Building-up Differential Homotopy Theory,  2023

Abstract (pdf file 58kb, updated 03/03/23)                                      “***” indicates an online lecture.

Tuesday 7th March

Title: Smooth maps on convex sets

By:    Yael Karshon

         (The University of Toronto, Tel-Aviv University)


There are several notions of a smooth map from a convex set to a Cartesian space Someofthese notions coincide, but not all of them do. We construct a real-valued function on a convex subset of the plane that does not extend to a smooth function on any open neighbourhood of the convex set, but that for each k extends to a Ck function on an open neighbourhood of the convex set. It follows that the diffeological and Sikorski notions of smoothness on convex sets do not coincide. We show that, for a convex set that is locally closed, these notions do coincide. With the diffeological notion of smoothness for convex sets, we then show that the category of diffeological spaces is isomorphic to the category of so-called exhaustive Chen spaces. This is joint work with Jordan Watts.

Title: Formal connections and the category of braid cobordisms

By:    Toshitake Kohno

         (Meiji University, the university of Tokyo)


The purpose of this talk is to describe an application of higher holonomy functors to the category of braid cobordisms. The theory of 2-connections on principal 2-bundles and their two dimensional holonomy has been developed by Baez, Schreiber and others. We obtain a categorical representation of the path 2-groupoid to the 2-Lie group. On the other hand, in the framework of the theory of iterated integrals, K.-T. Chen studied formal connections with values in the ring of non-commutative formal power series related to the cobar construction for the purpose of describing homology of loop spaces. We describe a universal form of two dimensional holonomy in terms of formal connections. We apply this method to the 2-category of braid cobordisms, where the 2-morphismsaresurface spanning braids in 4-space. Weobtain a power series of Lie type representation these 2-morphisms, which is considered as a 2-category version of the Kontsevich integral.

Title: Classification of locally standard torus actions

By:    Shintaro Kuroki

         (Okayama University of Science)


An action of a torus T on a manifold M is called a locally standard if the stabilizer is a sub-torus and the non-zero isotropy weights are a basis to its weight lattice at each point. In this case, we can get the following three data: 1. the orbit space M/T which admits the structure of a smooth manifold with corners; 2. the label λ, called unimodular labeling, on the facets of M/T by the isotropy weights on the codimension 2 invariant submanifolds; 3. the degree 2 cohomology class c of M/T with the coefficients of the integral lattice of Lie algebra of T (this represents the “twistedness” of M over M/T). In this talk, conversely, we show that three data recover the manifoldM withthelocally standard T-action up to equivariant diffeomorphism. This talk is the joint work with Yael Karshon.

Title: Introduction to Hypothesis H for Mathematicians ***

By:    Urs Schreiber

         (New York University in Abu Dhabi)


The key open question of contemporary mathematical physics is the elucidation of the currently elusive fundamental laws of “non-perturbative” states  ̶ranging from bound states as mundane as nucleons but more generally of quarks confined inside hadrons (declared a mathematical “Millennium Problem” by the Clay Math Institute), over topologically ordered quantum materials (currently sought by various laboratories), all the way to the ultimate goal of fundamentally understanding background-free quantum gravity and “grand unification”. Now, the foremost non-perturbative effect in quantum physics is “flux quantization”; and I begin by explaining in detail howthis findsitsnatural mathematicaldefinition indifferential cohesive homotopy theory. By going through key exampled  ̶as a fun exercise in cohesive homotopy theory ̶ I explain how to systematically derive from this: 1. magnetic flux quantization (experimentally seen in superconductors), and then by just the same logic: 2. the widely expected Hypothesis K that “RR-flux” is quantized in topological K-theory, and in evident non-abelian generalization: 3. our novel Hypothesis H that “G-flux” is quantized in unstable Cohomotopy (i.e.: framed Cobordism). Depending on time and interest, I may close by indicating (A) how coupling to gravity enhances these flux quantization laws to *T*wisted & *E*quivariant & *D*ifferential (TED) refinements of these cohomology theories and (B) how Hypothesis H explains anyonic topological order controlled by KZ-monodromy in bundles of conformal blocks. This is joint work with Hisham Sati. Slides will be available at:

Title: Homotopy types of higher smooth spaces ***

By:    Severin Bunk

         (The University of Oxford)


Differential homotopy theory refines ordinary homotopy theories of spaces. In particular, it encodes homotopical information using smooth, and possibly higher categorical, versions of spaces. To make a connection with classical homotopy theory, it is important to assign to any space in differential homotopy theory an ’underlying (ordinary) space’, and this should be achieved in a homotopically meaningful way. In this talk I will demonstrate methods to establish this connection and, if time permits, use these to compute an example homotopy type of a smooth space which does not have an underlying set.