Oren Ben-Bassat

What is a p-adic TQFT?

I will report on joint work with my postdoc Nadav Gropper (University of Haifa, University of Pennsylvania). I will describe some p-adic, arithmetic "cobordism" categories and explain a correspondence between TQFTs mapping from these to vector spaces and extended Frobenius algebras. In the example of finite group gauge theory, I will do some explicit computations in the 2-dimensional case and explain the connection to Galois groups of finite extensions of the p-adic rational numbers. At the end, I will comment on how derived analytic geometry can be used to understand the continuum limit converging towards the case of p-adic Lie gauge groups.

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Dario Beraldo

On Bloch conductor conjecture

Let X-->S be a family of algebraic varieties parametrized by a trait S, possibly of mixed characteristic. Bloch conductor formula is a conjecture (proved in certain cases) that describes the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture: our methods use derived and non-commutative algebraic geometry. This is a joint work with Massimo Pippi.

Geometric Langlands correspondence

This talk is a continuation of Nick Rozenblyum's talks.

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Emma Brink

Condensed Group Cohomology

For a group object in an ∞-topos, group cohomology can be defined as (derived) fixed points. In my talk, I will compare condensed group cohomology of a topological group with its continuous group cohomology (defined via continuous cochains).
The theories coincide for locally profinite groups and solid (e.g. locally profinite) coefficients. But in general, condensed group cohomology is a more refined invariant.
On solid coefficients with trivial G-action, condensed group cohomology can be identified with the condensed cohomology of a classifying space of principal G-bundles, which agrees with its sheaf cohomology in many cases.
Although different from condensed group cohomology, continuous group cohomology with solid coefficients can be described as Ext groups in the condensed setup for a broad class of groups.

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Sarunas Kaubrys

Exponential map and Donaldson-Thomas theory

For a derived Artin stack Pantev-Toen-Vaquie-Vezzosi have defined the notion of a (n)-shifted symplectic structure, generalizing usual symplectic forms on smooth varieties. To any (oriented) (-1)-shifted symplectic stack one can attach a perverse sheaf called the DT sheaf. This sheaf can be viewed as a categorification of the classical Donaldson-Thomas invariants of the moduli space of coherent sheaves on a 3-Calabi-Yau variety. If X is a (0)-symplectic stack, one can consider the shifted cotangent bundle T^*[-1]X or the loop stack LX=Map(S^1,X), both of which have a natural (-1)-shifted symplectic structure. There is a natural exponential map between the (formal completions of) the (-1)-shifted cotangent bundle and the loop stack. In this talk I explain how this exponential map preserves the given symplectic structures. As an application, this leads to a computation of the cohomology of the DT sheaf of the moduli space of local systems on the product of a Riemann surface times S^1. In the special case of the 3 torus this also leads to a topological mirror symmetry type result for SLn and PGLn invariants.

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David Kern
slides

Higher-categorical logarithmic structures from higher Brauer groups

I will explain how to obtain the notion of derived log structures from only the Picard stack, using the language of internal colimits (in the topos of derived stacks) seen as internal gradings. Generalising to the higher Brauer stacks, we can define a concept of logarithmic structures in a new world of n-categorical geometry. This is joint work in progress with David Rydh.

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Adeel Khan

Derived specialization and virtual phenomena on moduli spaces

One motivation for derived algebraic geometry is the “hidden smoothness” exhibited by many moduli spaces even when their classical truncations are singular. It was hoped moreover that this hidden smoothness would provide the conceptual explanation for the existence of various “virtual” phenomena on these moduli spaces. In this talk I will discuss a general pattern for extracting virtual invariants from derived schemes and stacks, based on the technique of specialization to the normal bundle in derived algebraic geometry.

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Jeroen Hekking

Normal deformations in derived geometry

The deformation to the normal bundle is a natural incarnation in derived geometry of the classical deformation to the normal cone. The quasi-smooth case was first introduced by Khan & Rydh in 2018. In this talk, we will construct the normal deformation for any morphism of stacks using Weil restrictions, and study some of its basic properties (joint with Khan & Rydh). If time permits, I will say a few words about a generalisation to other kinds of derived geometries (joint with Ben-Bassat). Some applications of this theory will be discussed in Adeel’s talk.

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Catrin Mair

Condensed Shape of a Scheme

The ∞-category Cond(Ani) of condensed anima combines homotopy theory with the topological space direction of condensed sets. For example, we can recover the "Shape" of a sufficiently nice topological space from the corresponding condensed anima. My talk will focus on a joint refinement of the étale homotopy type and the pro-étale fundamental group of a scheme, realised as an object in Cond(Ani). This condensed version of a homotopy type, which I will refer to as condensed shape, is closely related to the work of Barwick, Glasman and Haine in the Exodromy paper.

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Sofía Marlasca Aparicio

Ultrasolid Homotopical Algebra

We present the theory of ultrasolid modules over a field (first proposed by Dustin Clausen), which generalises the solid modules over Q or F_p of Clausen and Scholze. Ultrasolid modules are a notion of complete modules over a discrete field. We build some basic results in ultrasolid commutative algebra and study its derived variants. Many results mimic the classical theory and we finally apply this to obtain an extension of the Lurie-Schlessinger criterion, which says that any formal moduli problem with coconnective tangent fibre is representable by a suitable ultrasolid derived algebra.

