Preprints, J.F. Jardine

Local data structures

26 pages (2023)
Abstract : Local data structures are systems of neighbourhoods within data sets. Specifications of neighbourhoods can arise in multiple ways, for example, from global geometric structure (stellar charts), combinatorial structure (weighted graphs), desired computational outcomes (natural language processing), or sampling. These examples are discussed, in the context of a theory of neighbourhoods. This theory is a step towards a mathematical understanding of clustering for large data sets. These clusters can only be approximated in practice, but approximations can be constructed from neighbourhoods via patching arguments that are derived from the Healy-McInnes UMAP construction. The patching arguments are enabled by changing the theoretical basis for data set structure, from metric spaces to extended pseudo metric spaces.

Layers and stability

13 pages (2021)
Abstract : The hierarchy associated to clusters in the HDBSCAN algorithm has layers, which are defined by cardinality. The layers define a subposet of the HDBSCAN hierarchy, which is a strong deformation retract and admits a stability analysis. That stability analysis is introduced here. Cardinality arguments lead to sharper stability results than one sees for branch points.

Stability for UMAP

13 pages (2020)
Abstract : This paper displays the Healy-McInnes UMAP construction V(X,N) as an iterated pushout of Vietoris-Rips objects associated to extended pseudo metric spaces (ep-metric spaces) that are defined by choices of neighbourhoods of the elements of a finite set X. An inclusion of X in another finite set Y defines a map of UMAP systems from V(X,N) to V(Y,N') in the presence of a compatible system of neighbourhoods N' for Y. There is also a map of ep-metric spaces from (X,D) to (Y,D'), where D and D' are are colimits (global averages) of the ep-metrics defined by the neighbourhood systems for X and Y. We prove a stablity result for the restriction of this ep-metric space map to global components. This stability result translates, via excision for path components, to a stability result for global components of the UMAP systems.

Metric spaces and homotopy types

17 pages (2020)
Abstract : By analogy with methods of Spivak, there is a realization functor which takes persistence diagram Y in simplicial sets to an extended pseudo-metric space Re(Y). The functor Re has a right adjoint, called the singular functor, which takes an ep-metric space Z to a persistence diagram S(Z). We give an explicit description of Re(Y), and show that it depends only on the 1-skeleton of Y. If X is a finite data set, then there is an isomorphism between the realization of the Vietoris-Rips system V(X) and the finite metric space X. The persistence diagrams V(X )and S(X) are sectionwise weakly equivalent for all such data sets X.

Directed persistence

10 pages (2020)
Abstract:  Oriented Vietoris-Rips complexes arise naturally from listings of the underlying data sets. The path category (fundamental category) invariant for a finite oriented simplicial complex is computable, via an algorithm that is recalled here. A stability theorem for oriented complexes is proved, which specializes to a stability result for path categories.

Branch points and stability

8 pages (2020), arXiv 2003.06285 [math.AT], revised May 3, 2020
Abstract:  The hierarchy and branch point posets for a data set each have a calculus of least upper bounds. Upper bounds are used to show that the map of branch points associated to an inclusion of data sets is a controlled homotopy equivalence, where the control is defined by an upper bound relation that is associated to the Hausdorff distance between the data sets.

This preprint is a sequel to the paper Stable components and layers (arXiv:1905.05788 [math.AT]), which is to appear in Canad. Math. Bull.

Persistent homotopy theory

21 pages (2020), arXiv 2002.10013 [math.AT], revised Oct 26, 2020
Abstract:  Vietoris-Rips and degree Rips complexes are represented as homotopy types by their underlying posets of simplices, and basic homotopy stability theorems are recast in these terms. These homotopy types are viewed as systems (or functors), which are defined on a parameter space. The category of systems of spaces admits a partial homotopy theory that is based on controlled equivalences, suitably defined, that are the output of homotopy stability results.

Data and homotopy types

12 pages (2019), arXiv 1908:06323 [math.AT]
Abstract:  This paper presents explicit assumptions for the existence of interleaving homotopy equivalences of both Vietoris-Rips and Lesnick complexes that are associated to an inclusion of data sets. Consequences of these assumptions are investigated on the space level, and for corresponding hierarchies of clusters and their sub-posets of branch points. Hierarchy and branch point posets admit a calculus of least upper bounds, which is used to show that the map of branch points associated to the inclusion of data sets is a controlled homotopy equivalence under such assumptions.
NB: This paper will not be published. Its content has been expanded and enhanced in the newer papers Persistent homotopy theory and Branch points and stability.

