Title: Marginally trapped submanifolds
Abstract: Trapped surfaces were introduced in the sixties by R. Penrose in order to prove the existence of singularities of solutions to the Einstein equations of General Relativity (GR). The close concept of marginally trapped surfaces arises in the description of the apparent horizon of a black hole. This concept has become a central one in GR, being related to singularity theorems and Penrose inequalities, among others. Geometrically, a surface is said to be marginally trapped if its mean curvature vector is light-like (i.e., its squared norm vanishes). Analytically, it is an underdetermined, non-linear, elliptic PDE. In this talk I will first introduce the concepts of mean curvature vector and marginally trapped submanifold. Then I will report on recent work that provide local, explicit descriptions of marginally trapped submanifolds of co-dimension two in several natural pseudo-Riemannian manifolds (de Sitter and anti de Sitter spaces, Robertson-Walker spaces,...). These results are based on the canonical structure of the space of light-like geodesics of a given pseudo-Riemannian manifold. This is joint work with Yamile Godoy (University de Córdoba, Argentina) and Nastassja Cipriani (KU Leuven, Belgium).
Title: Representations and composition of Dirac structures for infinite-dimensional systems
Abstract: Dirac structures are used to formalize the power-conserving interconnection structure of physical systems. We approach the Dirac structures for in finite dimensional systems using the theory of linear relations on Hilbert spaces. Several representations of Dirac structures are presented and related to each other constructively. Necessary and sufficient conditions for preserving the Dirac structure under interconnection are also discussed. Several examples will illustrate the theory.
Title: The geometric Bohr topos
Abstract: The Bohr topos is a construction that has been introduced in Nijmegen as part of the program aiming at developing a physical theory based on topos theory. It is constructed to represent the "spaces" of measures that can be performed on some (quantum) physical system, but from this perspective it has some defect: it has some unwanted additional points, the topology on it is too algebraic, the construction is non-geometric etc... Using tools from my thesis, I have constructed a "geometric" version of it, which avoid most of these issues and might allow to push this approach a little further. In this talk I will explain this construction in a way avoiding any specific knowledge of topos theory.
Title: Encoding equivariant commutativity via operads
Abstract: I will discuss ongoing joint work with Javier J. Gutiérrez to study the types of equivariant commutativity which arose in the recent work of Hill-Hopkins-Ravenel on the Kervaire problem. We will recall what commutativity means in this context, why it is important, and how the N∞-operads constructed by Blumberg and Hill capture this notion. We then give the first general construction of N∞- operads, and in doing so we characterize the equivariant homotopy type of the spaces of these operads and their natural generalizations. Doing so requires us to build Rezk-style model structures on the category of G-equivariant operads for any family of subgroups of G x Σ_{n}. If there is time at the end we'll discuss how our results relate to questions of rectification and of Bousfield localization.
Title: Dendroidal sets and noncommutative spaces
Abstract: There are two prominent paradigms of noncommutative geometry. One of them was introduced by Connes that builds upon the theory of C^{*}-algebras. The other viewpoint (largely due to Manin-Kontsevich and several others) is inspired by algebraic geometry and hence it is almost devoid of analysis. It has been a challenge (albeit of academic nature) to reconcile the two viewpoints. Using the theory of dendroidal sets (introduced by Moerdijk-Weiss) we are going to explain the first step towards this goal. The attempt is new and it is a humble suggestion to initiate a dialogue between the two schools of thought.
Title: A characterization of quasicategorical t-structures
Abstract: We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. A t-structure on a stable ∞-category C is equivalent to a "normal torsion theory" ([CHK]) F on C, i.e., to a factorization system F=(E,M) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts. We will re-interpret the main classical results in the theory of t-structures in terms of normal torsion theories (abelianity of the heart, semiorthogonal decomposition, recollements).
Title: Can we do homotopy theory in constructive mathematics?
Abstract: All the results of existence of model structures that we know of seem to rely on non-constructive arguments, and in fact it is not clear at all whether or not it is possible to construct the ordinary model category structure on simplicial sets within constructive mathematics. In this talk I will give motivations (mostly coming from topos theory and infinity-topos theory) for understanding if it is possible to construct such model structures in constructive mathematics, and I will present some recent results in this direction (this is still a work in progress). The general long term goal would be to obtain constructive notions of homotopy types, higher categories and higher toposes which give back, when interpreted in the internal logic of an ordinary topos, the notions of stack of infinity groupoids, stack of infinity categories, and infinity toposes over that topos. This will be explained in more detail during the talk.
Title: Minimal fibrations in Reedy-like model categories.