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Valerio Melani

Beilinson-Drinfeld affine Grassmannians for surfaces

Let G be a complex affine algebraic group. If C is a smooth algebraic curve and x is a point in C, the affine Grassmannian is an algebro-geometric object that study G-bundles on C together with a trivialization outside x. A particularly useful version is the so-called Beilinson-Drinfeld affine Grassmannian, where the point x is allowed to move, and we can even allow multiple points x. In this talk we present possible analogs for the Beilinson-Drinfeld affine Grassmannian, in the case where the curve is replaced by a smooth projective surface, and the trivialization data are given with respect to flags of closed subschemes. Based partly on a joint work with B. Hennion and G. Vezzosi, and partly on a joint work in progress with A. Maffei and G. Vezzosi.

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Matteo Montagnani

Smooth and proper categories in non-archimedean geometry

Toen and Vaquie proved that the category of perfect complexes of a smooth and proper complex analytic varieties is smooth and proper if and only if the variety is algebraic. I will describe how to adapt this proof to obtain an analog result in the non-archimedean and formal setting. In the end I will give an idea on how we can use the theory of condensed mathematics developed by Scholze and Clausen to characterize smooth and proper rigid analytic varieties.

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Devarshi Mukherjee

K-theory for analytic spaces

We introduce a version of algebraic K-theory and related localising invariants for bornological algebras, using Efimov's recently introduced continuous K-theory. In the commutative setting, our invariant satisfies descent for various topologies that arise in analytic geometry. If time permits, I will also discuss a version of the Grothendieck-Riemann-Roch Theorem for analytic spaces.

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Simone Murro
slides

A pathway to noncommutative Gelfand duality

The duality between algebraic structures and geometric spaces is of paramount importance in mathematics and physics, because provides a dictionary to describe manifolds and variaties in a purely algebraic fashion. In his seminal paper, Gelfand showed that a topological space can be functorially reconstructed from its Banach algebra of continuous functions. Conversely, the Gelfand spectrum of the algebra of continuous functions is homeomorphic to the underlying topological space. The goal of this talk is to constrcut a sufficiently robust notion of spectrum for general rings that allows one to implement a non-commutative analog of Gelfand duality. Our notion of spectrum, although formally reminiscent of the Grothendieck spectrum, is new ; in particular, it does not always agree with the Grothendieck spectrum.

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Jonathan Pridham
slides

Derived analytic geometry via entire functional calculus and dagger affinoids

Differential graded algebras with entire functional calculus (EFC-DGAs) give a very algebraic approach to derived analytic geometry, pioneered for more general Fermat theories by Carchedi and Roytenberg. They are well-suited to modelling finite-dimensional analytic spaces, and classical theorems in analysis ensure they give a largely equivalent theory to Lurie's more involved approach via pregeometries. DG dagger affinoid spaces provide a well-behaved class of geometric building blocks whose homotopy theory is governed by the underlying EFC-DGAs. Time permitting, I might also say a little about the challenges of establishing non-commutative generalisations.

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Nick Rozenblyum

Geometric Langlands correspondence

In these talks, I will describe the geometric Langlands conjecture and explain some of the ingredients involved in its proof, focusing on the derived geometric aspects. This is joint work with Arinkin, Beraldo, Campbell, Chen, Gaitsgory, Faergeman, Lin, and Raskin.

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Pelle Steffens

Representability of (compactified) elliptic moduli problems

It is well known that derived differential geometry is the correct framework for the study of moduli spaces of solutions of nonlinear PDEs on manifolds. In this talk I will discuss recent work showing how to use nonlinear Fredholm analysis to prove the representability of such derived moduli stacks for elliptic systems, parametrised by a smooth (C^∞)-stack. I will also comment on ongoing work (joint with John Pardon) on derived differential geometry relative to the (pre)geometry of logarithmically smooth manifolds with corners and the elliptic representability results in that setting, which cover the compactified moduli spaces that appear in symplectic topology and gauge theory.

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Jay Swar

Relative Symplectic Structures on Syntomified Moduli

Syntomification is a procedure for "geometrizing" the crystalline part of p-adic cohomology theories for nice enough p-adic spaces. The input is a (bounded formal) p-adic scheme and the output is a stack over a syntomic A^1/G_m. Focusing on certain moduli of Galois representations, we'll discuss the induced relative structures on the syntomified stack coming from arithmetic dualities.

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Nikola Tomic

A (-1)-shifted coisotropic morphism.

In this talk I will present some new investigations on shifted Poisson geometry, namely I will prove that the inclusion of the classical critical locus into the derived critical locus has a (-1)-coisotropic structure. This uses ideas of BV formalism and all the tools developped by Melani-Safronov on coisotropic structures. It also sheds the light on how to define shifted coisotropic reductions.

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Pietro Vanni
slides

Tempered functions in derived analytic geometry

Let R be a nonarchimedean Banach ring. Tempered power series are power series with coefficients in R that have bounded log-growth. They naturally form an algebra. We study this algebra in the framework of derived analytic geometry of ind-Banach modules over R. We describe how this perspective can be used to interpret the classical transfer theorem for log-growth of solutions of p-adic differential equations. Moreover we illustrate how one can construct a tempered version of convergent rigid cohomology of a smooth variety in characteristic p. This is a joint work with Federico Bambozzi and Bruno Chiarellotto.

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