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Welcome to the STRUCTURES wiki.

This wiki is currently centered around the exploratory project EP Mathematics and Data within STRUCTURES and is meant to provide a platform for collaborative work on data analysis across the fields of pure mathematics and natural sciences.

Purpose

The purpose of this wiki is to provide a public platform where people can learn about recent approaches to data analysis via topological and geometrical techniques. We take two complementary approaches to achieve this:

• The foundations of topology and geometry are well known to most mathematicians, but less so to researchers in the natural sciences, who we believe might benefit greatly from incorporating related techniques into their toolbox. To remedy this we collect the most important elements of the underlying theory and provide introductory articles to the relevant subjects.
• Analysing and handling real-life data is an art well-developed in the natural sciences, but may at times seem enigmatic to mathematicians, who usually are not exposed to large sets of data on a daily basis. It is apparent that an implementation of the theoretical ideas on computers is paramount for applications, so we provide a list of existing software and offer a number of short tutorials. We furthermore present case studies of real-life applications.

We hope that in establishing this wiki from these two perspectives, researchers from both mathematics and natural sciences are provided with the relevant background to easily approach each other, laying ground for fruitful collaborations.

Structure of this wiki

In this wiki a brief introduction to computational topological methods shall be given, beginning with an example analysis. A primer on computational topology theory follows, including the most important theorems of the topological constructions undertaken, leading over to a few recent research examples, in which these methods are applied. Subsequently, several numerical examples are given, which might serve as points of entry for individual first steps with computational topology methods.

• What is Topological Data Analysis? - A Primer
• Introduction to Topological and Geometric Methods in Data Analysis
• Literature
• List of Software
• Case Studies

Activities

• Lectures
  • Geometric Methods in Data Analysis - Andreas Ott [1]
• Seminars
  • High-dimensional geometry and probability with view towards applications in data science - Jan Swoboda [2]
  • Topological Data Analysis and Persistent Homology - Johannes Krebs [3]
• Workshops
  • Geometry, Topology, and Computation - Peter Albers, Roman Sauer (Karlsruhe), Anna Wienhard June 12 - 14, Heidelberg (Mathematikon)
• Summer School
  • Persistent Homology and Barcodes - Peter Albers, Leonid Polterovich (Tel Aviv), Kai Zehmisch (Gießen) August 5-9, JLU Gießen - Schloß Rauischholzhausen

Disclaimer

One goal of this wiki is to help make information widely available and usefully related to other information. In this users and contributors are expected to follow traditional academic practice. Whenever useful, references and pointers to related literature are given, providing further information to the reader.

Nonetheless, this wiki is meant neither to be complete nor original.

Contributing to this wiki

Everyone is invited to contribute to this wiki. If you are interested, get in touch with us via mail:

structures-hiwi@mathi.uni-heidelberg.de