Mathematics

What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient statistic. We therefore propose that the correct invariant is not the persistence diagram of X, but rather the collection of persistence diagrams of many small subsets. This invariant, which we call "distributed persistence," is trivially parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of point clouds (with the quasi-isometry metric) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is a global quasi-isometry. This is a much stronger property than simply being injective, as it implies that the inverse of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the quasi-isometry bounds depend on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying certain covering properties that arise with high probability when randomly sampling subsets. These theoretical results are complemented by two synthetic experiments demonstrating the use of distributed persistence in practice.

The present note surveys my research related to generalizing notions of abelian group theory to non-commutative case and applying them particularly to investigate fundamental groups.

The paper focuses on investigating how certain relations between strict n -categories are preserved in a particular implementation of (∞,n) -categories, given by saturated n -complicial sets. In this model, we show that the (∞,n) -categorical nerve of n -categories is homotopically compatible with 1 -categorical suspension and wedge. As an application, we show that certain pushouts encoding composition in n -categories are homotopy pushouts of saturated n -complicial sets.

In this paper, we introduce the notion of G ∞ -ring spectra. These are globally equivariant homotopy types with a structured multiplication, giving rise to power operations on their equivariant homotopy and cohomology groups. We illustrate this structure by analysing when a Moore spectrum can be endowed with a G ∞ -ring structure. Such G ∞ -structures correspond to power operations on the underlying ring, indexed by the Burnside ring. We exhibit a close relation between these globally equivariant power operations and the structure of a β -ring, thus providing a new perspective on the theory of β -rings.

We develop the foundations of G -global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the G -global algebraic K -theory of small symmetric monoidal categories with G -action, unifying G -equivariant algebraic K -theory, as considered for example by Shimakawa, and Schwede's global algebraic K -theory. As an application of the theory, we prove that the G -global algebraic K -theory functor exhibits the category of small symmetric monoidal categories with G -action as a model of connective G -global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic K -theory.

We construct effective GKM T 3 -actions with connected stabilizers on the total spaces of the two S 2 -bundles over S 6 with identical GKM graphs. This shows that the GKM graph of a simply-connected integer GKM manifold with connected stabilizers does not determine its homotopy type. We complement this by a discussion of the minimality of this example: the homotopy type of integer GKM manifolds with connected stabilizers is indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lower number of fixed points. Regarding geometric structures on the new example, we find an almost complex structure which is invariant under the action of a subtorus. In addition to the minimal example, we provide an analogous example where the torus actions are Hamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integer GKM manifolds.

We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie-Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish at a prime p . Combined with recent results on the relationship between embedding calculus and finite-type theory, we deduce that the (n+1) -st Goodwillie-Weiss approximation is a p -local universal Vassiliev invariant of degree ≤n for every n≤p+1 .

In his classical textbook on algebraic topology Edwin Spanier developed the theory of covering spaces within a more general framework of lifting spaces (i.e., Hurewicz fibrations with unique path-lifting property). Among other, Spanier proved that for every space X there exists a universal lifting space, which however need not be simply connected, unless the base space X is semi-locally simply connected. The question on what exactly is the fundamental group of the universal space was left unanswered. The main source of lifting spaces are inverse limits of covering spaces over X , or more generally, over some inverse system of spaces converging to X . Every metric space X can be obtained as a limit of an inverse system of polyhedra, and so inverse limits of covering spaces over the system yield lifting spaces over X . They are related to the geometry (in particular the fundamental group) of X in a similar way as the covering spaces over polyhedra are related to the fundamental group of their base. Thus lifting spaces appear as a natural replacement for the concept of covering spaces over base spaces with bad local properties. In this paper we develop a general theory of lifting spaces and prove that they are preserved by products, inverse limits and other important constructions. We show that maps from X to polyhedra give rise to coverings over X and use that to prove that for a connected, locally path connected and paracompact X , the fundamental group of the above-mentioned Spanier's universal space is precisely the intersection of all Spanier groups associated to open covers of X , and that the later coincides with the shape kernel of X . Furthermore, we examine in more detail lifting spaces over X that arise as inverse limits of coverings over some approximations of X .

By utilizing the idea of Colombeau's generalized function, we introduce a notion of asymptotic map between arbitrary diffeological spaces. The category consisting of diffeological spaces and asymptotic maps is enriched over the category of diffeological spaces, and inherits completeness and cocompleteness. In particular, the set of asymptotic functions on a Euclidean open set include Schwartz distributions and form a Colombeau type smooth differential algebra over Robinson's field of asymptotic numbers. To illustrate the usefulness of our machinery, we show that homotopy extension property can be established for smooth relative cell complexes if we exploit asymptotic maps instead of smooth ones.

We investigate the correspondence between generalized persistence modules and graded modules in the case the indexing set has a monoid action. We introduce the notion of an action category over a monoid graded ring. We show that the category of additive functors from this category to the category of Abelian groups is isomorphic to the category of modules graded over the set with a monoid action, and to the category of unital modules over a certain smash product. Furthermore, when the indexing set is a poset, we provide a new characterization for a generalized persistence module being finitely presented.