10:00am - 12:00pmAlgebraic geometry in topological data analysis
Chair(s): Nina Otter (UCLA, United States of America)
In the last 20 years methods from topology, the mathematical area that studies “shapes", have proven successful in studying data that is complex, and whose underlying shape is not known a priori. This practice has become known as topological data analysis (TDA). As additional methods from topology still find their application in the study of complex structure in data, the practice is evolving and expanding, and now moreover draws increasingly upon data science, computer science, computational algebra, computational topology, computational geometry, and statistics.
While ideas from category theory, sheaf theory and representation theory of quivers have driven the theoretical development in the past decade, in the last years ideas from commutative algebra and algebraic geometry have started to be used to tackle some theoretical problems in TDA. The aim of the minisymposium is to seize this momentum and to bring together experts in algebraic geometry and researchers in topological data analysis to explore new avenues of research and foster research collaborations.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
High-throughput topological screening of nanoporous materials
Kathryn Hess
EPFL, Switzerland
Thanks to the Materials Genome Initiative, there is now a database of millions of different classes of nanoporous materials, in particular zeolites. In this talk I will sketch a computational approach to tackle high-throughput screening of this database to find the the best nano-porous materials for a given application, using a topological data analysis-based descriptor (TD) recognizing pore shapes. For methane storage and carbon capture applications, our method enables us to predict performance properties of zeolites. When some top-performing zeolites are known, TD can be used to efficiently detect other high-performing materials with high probability. We expect that this approach could easily be extended to other applications by simply adjusting one parameter: the size of the target gas molecule.
Sampling real algebraic varieties for topological data analysis
Parker Edwards1, Emilie Dufresne2, Heather Harrington3, Jonathan D. Hauenstein4
1University of Florida, United States of America, 2University of York, 3University of Oxford, 4University of Notre Dame
I will discuss an adaptive algorithm for finding provably dense samples of points on a real algebraic variety given the variety's defining polynomials as input. Our algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling and it also employs geometric heuristics to reduce the size of the sample. As persistent homology methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying these methods more feasible. I will also present results of applying persistent homology to point samples generated by an implementation of the algorithm.
How wild is the homological clustering problem?
Ulrich Bauer
TU Munich, Germany
Connected components form the basis of many clustering methods, often requiring a choice of two parameters (geometric scale and density). Applying 0th homology yields a diagram of vector spaces reflecting the connected components, with surjections in the scale parameter direction. This motivates the study of the parameter landscape by means of quiver representations: indecomposable summands can be interpreted as topological features. We identify all cases where the set of possible indecomposables has a simple classification (finite type or tame). The result is obtained using tilting theory and a novel equivalence theorem on cotorsion-torsion triples, whose development has been motivated by the clustering problem.
Learning elliptic curves
Daniele Agostini
Humboldt-Universitaet zu Berlin
Elliptic curves are all homeomorphic as topological spaces, more precisely, they are all real tori of dimension two. However, they carry infinitely many different complex structures. The topological structure can be detected easily by the tools of persistent homology, but can we also recover the complex structure? In other words, can we "learn" an elliptic curve from data? In my talk I would like to address this question.
