry to generalize microlocal sheaf theory to singular spaces.  This has some interesting interactions with the shifted symplectic geometry of Pantev-Toën-Vaquié-Vezzosi, and time-permitting I will mention some applications to Donaldson-Thomas theory obtained in joint work in progress with Tasuki Kinjo.
 

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Mardi 18 Mai 14h - Yonatan Harpaz (Paris 13): The Massey vanishing conjecture for number fields

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Higher Massey products are operations which give information about the higher coherence structure of differential graded rings. When applied in the context of Galois cohomology with prime coefficients, these operations are conjectured to vanish (in a suitable sense) by Minác and Tân. This conjecture is known for triple Massey products and
for all higher Massey products in the case of local fields by work of Matzri, Efrat and Minác--Tân. In joint work with Olivier Wittenberg we prove the conjecture for all higher Massey products in the case of number fields. Our strategy involves using recent techniques concerning the local-global principle for rational points, applied to a suitable Diophantine avatar of the original problem.

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Mardi 25 Mai 14h - Carlos Simpson (Nice) : Apprentissage par machine des preuves: l'étude des preuves de la classification des F_1-algèbres nilpotents.

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Nous examinons l'application de "Deep learning" à la question de guidage d'une preuve. Il s'agit d'une question de Reinforcement Learning, sujet dans lequel des progrès substantiels ont été apportés récemment ("Alpha Go" etc). Ici on va l'appliquer à la question du choix des coupures dans une preuve de classification des semigroupes nilpotents de petit ordre. On montrera que la machine peut en effet apprendre à donner une bonne preuve.

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(Pour assister, merci de me contacter).

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Contact: bertrand.toen@math.univ-toulouse.fr