3:00pm - 5:00pmFrom algebraic geometry to geometric topology: crossroads on applications
Chair(s): Jose Carlos Gomez Larrañaga (CIMAT), Renzo Ricca (University of Milano-Bicocca), De Witt Sumners (Florida State University)
The purpose of the Minisymposium "From Algebraic Geometry to Geometric Topology: Crossroads on Applications" is to bring together researchers who use algebraic, combinatorial and geometric topology in industrial and applied mathematics. These methods have already seen applications in: biology, physics, chemistry, fluid dynamics, distributed computing, robotics, neural networks and data analysis.
(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)
Time-reversal homotopical properties of concurrent systems
Eric Goubault
École Polytechnique
Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which respect this directedness have been introduced to classify directed spaces. In this talk we study the properties of such invariants with respect to the reversal of the flow of time in directed spaces. Known invariants, natural homotopy and homology, have been shown to be unchanged under this time-reversal (as first noticed by e.g. K. Hess and L. Fajstrup). We show that these can be equipped with additional algebraic structure witnessing this reversal. Specifically, when applied to a directed space and to its reversal, we show that these refined invariants yield dual objects.These refined invariants, natural systems with a composition pairing, enjoy lots of interesting properties, as first noticed by Timothy Porter. We further refine natural homotopy by introducing a notion of relative directed homotopy and showing the existence of a long exact sequence of natural homotopy systems.Joint work with Philippe Malbos and Cameron Calk.
Efficient computation of multiparameter persistent homology
Abraham Martín del Campo Sánchez
CONACYT-CIMAT
We will present an efficient implementation of an algorithm to compute multiparametric persistent homology. The algorithm uses algebraic techniques and was originally proposed by Chacholski, Scolamiero, and Vaccarino. During the talk, we will explain the different reformulations of the definition of multiparametric persistence that give rise to the algorithm we corrected and implemented. This is joint work with Oliver Gäfvert (KTH) and Nina Otter (UCLA).
Classification of Streamline Topologies for Hamiltonian vector fields and its applications to Topological Flow Data Analysis
Takashi Sakajo
Kyoto University
We have developed a classification theory for structurally stable Hamiltonian vector fields on multiply connected planar domains in the presence of a uniform flow, which is a model of two-dimensional incompressible fluid flows. The theory enables us to assign a unique sequence of letters and a tree structure, called maximal words, and Reeb graphs, to every topological streamline structure of the Hamiltonian vector fields. They are intuitively interpretable to those who are not familiar with mathematics. An automatic conversion algorithm is now available as a computer software, and it is thus applicable to massive flow pattern data obtained by numerical simulations and/or physical measurements in fluid science, engineering and medical studies. By extracting global topological information from flow data, one is expected to figure out latent knowledge that are not recognized by experts in those fields so far. In addition, we have also developed a mathematical theory describing all possible global transitions of streamline topologies, without exceptions, through marginal structurally unstable Hamiltonian vector fields in terms of the changes of the sequence of letters. By simply comparing them, we predict the change of global flow patterns that could possibly happen in future.
Based on the classification theory, we introduce a new way of topological data analysis, called Topological Flow Data Analysis (TFDA). Owing to TFDA, long-time evolutions of flows (or Hamiltonian vector fields), whose data size often exceeds more than giga-bytes, is drastically compressed into a small size of text data expressing the change of streamline topologies, which is amenable to statistical and/or time-series analysis, and machine learning for global topological information with ease. We show some applications to medical images of heart flows and flow patterns in meteorology. We also show another example illustrating that TFDA is available to create a data-driven model predicting a complex flow phenomenon.
Robot motion planning and equivariant cohomology
Michael Farber
Queen Mary, University of London
The task of creating software AI for autonomous robots has an interesting topological aspect. A motion planning algorithm for a mechanical system can be represented by a section of a specific fibration and the complexity of such a section is measured by numerical invariants TC(X) and TC_r(X). Here X denotes the configuration space of the system and r>1 is an integer. I will describe recent results about computing the numbers TC(X) and TC_r(X) in the case when the space X is aspherical, i.e. has vanishing higher homotopy groups. The talk will include some results obtained jointly with S. Mescher, M. Grant, J.Oprea and G. Lupton.