# Computational Algebraic Topology

### Dmitry N. Feichtner-Kozlov

Universität Bremen, Germany### Gunnar Carlsson

Stanford University, United States

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## Abstract

The workshop was conducted jointly with a workshop in statistical learning theory. There was substantial interaction between the two groups, both formally in terms of talks attended by members of both groups, as well as via informal discussions. The intellectual themes which were presented during the workshop are described below. \vspace{0.3cm} \noindent {\bf Sensor nets and engineering applications:} \vspace{0.3cm} \noindent In the opening talk R.\ Ghrist spoke about the topology necessary to develop methods for determining intruders have entered a net of sensors, and for counting their number. Ghrist, jointly with V. de Silva and Y. Baryshnikov, has developed techniques based directly on homological calculations as well as on integrals over Euler characteristics which hold promise for implementable algorithms. In order for such algorithms to be maximally useful, one must develop error insensitive methods, which will require more probabilistic methods to be included within the algebraic topological framework. \pagebreak \noindent {\bf Combinatorial applications:} \vspace{0.3cm} \noindent Several presentations at the workshop elaborated on the subject of combinatorial algebraic topology. D.\ Kozlov has given a~~survey talk, which has set the accents on the subject, tying together structures, methods, and applications, as these are present at the current state of the development. Talks by R. Jardine and M. Raussen concerned the combinatorial and computational aspects of homotopy theory, finding applications of such abstract notions as Quillen's closed model category. K. Knudson gave an~~interesting account of connections between persistent homology and discrete morse theory. Finally, the talk of E.\ Babson dealt with more probabilistic aspects and served as a~~bridge to the presentations of M. Kahle and P.\ Bubenik. \vspace{0.3cm} \noindent {\bf Dynamical systems:} \vspace{0.3cm} \noindent K. Mischaikow and S. Day spoke about the use of algebraic topology to understand the qualitative structure of dynamical systems. Mischaikow introduced his paradigm of building databases of dynamical systems based on choices of parameter values. His methods permit the construction of partitions of parameter space within which the qualitative structure remains the same. In addition, Conley index methods, or rather their computational versions, are used to prove the existence of fixed points, recurrent points, and invariant subsets within a given region in a spatial domain. \vspace{0.3cm} \noindent {\bf Data analysis:} \vspace{0.3cm} \noindent G. Carlsson and V. de Silva spoke about applications of various kinds of diagrams to understand the qualitative geometric nature of data sets. For example, persistence diagrams allow one to recover Betti numbers of sublevel sets of a probability distribution, multidimensional persistence allows one to study sublevel sets of various functions as well, and the analysis of structure theorems for certain kinds of quivers permits one to extend the bootstrap methods to clustering, Betti numbers, as well as to perform dynamic clustering (i.e. clustering over time). There are now viable computational methods for all of these applications. \vspace{0.3cm} \noindent {\bf Probabilistic methods:} \vspace{0.3cm} \noindent M. Kahle and P. Bubenik spoke about the beginnings of stochastic algebraic topology. Work at the level of zeroth Betti numbers has already been carried out by M. Penrose, under the heading of ``geometric random graphs". What is now needed is an extension of this work to higher dimensional homology groups, as well as to the barcodes which arise in persistent homology. Ultimately, precise results along these lines will open up the possibility of direct evaluation of significance of various qualitative observations given a null hypothesis. \vspace{0.3cm} \noindent There were also several talks more centered at applications, such as vision recognition (J. Giesen) and material science (R.\ MacPherson). \vspace{0.3cm} \noindent All things considered, the workshop was a~~great success in terms of scientific interaction, both within this group, as well as with the researchers in statistical learning theory, as was witnessed by many involved discussions, which often lasted well into the late evenings.