A model for the E3 fusion-convolution product of constructible sheaves on the affine Grassmannian

In this paper we provide a detailed construction of an associative and braided convolution product on the category of equivariant constructible sheaves on the affine Grassmannian through derived geometry. This product extends the convolution product on equivariant perverse sheaves and is constructed as an $E_3$-algebra object in $\infty$-categories. The main tools amount to a formulation of the convolution and fusion procedures over the Ran space involving the formalism of 2-Segal objects and correspondences from Dyckerhoff and Kapranov, "Higher Segal Spaces I", and Gaitsgory and Rozenblyum, "A Study in Derived Algebraic Geometry I"; of factorising constructible cosheaves over the Ran space from Lurie, "Higher Algebra" Chapter 5; and of constructible sheaves via $\infty$-categorical exit paths

Whitney stratifications are conically smooth

In their paper "Local structures on stratified spaces", Ayala, Francis and Tanaka introduced the notion of conically smooth structure on stratified spaces. This is a very well behaved analogous of a differential structure in the context of manifold stratified topological spaces, satisfying good properties such as the existence of resolutions of singularities and handlebody decompositions. In this paper we prove that any Whitney stratified space admits a conically smooth structure, as conjectured by Ayala, Francis and Tanaka themselves, thus establishing a connection between this new theory and the classical examples of stratified spaces from differential topology.Comment: 15 page

The derived Brauer map via twisted sheaves

Let $X$ be a quasicompact quasiseparated scheme. The collection of derived Azumaya algebras in the sense of To\"en forms a group, which contains the classical Brauer group of $X$ and which we call $Br^\dagger(X)$ following Lurie. To\"en introduced a map $\phi:Br^\dagger(X)\to H^2_{et}(X,\mathbb G_m)$ which extends the classical Brauer map, but instead of being injective, it is surjective. In this paper we study the restriction of $\phi$ to a subgroup $Br(X)\subset Br^\dagger(X)$, which we call the "derived Brauer group", on which $\phi$ becomes an isomorphism $Br(X)\simeq H^2_{et}(X,\mathbb G_m)$. This map may be interpreted as a derived version of the classical Brauer map which offers a way to "fill the gap" between the classical Brauer group and the cohomogical Brauer group. The group $Br(X)$ was introduced by Lurie by making use of the theory of prestable $\infty$-categories. There, the mentioned isomorphism of abelian groups was deduced from an equivalence of $\infty$-categories between the "Brauer space" of invertible presentable prestable $\mathcal O_X$-linear categories, and the space $Map(X,K(\mathbb G_m,2))$. We offer an alternative proof of this equivalence of $\infty$-categories, characterizing the functor from the left to the right via gerbes of connective trivializations, and its inverse via connective twisted sheaves. We also prove that this equivalence carries a symmetric monoidal structure, thus proving a conjecture of Binda an Porta.Comment: 23 page

A study of the spherical Hecke category via derived algebraic geometry

My thesis project lies at the interface of algebraic geometry, topology and representation theory. I focused on using tools from homotopy theory, and particularly derived algebraic geometry and infinity-category theory, to provide significant generalizations of classical results in the area of the Geometric Langlands Program. In the first chapter, I proved that under certain conjectures the infinity-category Sph(G) associated to a reductive group G admits an E3-monoidal structure extending the symmetric monoidal convolution product of perverse sheaves. In the second chapter, I proved with Marco Volpe a conjecture of Ayala, Francis and Rozenblyum saying that every stratified space satisfying Whitney's conditions admits a conically smooth structure. In the third chapter, I proved with Michele Pernice a conjecture by Federico Binda and Mauro Porta establishing a relationship between the notions of Gm-gerbe and derived Azumaya algebra.Mon projet de thèse se situe à l'interface de la géométrie algébrique, de la topologie et de la théorie des représentations. Je me suis concentré sur l'utilisation d'outils de la théorie de l'homotopie, en particulier la géométrie algébrique dérivée et la théorie des infini-catégories, pour fournir des généralisations significatives des résultats classiques établis dans le Programme Géométrique de Langlands. Dans le premier chapitre, j'ai prouvé que, sous certaines conjectures, l'infini-categorie Sph(G) associé à un groupe reductif G admet une E_3- structure monoïdale étendant le produit de convolution monoïdale symétrique des faisceaux pervers. Dans le deuxième chapitre, j'ai prouvé avec Marco Volpe une conjecture de Ayala, Francis et Rozenblyum disant que tout espace stratifié satisfaisant les conditions de Whitney admet une structure coniquement lisse. Dans le troisième chapitre, j'ai prouvé avec Michele Pernice une conjecture de Federico Binda et Mauro Porta établissant une rélation entre la notion de Gm-gerbe et de algèbre de Azumaya derivée