Complexity reduction for path categories

14 pages (2016), arXiv: 1909.08433 [math.AT]
Abstract: This paper displays complexity reduction techniques for calculations of path categories (or fundamental categories) P(K) for finite simplicial and cubical complexes K. The central technique involves identifying inclusions of complexes for which the induced functor of path categories is fully faithful. Refinements of cubical complex structures are discussed. A first method for parallelizing the calculation of path categories for cubical complexes is introduced.

Pro-equivalences of diagrams

19 pages (2016), arXiv: 1909.08429 [math.AT]
Abstract: This paper presents a model structure for natural transformations of diagrams of simplicial presheaves of a fixed shape, in which the weak equivalences are defined by analogy with pro-equivalences between pro-objects.

Path categories and quasi-categories

38 pages (2015), arXiv: 1909.08419 [math.AT]
Abstract: This paper was written in support of the paper The Local Joyal Model Structure, by Nicholas Meadows. Meadows' paper constructs a presheaf-theoretic version of Joyal's quasi-category model structure, for which a map of simplicial presheaves is a weak equivalence if it is a stalkwise categorical weak equivalence.
The present paper is an exposition of the Joyal structure, which displays proofs of technical details that are used in the Meadows paper. It was also written with a view to setting up potential applications in concurrency theory. The main new technical result is a characterization of weak equivalences of quasi-categories as maps which induce equivalences of a certain infinite family of naturally defined groupoids.

 The Barratt-Priddy-Quillen Theorem

8 pages (2009)
Abstract: This preprint is a polished version of notes for a colloquium given at the University of Bremen in November, 2007. It gives a combinatorial proof of the Barratt-Priddy-Quillen theorem which asserts that the stable homotopy groups of spheres are isomophic to the homotopy groups of the spaces obtained by applying the Quillen plus construction to the classifying space of the infinite symmetric group. This was my take on the subject at the time that the lecture was given, but the ideas are essentially the same as those found in the following paper:
Christian Schlichtkrull. The homotopy infinite symmetric product represents stable homotopy. Algebr. Geom. Topol. 7, 1963-1977, 2007.

E_2 model structures for presheaf categories

35 pages (2001)
Abstract: The paper gives E_2 model structures in the style of Dwyer-Kan-Stover and Goerss-Hopkins for categories of simplicial objects in pointed simplicial presheaves, presheaves of spectra and presheaves of symmetric spectra on a small Grothendieck site. Analogs of these results for unstable and stable motivic homotopy theory are also displayed and proved. The key technical device is a bounded approximation technique for objects in the respective categories, which ultimately depends on cardinality count methods previously seen in localization theory.
Note: This paper will not be published.

Cohomological invariants associated to symmetric bilinear forms

35 pages (1998)
Abstract: This paper reviews the construction of the Hasse-Witt and Stiefel-Whitney classes for an orthogonal representation of a Galois group, and then gives a simplicial presheaf theoretic demonstration of the Frohlich-Kahn-Snaith formula for the Hasse-Witt invariant of the associated twisted form. A Steenrod squares argument is used to show that this formula has an analogue in degree 3. The mod 2 étale cohomology of the classifying simplicial scheme of the automorphism group of an arbitrary non-degenerate symmetric bilinear form is calculated, and the relation of this cohomology ring with the Hasse-Witt classes of the Frohlich twisted form is discussed.
This is a corrected version of a paper that has been published (Expositiones Math. 10 (1992), 97-134). There is a bad printer error which makes the introduction of the published version completely unintelligible.

A rigidity theorem for L-theory

8 pages (1983)
Abstract: This paper proves a rigidity theorem for mod l Karoubi L-theory, and then uses it to calculate the mod l Karoubi L-groups of algebraically closed fields. All proofs and calculations given here depend on the homotopy theory of simplicial sheaves.
This paper was written in 1983, and is unpublished. Karoubi proved and published similar results from a different point of view - see this preprint for further details.

 Algebraic homotopy theory, groups and K-theory

 Thesis, University of British Columbia (1981)