Session Information (Location: Grand Ballroom)

Topology: New Trends Applications and Theory

Session Organizers: Javier Arsuaga (UC Davis) and Robin Wilson (Cal Poly Pomona)
Session Speakers: Nicolas Garcia Trillos (Brown University), Fabiola Manjarrez (National Autonomous University of Mexico), Angélica Osorno (Reed College), Jose Perea (Michigan State University), and Luis Valdez-Sanchez (University of Texas at El Paso).

LOCATION: Grand Ballroom

2:00 - 2:25 Jose Perea (Michigan State University)
2:30 - 2:55 Fabiola Manjarrez (National Autonomous University of Mexico)
3:00 - 3:30 Break
3:30 - 3:55 Luis Valdez-Sanchez (University of Texas at El Paso)
4:00 - 4:25 Nicolas Garcia Trillos (Brown University)
4:30 - 4:55 Angélica Osorno (Reed College)

Nicolas Garcia Trillos (Brown University)
Title: Graph-based methods in machine learning and their continuous counterparts.
Abstract: Many machine learning procedures aimed to extract information from data can be defined as precise mathematical objects that are constructed in terms of the data. It is often assumed that the data is “big” in complexity but also in quantity, and in this “large amount of data’’ setting, a basic mathematical concept that one can explore is that of closure of a given class of statistical procedures (i.e. what are the limiting procedures as the number of data points available goes to infinity.) In this talk, I will explore this notion in the context of graph-based methods. Examples of such methods include minimization of Cheeger cuts, spectral clustering, and graph-based bayesian semi-supervised learning, among others.

Fabiola Manjarrez (National Autonomous University of Mexico)
Title: Genera of tunnel number one satellite knots
Abstract: Tunnel number one satellite have been characterized by a 4-tuple of integers. In this talk I will show an algorithm to compute the genus of such knots from the 4-tuple. This is joint work with Mario Eudave, Enrique Ramírez and Jesús Rodríguez.

Angélica Osorno (Reed College)
Title: Systems of fixed points and equivariant homotopy theory
Abstract: I will describe the classical result of how to recover the homotopy theory of a space with an action of a group G from the homotopy theory of its system of fixed points. Then I will describe the analogous recent result of Guillou and May for genuine equivariant G-spectra, I will then show how to use this result to construct a new equivariant infinite loop space machine, whose input data is in terms of fixed points. This is joint work with Anna Marie Bohmann.

Jose Perea (Michigan State University)
Title: Topological Time Series Analysis
Abstract: Time varying observations are ubiquitous in today's data rich world; examples include real-valued time series (like sound and speech), video data, dynamic networks, etc. I will show in this talk how techniques from nonlinear time series analysis, dynamical systems, and computational topology can be combined to generate topological summaries describing the dynamics underlying such systems. Several applications in areas such as biology, speech sciences and artificial intelligence will be discussed.

Luis Valdez-Sanchez (University of Texas at El Paso)
Title: Surgery on knots in the 3-sphere
Abstract: Any closed and orientable 3-manifold can be constructed by performing surgery on the knot components of some link in the 3-sphere. More specifically, by Thurston's work, the complement of any knot in the 3-sphere is either a Seifert, toroidal, or hyperbolic 3-manifold, and all but finitely many (exceptional) surgeries on a knot with hyperbolic complement yield hyperbolic 3-manifolds. In this talk we give a brief overview of the surgery construction on knots in the 3-sphere and some of the old and current topological and geometric questions that arise from it.

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