Representations in higher structures: Titles and abstracts

I will present aspects of my joint work with Anel, Finster and Joyal on the connection between higher topos theory and calculus in homotopy theory, specifically Goodwillie calculus and orthogonal calculus by Weiss.

Higher topoi are a convenient framework for (unstable) homotopy theory. After a (very) brief introduction I will outline how we can view topoi as analogues of commutative rings. We can construct analogues of many notions from classical algebra like ideals, radicals and adic completion. Then I will mention how the Goodwillie tower and the orthogonal tower are special cases (of the analogoues) of adic completion towers. If there is time, I will mention a generalized Blakers-Massey theorem whose special case for the Goodwillie tower was conjectured by Goodwillie and was not known for the orthogonal tower.
 

We construct the smooth higher group of symmetries of any higher geometric structure on manifolds. Via a universal property, this classifies equivariant structures on the geometry. We present a general construction of moduli stacks of solutions in higher-geometric field theories and provide a criterion for when two such moduli stacks are equivalent. We then apply this to the study of generalised Ricci solitons, or NSNS supergravity: this theory has two different formulations, originating in higher geometry and generalised geometry, respectively. These formulations produce inequivalent field configurations and inequivalent symmetries. We resolve this discrepancy by showing that their moduli stacks are equivalent. This is joint work with C. Shahbazi.

Floer cohomology comes in various flavours and has developed into a primary tool in low-dimensional topology. In this talk, I will discuss an equivariant version of Seiberg–Witten–Floer cohomology for finite group actions on rational homology 3-spheres. This gives rise to a series of numerical invariants, generalizing the Ozsvath–Szabo d-invariant. I will survey basic properties and applications of these invariants in knot theory and as obstructions to equivariant embeddings of 3-manifolds.

Given any integer n, there is a universal enveloping functor from (homotopy) Lie algebras to little n-disks algebras, which are higher analogues of the classical universal enveloping associative algebra functor. These functors can be nicely modeled by factorization algebras and in this talk we wish to study higher version of classical results: namely to exhibit their center and deformation complex (given by their derived center).

The cobar construction induces a Koszul duality between algebras and coalgebras, providing an equivalence of suitable homotopy theories of augmented differential graded algebras and differential graded conilpotent coalgebras.

I will talk about far-reaching generalisation of this result to categorical Koszul duality, introducing a category of coalgebras Quillen equivalent to differential graded categories. This equivalence is moreover quasi-monoidal and by constructing internal homs of certain coalgebras we can construct a concrete closed monoidal model for dg categories. In particular this gives natural descriptions of mapping spaces and internal homs between dg categories.

This is joint work with A. Lazarev.

I will explain a method of differentiation of higher groupoids to their infinitesimal counterparts. Higher groupoid objects in a category C with a Grothendieck pretopology were first introduced by Henriques and Zhu in terms of Kan simplicial objects in C. In 2006, Severa has argued that the L_oo-algebroid of a higher Lie groupoid G is given by the inner hom in the category of simplicial supermanifolds from the nerve of the pair groupoid of R^0|1 to G. Using the language of categorical ends, I will generalize this to groupoids in categories with an abstract tangent functor (in the sense of Rosicky) and a Grothendieck pretopology. If time permits, I will discuss possible applications to geometric deformation theory. This is joint work with Christian Blohmann.

In joint work with Konrad Waldorf and Peter Kristel, we construct a representation for the String 2-group on a von Neumann algebra. This uses a certain strict model for the string group, developed together with Konrad Waldorf.

It is well known that multiplicative principal bundles with abelian structure group correspond to central Lie group extensions. A similar result holds for multiplicative bundle gerbes and central extensions of Lie 2-groups. Transgression of bundle gerbes to principal bundles over loop spaces connects these two concepts. In this talk, I will present the problems that appear when trying to generalize these results to the non-central case and our attempts to resolve those. As an application, we propose an approach to associate Tate-K-theory classes to 2-vector-bundles.

