TBA

• Vistoli

Synthetic Betti realization

• Haine

Let X be a complex variety. In Deligne’s work on Hodge theory, he explained how to use the algebraic structure of X to endow the rational Betti cohomology H∗(X(C); Q) with a functorial increasing weight filtration. Deligne’s construction of the weight filtration is quite ingenious, and it isn’t a priori clear if the weight filtration can be refined to a filtration on the integral Betti cohomology H∗(X(C); Z). Much later, Gillet and Soulé showed that this is the case. In this talk, we’ll explain joint work with Piotr Pstrągowski that shows that for any complex-orientable cohomology theory A∗, there is a functorial weight filtration on A∗(X(C)) that recovers the Deligne and Gillet–Soulé filtrations in the cases of rational and integral cohomology. To do this, we give a new description of the the ∞-category SH(C) of motivic spectra over C as sheaves on a subcategory of compact pure motives; this generalizes Bondarko’s weight structure on MGL-modules. From this description we also refine Betti realization to a functor landing in Pstrągowski’s ∞-category of synthetic spectra, extending a construction of Pstrągowski for cellular motivic spectra to all motivic spectra.

Arcs with values in the projective line

• Maffei

Motivated by a construction of a work of Melani, Nocera and Vezzosi on G torsors on surfaces we study the space of arcs with values in the projective line.

On the Bloch conductor formula

• Beraldo

I will discuss a proof of a generalization of the Bloch conductor conjecture, obtained in joint works with Massimo Pippi. The proof exploits several innovative ideas of Bertrand Toen and Gabriele Vezzosi.

On Quillen’s finiteness conjecture for the cotangent complex in characteristic two

• Schürg

Quillen made two finiteness conjectures on the cotangent complex. The first is now known as Avramov’s theorem, the second is still open in general. In characteristic two, Turner proved Quillen’s second conjecture over a Cohen-Macaulay base. In the talk, I’ll outline a strategy to improve Turner’s approach to an arbitrary base using tools from derived algebraic geometry. This is work in progress.

TBA

• Robalo

Integral structures on Hodge atoms

• Pantev

I will describe a novel formalism for defining and specifying integral structures on the tt^*-geometries that parametrize massive vacua in N=2 QFTs. I will focus on non-archimedean variants of these structures which arise naturally in complex geometry and mirror symmetry and are known as F-bundles, and on their basic pieces known as Hodge atoms. I will explain a construction of the Gamma-corrected integral structure on the A-model non-archimedean F-bundle and will discuss the special features of the Gamma-conjecture that are important in this setting. This is a report on a joint work in progress with Katzarkov, Kontsevich and Yu.

TBA

• Scherotzke

We prove a Hochschild--Konstant--Rosenberg (HKR) theorem for arbitrary derived Deligne--Mumford (DM) stacks. To this aim, we introduce the notion of orbifold inertia stack of a derived DM stack; this supplies a derived enhancement of the classical inertia stack, which does not always coincide with the classical truncation of the free loop space. We show that, in characteristic $0$, given a derived DM stack, the shifted tangent bundle of its orbifold inertia stack is equivalent to its free loop space. As applications, we determine the Hochschild homology and Hochschild cohomology of interesting derived DM stacks, for instance, weighted projective lines, root stacks, quotients by algebraic groups, mapping stacks. This is joint work with Lie Fu, Mauro Porta and Nicolo Sibilla.

Poster Session

Categorical topological algebra: Tate vs solid

• Pourcelot

In formal geometry, one naturally encounters complete algebras of various kinds. This raises the question of how to deal with these topological objects, at the derived level. I will review two categorical frameworks which answer this question: - the first one is that of Tate modules (introduced by Drinfeld following Beilinson, and Hennion at the derived level); - the second is the theory of solid modules of Clausen-Scholze, as part of condensed mathematics. Then I will explain the rather subtle relation between the two (∞-)categories of modules. To conclude, I will mention some consequences and first steps towards the theory of shifted symplectic structures in such contexts. This talk reports on joint work with Valerio Melani and Gabriele Vezzosi.