Abstract: A classical fact in the homotopy theory of simplicial sets says that any Kan fibration is homotopy equivalent to a fiber bundle. A proof of this uses the so-called minimal fibrations, which provide rigid models for maps between simplicial sets. In this talk, I will discuss an extension of the theory of minimal fibrations to more general "Reedy-like" model categories. I will also provide some simple applications, for example in the context of dendroidal sets. This is a joint work with I. Moerdijk.
Title: Models for singularity categories and applications to knot homology
Abstract: This is a summary of the results of my PhD project. In the first part, I will discuss model categorical enhancements for singularity categories. The basis is Hovey's description of abelian model structures in terms of cotorsion pairs and deconstructible classes, and the technical heart of our constructions are general results about localization and transfer of abelian model structures. I will also explain how classical recollements relating the singularity category to the derived category can be lifted to the level of model categories. In the second part, I will use these results in the special case of matrix factorizations to prove that Khovanov-Rozansky homology, an invariant of oriented links categorifying the quantum sl(k) link invariant, can be described in terms of stable Hochschild homology of Rouquier complexes of Soergel bimodules.
Title: Shuffle operads, the Loday-Pirashvili category, and tensoring with Perm
Abstract: The shuffle operads are an intermediate notion between symmetric and non-symmetric operads. We will give some intuition and a conjecture about them, which will involve the operad called Perm. We will also state a theorem relating this operad and the Loday-Pirashvili category of linear maps.
Title: A Dwyer-Kan model structure for coloured PROPs
Abstract: It is well known that the category of (small) simplicial categories admits a model structure (established by by Julie Bergner) that models the homotopy theory of (∞,1)-categories. Clemens Berger and Ieke Moerdijk, and more recently Fernando Muro, proved the existence of a model structure on the category of small categories enriched over a suitable monoidal model category V, in which the weak equivalences are the Dwyer-Kan weak equivalences (i.e., equivalence of categories "up to homotopy"). In the case V=sSets, they recover Bergner's result. Coloured PROPs can be regarded as a generalization of categories (and operads) where the morphisms are allowed to have multiple inputs and outputs. Coloured PROPs form a 2-category, with a well defined notion of equivalence between PROPs. In this talk, I will outline how to construct a model structure on the category of coloured PROPs enriched in a suitable monoidal model category V, in which the weak equivalences are the Dwyer-Kan weak equivalences (i.e., the homotopy analogous of equivalences of PROPs).
Title: Towers and fibered products of localized model structures
Abstract: Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products of model categories. As applications we consider Postnikov towers of model categories and Bousfield arithmetic squares of spectra. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization. This is a joint work with C. Roitzheim.
Title: Quanta of Geometry: Physical applications.
Abstract: When the orientability condition in noncommutative geometry is applied to maps from a four-manifold to a four sphere, a higher degree Heisenberg relation appears involving the Dirac operator and the Feynman slash coordinates. I show that this leads to the quantization of volume and fixes the algebra associated with the four-dimensional manifold. The Planck spheres are the quanta from which the space-time manifold is constructed. Remarkably, this algebra is identical to the one that arises in the noncommutative construction of the Standard Model. Noncommutative geometry thus offers a unified picture from low energies all the way up to Planckian energies where space-time fragments into quantum spheres.
Title: Cellular properties of Nilpotent spaces
Abstract: Nilpotent spaces are natural generalization of simply connected ones and include any connected topological group. We describe a method to approach certain localizations and co-localizations of these spaces. In particular their Postnikov sections P_{n}X have nice properties such as: If a generalized homology theory h_{*} vanishes on X it must vanish on P_{n}X for any natural n. This e.g. greatly restricts the possible fundamental and higher homotopy groups of such h-acyclic spaces.
Title: The higher Morita category of E_{n}-algebras.
Abstract: I will discuss a construction of a higher category of E_{n}-algebras and iterated bimodules, generalizing the classical bicategory of algebras and bimodules. This leads to generalizations of the Picard and Brauer groups, which have been studied in stable homotopy theory as interesting invariants of ring spectra, and should also lead to an "algebraic" construction of factorization homology as an extended topological quantum field theory.
Title: Augmented directed complexes as a model for homotopy types?
Title: Spaces of smooth embeddings and operads.
Abstract: I will discuss an operadic description of the homotopy theoretical obstructions to lifting smooth immersions into smooth embeddings. This characterisation, applied to the case of (high-dimensional variants of) long knots, strengthens recent results of Arone-Turchin and Dwyer-Hess. Joint work with Michael Weiss.