10:00am - 12:00pmFrom 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)
Momentum of vortex tangles by weighted area information
Renzo L. Ricca
University of Milano-Bicocca,
A method based on the interpretation of classical linear and angular momentum of vortex dynamics in terms of weighted areas of projected graphs of filament tangles has been introduced to provide an accurate estimate of physical information when no analytical description is available [1,2]. The method is implemented here for defects governed by the Gross-Pitaevskii equation [3]. New results based on direct application of this method to determine the linear momentum associated with interacting vortex rings, links and knots are presented and discussed in detail. The method can be easily extended and adapted to more complex systems, providing a useful tool for real-time diagnostics of dynamical properties of networks of filamentary structures.
This is joint work with Simone Zuccher (U. Verona).
[1] Ricca, R.L. (2008) Momenta of a vortex tangle by structural complexity analysis. Physica D 237, 2223-2227.
[2] Ricca, R.L. (2013) Impulse of vortex knots from diagram projections. In Topological Fluid Dynamics: Theory and Applications (ed. H.K. Moffatt et al.), pp. 21-28. Procedia IUTAM 7, Elsevier.
[3] Zuccher, S. & Ricca, R.L. (2019) Momentum of vortex tangles by weighted area information. Submitted.
Alexandrov spaces and topological data analysis
Fernando Galaz-García
KIT
Alexandrov spaces (with curvature bounded below) are metric generalizations of complete Riemannian manifolds with a uniform lower sectional curvature bound. In this talk I will discuss the geometric and topological properties of these metric spaces and how they arise in the context of topological data analysis.
Geometrical and topological analysis of chromosome conformation capture data
Javier Arsuaga
University of California, Davis
Despite the impressive development of methods to analyze Chromosome Conformation Capture (CCC) data, the topology of any genome still remains unknown. The output of a CCC experiment is a matrix of pairwise contact probabilities between genomic loci from which a map of distances, called distance map, can be obtained.
In this work we use distance geometry and random knotting arguments to derive some rigorous results for the interpretation of distance maps. In particular we provide a rigorous characterization of the distance map of a knot and of some of its symmetries. We end by presenting a key result that shows that in the presence of noise the topology of a chromosome cannot be recovered from a distance map.
Joint work with: K. Ishihara, K. Lamb, M. Pouokam, K. Shimokawa, and M. Vazquez
Asymptotic behavior of the homology of random polyominoes
Érika Roldán-Roa
The Ohio State University
In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extended, using the same techniques, to other polyforms and higher dimensions.