Cat 2015 Topological Data Analysis: New Developments and Challenges

s _______________________________________________________________________________________ Ulrich Bauer Title:Induced Matchings and the Algebraic Stability of persistence Barcode Abstract: We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f. We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f. As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between their barcodes. Our main result is based a structure theorem for submodules and quotients of persistence modules, and yields a novel “single-morphism” characterization of the interleaving relation on persistence modules. Gunnar Carlsson Colloquium title:The Shape of Data Colloquium abstract: There has been a great deal of attention paid to "Big Data" over the last few years. However, often as not, the problem with the analysis of data is not as much the size as the complexity of the data. Even very small data sets can exhibit substantial complexity. There is therefore a need for methods for representing complex data sets, beyond the usual linear or even polynomial models. The mathematical notion of shape, encoded in a metric, provides a very useful way to represent complex data sets. On the other hand, Topology is the mathematical sub discipline which concerns itself with studying shape, in all dimensions. In recent years, methods from topology have been adapted to the study of data sets, i.e. finite metric spaces. In this talk, we will discuss what has been done in this direction and what the future might hold, with numerous examples. Title: Structures on Spaces of Persistence Barcodes Abstract: There are a number of situations where databases of "unstructured data" contain elements which themselves have geometric structure. In this situation, it becomes important to study not only individual barcodes, but there placement in a space of such barcodes. There are different notions of these spaces and we will talk about a number of possibilities for them, including some theorems which can be useful in translating topological methods into a more traditional machine learning framework. Frédéric Chazal Title: Persistent homology for geometric complexes: stability and statistical aspects Abstract:: Persistent homology plays a fundamental role in Topological Data Analysis to extract multiscale topological features from data. In this talk we will present stability results for the persistent homology of filtered simplicial complexes built on top of totally bounded metric spaces. We will show how these results can be exploited to obtain statistical properties of persistence information in Topological Data Analysis.: Persistent homology plays a fundamental role in Topological Data Analysis to extract multiscale topological features from data. In this talk we will present stability results for the persistent homology of filtered simplicial complexes built on top of totally bounded metric spaces. We will show how these results can be exploited to obtain statistical properties of persistence information in Topological Data Analysis.