Ext algebras of subvarieties, formal geometry, and formality theorems

• Calaque

In this talk, I will present ongoing work with Alex Vitanov aimed at describing the Ext algebra of the sheaf associated with a closed embedding of smooth algebraic varieties. I will first recall the case of the diagonal embedding, and then explain the two main tools we use: - methods from formal geometry in a relative setting; - a formality theorem in the presence of branes. If time permits, I will conclude with a conjecture (which actually constitutes the main motivation for this work).

Special Talk

Deformation of algebraic cycles and the motive of the affine line

• Elmanto

Bertrand and Gabriele introduced to the world a derived algebraic geometry way of thinking about coefficients for cohomology theories, starting with their treatment of chern characters, mixed graded complexes and sheaves on the loop space. Taking inspiration from this story, I want to present a way to describe the deformation theory of motivic cohomology and algebraic cycles using a non-representable version of Dieudonné theory which, in positive and mixed characteristics, takes as input prismatic F-gauges and produces modules over (a version of) the Raynaud ring; this was first understood by Ekedahl and later revisited by Jacob Lurie and Yuanning Zhang. Our main results generalize Bloch and Stienstra's calculation of the deformation functor of algebraic cycles and is a motivic refinement of McCandless' work on cyclotomic spectra and Cartier modules. This is joint work in progress with Shubhodip Mondal.

Height one phenomena in p-adic Hodge theory

• Moulinos

The syntomification of the integers (Z_p^syn) acts as a parameter space for p-adic cohomology theories. By the work of Drinfeld and others, it is known to carry a canonical formal group, and hence a map to the moduli stack of formal groups. Pulling back along this map gives rise to a certain line bundle equipped with a distinguished section "v_1". On the p-completed moduli stack of formal groups, the nonvanishing of v_1 defines the substack parametrizing "height-1" formal groups. Yet the preimage of this locus on Z_p^syn, when computed in p-adic formal stacks is empty. There is a mismatch here because inverting v_1 on syntomic cohomology gives p-adic étale cohomology, but inverting it on the stack itself before passing to global sections gives nothing. I will explain how passing to analytic stacks brings this missing locus into view. By realizing the p-completed moduli stack of formal groups as an analytic stack, we construct an algebro-geometric shadow of K(1)-localization. Applied to syntomification, this produces an analytic stack whose dualizable quasi-coherent sheaves recover p-adic Galois representations, and more generally lisse étale sheaves on the generic fiber of a p-adic formal scheme. Time permitting, I will explain how a perfected version of this construction depends only on the generic fiber, providing a geometric counterpart to purity results for K(1)-localized K-theory. This is joint work in progress with Greg Andreychev.

TBA

• Nocera

Algebraic infty-categories, Theta-structures and tannakian duality

• Toen

(Joint with Nuiten and Vezzosi) I'll explain the words in the title and state a general tannakian duality relating certain infty-stacks ("tannakian gerbes") with certain algebraic symmetric monoidal infty-categories with Theta-structures ("tannakian Theta-categories"). Possible applications to motives and to the "exponential homotopy theory".

Deformation theory of the moduli superstack of stable maps

• Pavia

Supergeometry is the natural mathematical framework that accommodates the notion of supersymmetry in theoretical physics and string theory. In the last dozen years, this field of research has found a renewed popularity after the foundational work of Donagi and Witten, who proved that super moduli spaces of super Riemann surfaces cannot be straightforwardly recovered and studied starting from their bosonic ( = classical) reduction. The goal of this talk is to give an overview on the recent developments of the study of the moduli superstack of (punctured) stable maps, generalizing some well known properties of the moduli stack of stable maps from classical algebraic geometry. In particular, we will study its deformation/obstruction theory, and compute its virtual dimension. This is based on joint work with Ugo Bruzzo, Daniel Hernández Ruipérez, and Andrea Ricolfi.

TBA

• Simpson