# Topological Data Analysis: Developing Abstract Foundations (17w5108)

## Organizers

(Columbia University)

(Technische Universität München)

## Description

The Banff International Research Station will host the " Topological Data Analysis: Developing Abstract Foundations" workshop from July 30th to August 4th, 2017.

This workshop brings together two strands of research in applied algebraic topology. The first strand is the theory of topological persistence. Launched by Edelsbrunner, Letscher and Zomorodian in 1999 (with antecedents in work by Frosini in the early 1990s and by Bradley and Robins in the late 1990s), persistence has evolved through several phases: algorithmic, algebraic, category theoretic, sheaf theoretic. Persistence solves the problem of the topological instability of real-world data by adopting a multiscale framework where instability can be quantified and filtered out. Of course, it has raised many new questions. The second strand is statistical and probabilistic topology, characterized for instance by the sharp results of Kahle on random simplicial complexes, the inference theorems of Niyogi, Smale and Weinberger, and more recent work that focuses on classical questions of probability and statistics, including the construction of statistical summaries and sufficient statistics, addressing issues of dimension reduction, and the establishment of probabilistic limit theorems for topological quantities.

The two strands have quite different flavours, reflecting their origins in computer science, mathematics, probability and statistics. Our purpose is to bring focused attention to the foundations of statistical topology, motivated by the success of algebra and sheaf theory in topological persistence. We are driven by the question What is a random space?'' and we seek answers that reflect the subtlety and power of the work in both strands of applied algebraic topology.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides
an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disc
iplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineeri
ng Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional
de Ciencia y Tecnología (CONACYT).