Classifying topoi and groupoids

In this dissertation we examine the notion of classifying topos of a topological category, and its connections with the theory of groupoids. In the first chapter we introduce the notion of Grothendieck topos, and the general definition of the classifying topos BC of a topological category C. Then we make examples of the “classifying property”, starting with the classical theorem that connects the classifying space of a group with the principal G-bundles on a topological space, and then studying the classifying topos of a group Sh(G) (namely, the category of right G-sets), proving a theorem by Diaconescu that rephrases the classifying property of the classifying space in a categorical context. Also, we explain the connections between topos cohomology and group cohomology. Also, a brief overview of the more general statement of the “classifying property”, in term of first-order theories, is given. In the second chapter we examine more in detail the notion of homotopy for a topos, in order to establish a connection with the homotopy of the classifying space. We present the theory of ́etale homotopy, starting with the example of ́etale coverings on a scheme and then generalising to the case of general Grothendieck topoi. We are then able to consider the homotopy progroups of a Grothendieck topos, and to state the the so-called toposophic Whitehead theorem, that connects isomorphisms in ( ́etale) homotopy with isomorphisms in (topos) cohomology. We then deepen the context of simplicial objects and sheaves on these, in order to define the nerve of a topological category. This allows us to define and study the clas- sifying space of a topological category, in a way that extends the case of groups. In the third chapter we prove the comparison theorem: for an s- ́etale topological category C, there is a weak homotopy equivalence between the topoi Sh(BC) and Sh(C). As an application, we restrict to the case of topological groupoids, and consider the Haefliger groupoid Γq. This groupoid “classifies foliations”, in the sense that the existence of certain foliations on an open manifold is equivalent to the existence of a lifting in a diagram involving the classifying space of Γq . We prove a theorem by Segal, following Moerdijk’s alternative proof, according to which Sh(Γq) can be replaced, up to homotoopy, by the classifying space of M(Rq), the monoid of smooth embeddings of Rq into itself. In the fourth chapter we consider a sort of “inverse question”: given a Grothendieck topos E, is it true that it can be represented as the classifying topos of a groupoid G? The answer is, in general, negative. It is positive when taking topoi with “enough points”. Also, every topos can be represented as the classifying topos of a “localic groupoid”. These results are due, respectively, to Butz and Moerdijk, and to Joyal and Tierney. We examine the proof of the first theorem and remark the use of a set-theoretic argument that fails in the proof of the second result. Finally, following another article by Moerdijk, we examine how localic groupoids (more precisely, localic groups) enter into the problem of Morita-equivalence of sites, underlying the differences with the standard case of discrete groups on one side (that have a simpler behaviour) and topological groups on the other (whose problems can more easily be solved in the localic context)

A study of the spherical Hecke category via derived algebraic geometry

Mon projet de thèse se situe à l'interface de la géométrie algébrique, de la topologie et de la théorie des représentations. Je me suis concentré sur l'utilisation d'outils de la théorie de l'homotopie, en particulier la géométrie algébrique dérivée et la théorie des infini-catégories, pour fournir des généralisations significatives des résultats classiques établis dans le Programme Géométrique de Langlands. Dans le premier chapitre, j'ai prouvé que, sous certaines conjectures, l'infini-categorie Sph(G) associé à un groupe reductif G admet une E_3- structure monoïdale étendant le produit de convolution monoïdale symétrique des faisceaux pervers. Dans le deuxième chapitre, j'ai prouvé avec Marco Volpe une conjecture de Ayala, Francis et Rozenblyum disant que tout espace stratifié satisfaisant les conditions de Whitney admet une structure coniquement lisse. Dans le troisième chapitre, j'ai prouvé avec Michele Pernice une conjecture de Federico Binda et Mauro Porta établissant une rélation entre la notion de Gm-gerbe et de algèbre de Azumaya derivée.My thesis project lies at the interface of algebraic geometry, topology and representation theory. I focused on using tools from homotopy theory, and particularly derived algebraic geometry and infinity-category theory, to provide significant generalizations of classical results in the area of the Geometric Langlands Program. In the first chapter, I proved that under certain conjectures the infinity-category Sph(G) associated to a reductive group G admits an E3-monoidal structure extending the symmetric monoidal convolution product of perverse sheaves. In the second chapter, I proved with Marco Volpe a conjecture of Ayala, Francis and Rozenblyum saying that every stratified space satisfying Whitney's conditions admits a conically smooth structure. In the third chapter, I proved with Michele Pernice a conjecture by Federico Binda and Mauro Porta establishing a relationship between the notions of Gm-gerbe and derived Azumaya algebra

Effects of pre‐operative isolation on postoperative pulmonary complications after elective surgery: an international prospective cohort study

We aimed to determine the impact of pre-operative isolation on postoperative pulmonary complications after elective surgery during the global SARS-CoV-2 pandemic. We performed an international prospective cohort study including patients undergoing elective surgery in October 2020. Isolation was defined as the period before surgery during which patients did not leave their house or receive visitors from outside their household. The primary outcome was postoperative pulmonary complications, adjusted in multivariable models for measured confounders. Pre-defined sub-group analyses were performed for the primary outcome. A total of 96,454 patients from 114 countries were included and overall, 26,948 (27.9%) patients isolated before surgery. Postoperative pulmonary complications were recorded in 1947 (2.0%) patients of which 227 (11.7%) were associated with SARS-CoV-2 infection. Patients who isolated pre-operatively were older, had more respiratory comorbidities and were more commonly from areas of high SARS-CoV-2 incidence and high-income countries. Although the overall rates of postoperative pulmonary complications were similar in those that isolated and those that did not (2.1% vs 2.0%, respectively), isolation was associated with higher rates of postoperative pulmonary complications after adjustment (adjusted OR 1.20, 95%CI 1.05-1.36, p = 0.005). Sensitivity analyses revealed no further differences when patients were categorised by: pre-operative testing; use of COVID-19-free pathways; or community SARS-CoV-2 prevalence. The rate of postoperative pulmonary complications increased with periods of isolation longer than 3 days, with an OR (95%CI) at 4-7 days or >= 8 days of 1.25 (1.04-1.48), p = 0.015 and 1.31 (1.11-1.55), p = 0.001, respectively. Isolation before elective surgery might be associated with a small but clinically important increased risk of postoperative pulmonary complications. Longer periods of isolation showed no reduction in the risk of postoperative pulmonary complications. These findings have significant implications for global provision of elective surgical care