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Monday 13 June
Program
9:00 - 9:50
Registration (Aula T1)
9:50 - 10:00
Opening (Aula T1)
10:00 - 11:00
Abstract: Foliations defined on algebraic varieties are rarely without singularities. These singularities can be studied using derived techniques via a notion of "derived foliations", in the same way than singularities of algebraic varieties can be studied using the notion of derived schemes. In this talk, I will present the notion of derived foliations and report on recent applications for the study of singular foliations. These include results in the complex case, as well as for foliations defined over base fields of arbitrary characteristics. (Joint with G. Vezzosi.)
Coffee Break
11:30 - 12:30
Abstract: We explain how shifted symplectic structures on derived stacks are connected to Calabi–Yau structures on differential graded categories. More concretely, we will show that the cotangent complex to the moduli stack of a differential graded category A is isomorphic to the moduli stack of the Calabi–Yau completion of A, answering a conjecture of Keller–Yeung.
This is joint work with Damien Calaque and Tristan Bozec: arxiv.org/abs/2006.01069.
15:30 - 16:30
Abstract: Configuration spaces consist in ordered collections of points in a given ambient manifold. Kontsevich and Tamarkin proved that the configuration spaces of Euclidean n-spaces are rationally formal, i.e., that their rational homotopy type is completely encoded by their cohomology. Their proofs use ideas from the theory of operads, and they prove the stronger result that the operads associated to configuration spaces of Euclidean n-spaces, called the little n-cubes operads and denoted En, are formal.
Voronov's Swiss-cheese operads encode the action of an En-algebra on an En – 1-algebra. Livernet and Willwacher proved that an enlarged version of this operad which encodes morphisms (rather than actions) is not formal. In this talk, I will explain why a higher codimensional version of the Swiss-cheese operad, which encodes a central derivation from an Em-algebra to an En-algebra, is formal for n – m ≥ 2. Moreover, I will sketch a proof of why Voronov's original version of the codimension one Swiss-cheese operad is non-formal (in joint work with R. V. Vieira).
Coffee Break
17:00 - 18:00
Abstract: Classical Cartan calculus concerns the action of vector fields acting on differential forms of a smooth manifold via Lie derivative and contraction. The key result is the Cartan magic formula which expresses the relation between the two actions. In the algebro-geometric setting, I will describe an interpretation of the Cartan calculus as Griffiths transversality for the Gauss–Manin connection on the universal deformation space. This naturally generalizes to the setting of derived algebraic geometry and also to Tamarkin–Tsygan calculus in noncommutative geometry, which involves the action of Hoschschild cohomology on Hochschild homology of a DG category. I will explain a very general theorem relating derived loop spaces and connections, which is used to give a precise relationship between the two. Time permitting, I will describe some applications to shifted symplectic structures on moduli spaces. This is joint work with Christopher Brav.
Tuesday 14 June
10:00 - 11:00
Unobstructedness of some Calabi–Yau varieties in characteristic p
Lukas Brantner
Lukas Brantner
Abstract: The Bogomolov–Tian–Todorov theorem asserts that every Calabi–Yau variety Z over an algebraically closed field k of characteristic zero is unobstructed. In joint work with Taelman, we use derived algebraic geometry to establish analogues of this result in characteristic p. More precisely, we show that if Z has degenerating Hodge–de Rham spectral sequence and torsion-free crystalline cohomology, then it is unobstructed. If Z is ordinary, we moreover prove that its deformation space is a formal torus, which implies unobstructedness and the existence of a canonical lift. Our work generalises ealier results by Achinger–Zdanowicz, Deligne–Nygaard, Ekedahl–Shepherd-Barron, Schröer, Serre–Tate, and Ward.
Coffee Break
11:30- 12:30
Abstract: A key example of 2-Segal spaces, or decomposition spaces, is the output of Waldhausen's S-construction when applied to an exact category. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we showed that this construction can be generalized to obtain an equivalence of homotopy theories; the most general input is that of an augmented stable double Segal space. Because many interesting examples of 2-Segal spaces additionally have a cyclic structure, a natural question is how to characterize which of these input structures correspond to cyclic 2-Segal spaces. In ongoing joint work with Stern, we are developing such a characterization.
15:30 - 16:30
Abstract: Triangulated categories are paramount in homotopy theory and homological algebra. However, they are lacking in some aspects, like the non-functoriality of cones and the non-existence of a well-behaved K-theory. This motivates the use of enhancements, such as model categories and DG-categories.
During the last twenty years, many mathematicians have addressed the question whether a given triangulated category has an essentially unique enhancement. There are remarkable results by Schwede and Shipley on the stable homotopy category, and Lunts, Orlov, Canonaco, Stellari, and Neeman for derived categories of abelian categories, just to cite a few names.
In this talk, we consider a totally different class of triangulated categories, typical from representation theory: those with a higher cluster tilting object stable under an iterated suspension. Despite their algebraic origin, the reasons for uniqueness are mainly homotopical. They actually lead to a complete classification of those enhanced triangulated categories in terms of algebraic data.
These results have been obtained jointly with Gustavo Jasso (Lund).
Coffee Break
17:00- 18:00
Abstract: The integration procedure associates an infinity-groupoid to a (complete/nilpotent) homotopy Lie algebra. This essential tool in deformation theory was introduced by Hinich and later refined by Getzler. Recently, a new method was developed by Robert-Nicoud and Vallette: it relies on the representation of the Getzler functor by an universal object. The goal of this talk is to generalize their method to the integration of curved absolute homotopy Lie algebras. "Absolute algebras" are a new type of algebraic structures that come naturally equipped with infinite summations, without having a priori an underlying topology. We will explain why these new objects appear and how to integrate them, generalizing the above cases. We will then develop applications to rational homotopy theory and to deformation theory.
