From Analysis to Homotopy Theory: Titles and Abstracts

We will see how the history of logic of the last hundred years has lead us precisely to this place where logic, homotopy theory, and category theory become one.

I will discuss the main points of my involvement in Uli's work on coarse geometry. I will focus on coarse homology theories and assembly maps.

Higher category theory is playing an increasingly important role throughout mathematics. While (oo,1)-category theory is sufficient for many purposes, its study naturally leads to (oo,2)-category theory. For certain applications, however, it is necessary to work at the level of (oo,oo)-categories. For instance, one gains the ability to perform fully lax constructions, though at the cost of increasing combinatorial complexity. We will consider certain presentations of the (oo,1)-category of (oo,oo)-categories, and various operations, such as the lax Gray tensor, join, and smash, which arise naturally in this setting.

Kreck’s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of Sⁿ x Sⁿ. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of S² x S². This is joint work with John Nicholson and Simona Veselá.

I begin by summarizing some things known about the signature of total spaces of various kinds of bundles. I will then focus on bundles over stably framed base manifolds M and explain a full calculation of the group of signatures of local systems of unimodular forms over such M by homotopy theoretic means. This calculation recovers known divisibility results of smooth manifold bundles (again over stably framed base) previously obtained via index theory. If time permits, I will also discuss some further results where both analytic and homotopy theoretic arguments can be given.

Wann sind mathematische Objekte (z.B. Zahlen, Funktionen, geometrische Gebilde) "gleich"? Diese harmlos erscheinende Frage stellt sich als erstaunlich tückisch heraus. Insbesondere ist es nötig, viele verschiedene "Gleichheiten" zu betrachten und zu vergleichen. In diesem Vortrag werden wir sehen, wie man über "Gleichheiten" in einer geometrischen Sprache sprechen kann und welche Rolle dies in der Programmierung bzw. Computerverifikation spielt.

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Topological insulators are certain materials or metamaterials whose mathematical description involves certain non-trivial deformation invariants. We explain a coarse geometric approach to the mathematical treatment of these materials.

I will start by reviewing my recent work with Dan Grady on the geometric cobordism hypothesis and locality of fully extended nontopological functorial field theories. I will then apply these results to explicitly compute, in terms of homotopy coherent representations of Lie groups,
the space of 2-dimensional fully extended functorial field theories with geometric structures like flat 2-dimensional Riemannian metrics, 2|1-dimensional super Euclidean structures, or conformal structures. If time permits, I will discuss further constructions of field theories (ongoing work with Dan Grady) that involve differential characteristic classes, index and eta-invariants, and quantization.

In this talk I shall explain how cyclic cohomology and K-theory can be used in order to define and investigate primary and secondary invariants of G-equivariant Dirac operators on a cocompact G-proper manifold, with G a connected real reductive Lie group. I
will first treat, rapidly, the case of cyclic cocycles associated to elements in the differentiable cohomology of G; I will then go on to delocalized cyclic cocycles, the main topic of this talk. I will explain the challenges in defining the delocalized eta invariant associated to the orbital
integral defined by a semisimple element g in G and in showing that such an invariant enters in an Atiyah-Patodi-Singer index theorem for cocompact G-proper manifolds. I will then move to a higher version of these results, using the higher delocalized cyclic cocycles defined by Song and Tang. This talk is based on published work with Hessel Posthuma and two preprints with Hessel Posthuma, Yanli Song and Xiang Tang.

It is a fundamental observation of Atiyah-Patodi-Singer that the correct setup for index theory on manifolds with corners involves non-local spectral boundary conditions. A convenient calculus which contains these operators and their inverses is Melrose calculus of b-pseudodifferential operators.

If one passes to general manifolds with corners, it becomes a problem to decide whether one can render a (pseudo)differential operator with elliptic principal symbol Fredholm by applying a smoothing perturbation in the calculus (which is the correct generalization of introducing non-local boundary conditions).

The obstruction of doing so is a boundary analytic index, taking values in the K-theory of the b-compact operator of the boundary.

Therefore, it becomes important and interesting to get information about this boundary K-theory group.