I will discuss the construction and meaning of three different homotopy theories on the category of simplicial coalgebras over a field of arbitrary characteristic. Over an algebraically closed field F, one of these provides a full and faithful model for the homotopy theory of spaces considered under a notion of weak equivalence generated by continuous maps inducing an isomorphism on fundamental groups and a F-homology isomorphism at the level of universal covers. This extends classical rational homotopy theory and p-adic homotopy theory by including the fundamental group in complete generality. The key idea is to use Koszul duality theory between coalgebras and algebras to induce a notion of equivalence on simplicial coalgebras that is stronger than quasi-isomorphism and completely captures the fundamental group.

The key open question of contemporary mathematical physics is the elucidation of the currently elusive fundamental laws of strongly-interacting “non-perturbative” quantum states — including bound states as mundane as nucleons (declared a mathematical “Millennium Problem” by the Clay Math Institute), as well as strongly-correlated topologically ordered quantum materials (currently sought by various laboratories as hardware for topological quantum computation).

The popular strategy of regarding such systems as located on branes inside a higher dimensional string-theoretic spacetime (the “holographic principle”) shows all signs of promise but has been suffering from the ironic shortcoming that also string theory has only really been defined perturbatively. But string theory exhibits a web of hints towards the nature of its non-perturbative completion, famous under the working title “M-theory”. Thus, mathematically constructing M-theory should imply a mathematical understanding of quantum brane worldvolumes which should solve non-perturbative quantum physics: the M-strategy for attacking the Millennium Problem.

After a time of stagnation in research towards M-theory, we have recently formulated and extensively tested a hypothesis on the precise mathematical nature of at least a core part of the theory: We call this "Hypothesis H" since it postulates that M-branes are classified by coHomotopy-theory in much the same way that D-branes are expected to be classified by complex K-theory (a widely held but just as conjectural belief which might analogously be called "Hypothesis K"). In fact, stabilized coHomotopy is equivalently F1-K-theory over the “absolute base field with one element”; and it is also equivalently framed Cobordism cohomology.

In this talk I'll try to give an introduction to (1.) the motivation and (2.) some consequences of Hypothesis H, following expository notes under development here:

ncatlab.org/schreiber/show/Introduction+to+Hypothesis+H.

This is joint work with Hisham Sati.

Quantum topologists are used to thinking about traces in the framework of pivotal tensor categories and thus in a two-dimensional context to which a two-dimensional graphical calculus can be associated. We explain that traces are already naturally defined for twisted endomorphisms of linear categories, i.e. in a one-dimensional context. The endomorphisms are twisted by the Nakayama functor which, for a module category over a monoidal category, is a twisted module functor and hence an inherently three-dimensional object. This naturally leads to a three-dimensional graphical calculus. This calculus also has applications to Turaev–Viro topological field theories with defects.

Dagger categories are categories equipped with a certain involution, such as the category of Hilbert spaces equipped with the adjoint. They play a crucial role in comprehending unitarity within topological field theory. Among topological field theories, the "invertible" theories stand out for their simplicity, where all operations in the involved symmetric monoidal categories are invertible. Such categories are referred to as Picard groupoids and have been previously classified by a student of Grothendieck. In this talk, I will delve into the classification of dagger involutions on Picard groupoids, along with the study of functors that preserve the dagger structure. As a corollary, we will unveil a classification of Hermitian and unitary invertible topological field theories. 

Multiplicative gerbes can be understood as group-like objects in the category of gerbes, and therefore their representations could be taken to be module objects in the category of gerbes. With this point of view, K. Waldorf, J. Blanco and myself, constructed the category of representations of multiplicative gerbes, and given certain conditions, we were able to show sufficient conditions for two multiplicative gerbes to have equivalent categories of representations. The key ingredient for our construction is the model for cohomology of topological groups developed by Graeme Segal in the seventies.