Wednesday 15 June
10:00 - 11:00
Abstract: Weight decompositions were first introduced in the setting of rational homotopy as a tool to study p-universal spaces. The same notion may be adapted to many other algebraic contexts and, in general, positive and pure weight decompositions have strong homotopical consequences, often related to formality. A main source of weights is algebraic geometry, either via the theory of mixed Hodge structures on singular cohomology or the theory of Galois actions in étale cohomology. In this talk, I will review such weight structures defined at the cochain level, together with some main homotopical implications related to formality over the rationals and also over finite fields. This is mostly joint work with Geoffroy Horel and some work in progress with Bashar Saleh.
Coffee Break
11:30 - 12:30
Weight two compactly supported cohomology of moduli spaces of curves
Thomas Willwacher
Thomas Willwacher
Abstract: We show that the weight two compactly supported cohomology of the moduli spaces of curves is computed by a graph complex that (essentially) also computes a part of the cohomology of higher dimensional knot spaces. As a result, we discover many new nonzero classes in the cohomology of the moduli space Mg, n. In particular, we can express the weight 2 part of Hc (Mg, 0) completely in terms of the weight 0 part of Hc (Mg, n) for n = 1, 2.
Free Afternoon
20:00
Conference Dinner
Thursday 16 June
10:00 - 11:00
Abstract: In order to incorporate ideas from algebraic topology in concrete contexts such as TDA and lattice TQFT, one needs effective constructions of concepts defined only abstractly or axiomatically. In this talk, I will discuss such constructions for certain invariants derived from the cup product on the cohomology of spaces or, more specifically, from an E∞-structure on their cochains. When allowing for the computation of finer cohomological invariants, these effective constructions also reveal combinatorial information that, as we will see, is connected to convex geometry and higher category theory.
Coffee Break
11:30- 12:30
Abstract: Derived categories have come to play a decisive role in a wide range of topics. Several recent developments, in particular in the context of topological Fukaya categories, arouse the desire to study not just single categories, but rather complexes of categories. In this talk, we will discuss examples of such complexes in algebra, topology, algebraic geometry, and symplectic geometry, along with some results involving them.
15:30 - 16:30
Abstract: The lax Gray tensor product is a nonsymmetric, biclosed monoidal structure on (∞, n)-categories, whose internal hom classifies functors and lax natural transformations. In this talk, we discuss the construction of the lax Gray tensor product in full generality. Previously it was constructed in the strict setting by several authors, in certain models of n-categories not known to satisfy unicity, and in dimension 2 by Campbell and Maehara.
Along the way, we encounter a new, unicity-satisfying model of (∞, n)-categories as presheaves on a fairly large site of computads, and prove a new theorem allowing to construct certain pushouts of (∞, n)-categories by purely strict n-categorical methods.
Coffee Break
17:00 - 18:00
Abstract: The forgetful functor from commutative rings to associative algebras is fully faithful. The same is not true in homotopical algebra: the natural functor from Ho(CDGA) to Ho(DGA) is easily seen not to be full. However, in this talk I will explain that it is in fact faithful, and it is moreover injective on isomorphism classes. This answers a folklore problem in rational homotopy theory. I will then explain a Koszul dual story about the universal enveloping algebra functor from Ho(DGLA) to Ho(DGA), making progress on the "isomorphism problem" for enveloping algebras. (Joint with R. Campos, D. Robert-Nicoud, F. Wierstra.)
Friday 17 June
10:00 - 11:00
Abstract: In this talk I will describe recent work which expresses the Hodge degeneration between de Rham and Hodge cohomology as a delooping of the filtered loop space E2 groupoid. I will explain what these terms mean and will describe how this groupoid arises from viewing the circle S1 as an iterated cogroupoid in topological spaces. As I will explain, this is a stronger bit of structure than the more commonly studied pinch map on S1. Time permitting, I will describe how Todd classes for smooth and proper schemes emerge out of this filtered loop space story, as well as describe implications for Hochschild cohomology.
Coffee Break
11:30 - 12:30
Abstract: The framework of extended topological field theories, where these are defined as symmetric monoidal functors from (∞, n)-categories of cobordisms to some target, can be used to describe classical (as opposed to quantum) topological field theories by taking the target to be various (∞, n)-categories of iterated spans. I will discuss some general constructions of such field theories where the target has as objects derived algebraic stacks equipped with symplectic or Poisson structures. This is joint work with Damien Calaque and Claudia Scheimbauer (in the symplectic case) and Valerio Melani and Pavel Safronov (in the Poisson case).
15:30 - 16:30
Abstract: An (∞, 1)-category is m-semiadditive if it admits well behaved "direct sums" not just indexed by finite sets (as for classical semiadditive 1-categories) but by m-finite ∞-groupoids. Harpaz showed that (∞, 1)-categories of decorated spans of m-finite ∞-groupoids are characterized universally as free (∞, 1)-categories with this m-semiadditivity property.
In this talk I will explain all the terms mentioned above and report on work in progress with Claudia Scheimbauer aimed at giving a similar universal characterization of iterated spans by introducing a notion of m-semiadditivity for (∞, n)-categories for all n.
Coffee Break
17:00 - 18:00
Abstract: I am going to report on a work in progress with Martina Rovelli showing that simplicial and globular approaches to (∞, n)-categories can be related via a Quillen adjunction. I will assume no familiarity with the terms of the first sentence (except maybe a "Quillen adjunction"), and the main objective of the talk is to explain the content of this sentence. Time permitting, I will also talk about further ideas towards the comparison of these models.
Farewell