It turns out that the relevant group depends only on the combinatorics of the faces. To this end, Uli Bunke introduced the conormal homology, whose rationalization is the range of a natural Chern character from the boundary K-theory.

For a wile, it was conjectured that this conormal homology might be integrally isomorphic to the boundary K-theory, and that it might be always a torsion-free abelian group (a fact if one has only faces of codimension one and two).

The talk will present the theory and cumulate in the proof that conormal homology, and boundary K-theory for manifolds with corners are completely arbitrary: everything what shows up as homology and K-theory of a finite simplicial complex also shows up as conormal homology and boundary K-theory of a manifold with corners.

Costello and Gwilliam developed a systematic approach to perturbative quantization of field theories that describes classical and quantum observables as factorization algebras on the space-time manifold. In the simple case of the linear sigma model whose fields are smooth maps from the real line to an inner product space V (aka the free boson), the algebra of quantum observables is the Weyl algebra of differential operators on V. For the free fermion, where V is replaced by its odd analog, the algebra of quantum observables is the Clifford algebra generated by V.

In this talk, based on joint work with Bill Dwyer, I will discuss how to enhance the free fermion by suitable boundary conditions to obtain a factorization algebra of ``bulk-boundary quantum observables'', which can be evaluated on open intervals as well as half-open intervals. The challenge is that there is a space of boundary conditions, each of which yields a bulk-boundary factorization algebra, but unlike for the free boson (where there are Dirichlet and Neumann boundary conditions), there is no distinguished boundary condition. Our main result is that a spin structure on V yields a distinguished bulk-boundary factorization algebra of quantum observables. 

Crystalline cohomology was invented by Grothendieck and developed by Berthelot as a substitute for l-adic étale cohomology of varieties in characteristic p when l equals p. It was originally constructed using the crystalline site, a site that not everybody loves. In this talk I will explain a simple, direct construction of crystalline cohomology based on the observation that every smooth variety in characteristic p lifts in a certain sense uniquely to characteristic 0. This is joint work with Moritz Kerz.

Motivated by applications to pulling back quantum cellular automata from one manifold to another, we study the existence of immersions between smooth 4-manifolds. We show that they lead to a very interesting partial order on closed 4-manifolds. Joint work with Mike Freedman, Daniel Kasprowski and Matthias Kreck, arXiv:2208.03064.

For d ≥ 4 and p a sufficiently large prime, we construct a lattice Γ ≤ PSp2d(ℚp), such that its universal central extension cannot be sofic if Γ satisfies some weak form of stability in permutations. In the proof, we make use of high-dimensional expansion phenomena and, extending results of Lubotzky, we construct new examples of cosystolic expanders over arbitrary finite abelian groups. This is joint work with Lukas Gohla.

Kummer surfaces are classical examples of K3 surfaces which may be obtained by resolving all singularities in a quotient of a complex two-torus by the center of SU(2). This construction is known as orbifolding, and it generalizes to other finite subgroups of SU(2). One may then construct invariants of K3 surfaces, like the complex elliptic genus, by orbifolding techniques that are borrowed from conformal field theory. These techniques are global, obscuring the contributions that arise from resolving the singularities. We show how to disentangle these orbifold data, arriving at a proposal for complex elliptic genera of the resolutions of simple surface singularities.

One perspective on Kasparov's category KK is to consider it as the receptacle for a universal homology theory on C*-algebras, making it a category of "motives" in this context. It is a consequence of work of Cuntz, Higson and Uuye that KK is a Dwyer-Kan localisation of the category of C*-algebras. As observed by Land and Nikolaus, this yields in particular an infinity-categorical refinement of KK.
In the algebraic/homotopy theoretic context, Blumberg, Gepner and Tabuada introduced the infinity-category of localising motives as an analogue of KK. Continuing the analogy, I will explain that this infinity-category is a Dwyer-Kan localisation of the infinity-category of small stable infinity-categories, essentially due to the existence of an appropriate cofibration category structure. This provides an alternative approach to constructing counterexamples to the non-connective theorem of the heart in algebraic K-theory. Joint work with Maxime Ramzi and Vladimir Sosnilo.