Department of Mathematics and Statistics,

University of Ottawa

**Coffee:** Common Room, before the colloquium at 3:30pm.

(Carleton-uOttawa transportation information)

- Speaker: Vin de Silva (Pomona College)
- Title: Persistent Cohomology & Circular Coordinates
- Abstract: The ideas in this talk represent a fusion of two different strands of research from the early 2000s: NLDR (nonlinear dimensionality
reduction) and PCT (point-cloud topology). The two fields meet in the problem of finding circular coordinates for a data set. This new technology has possible applications in signal processing and dynamical systems.

The idea behind NLDR is to take a high-dimensional data set, perhaps obtained as a collection of scientific measurements, and to find a small set of real-valued coordinates that reveal meaningful parameters of the data. The classical linear instance of this is principal components analysis (PCA). The paradigm was introduced by Josh Tenenbaum in the late 1990s. Two well-known algorithms, Isomap (Tenenbaum, dS, Langford) and LLE (Roweis, Saul) were published in 2000, and many other researchers have published NLDR algorithms since then. Each algorithm exploits a different aspect of the inherent geometry of the data, in order to construct the coordinates.

Roughly over the same time period, several groups of researchers began developing tools and techniques for applying algebraic topology to scientific data. Here the idea is to detect the topological structure of a set of high-dimensional observed data points. The difficulty is that data are inherently noisy, and topological invariants are extremely sensitive to local noise. The early breakthrough came in 2000, with the publication of the persistence algorithm of Edelsbrunner, Letscher and Zomorodian. This new framework gives robust versions of the classical invariants of algebraic topology (such as homology and betti numbers), that can be used to estimate the topology (or "shape") of a noisy data set.

The two fields meet in the following way. From the NLDR side, one can generalize from real-valued coordinates to more general coordinates. We focus on circle-valued coordinates (such as angles). To discover these coordinates, we exploit not the geometry but the topology of the data. In order to do this robustly, it is necessary to use a persistence framework. I will indicate how these calculations are carried out, and give some examples of how one can exploit the resulting coordinates to empirically study time-series data and dynamical systems.

My collaborators in this work are Mikael Vejdemo-Johansson, Dmitriy Morozov, and Primoz Skraba.

- Speaker: Richard Hoshino, Quest University Canada, Squamish, BC
- Title: Optimal Pricing for Distance-Based Transit Fares
- Abstract: Numerous urban planners advocate for differentiated transit pricing to
improve both ridership and service equity. Several metropolitan cities
are considering switching to a more "fair fare system", where passengers
pay according to the distance travelled, rather than a flat fare or zone
boundary scheme that discriminates against various marginalized groups.

In this presentation, we present a two-part optimal pricing formula for switching to distance-based transit fares: the first formula maximizes forecasted revenue given a target ridership, and the second formula maximizes forecasted ridership given a target revenue. Both formulas hold for all price elasticities.

Our theory has been successfully tested on the SkyTrain mass transit network in Metro Vancouver, British Columbia, with over 400,000 daily passengers. This research served Metro Vancouver's transportation authority as they recently decided to switch to distance-based fares, with the full implementation likely to take place sometime in 2019.

(This is joint work with Jeneva Beairsto, a fourth-year undergraduate student).BIO: Richard Hoshino is a mathematics professor at Quest University Canada in Squamish, British Columbia. Prior to his arrival at Quest in 2013, Richard was a post-doctoral fellow at the National Institute of Informatics in Tokyo (2010-2012), and was a mathematician here in Ottawa with the Canada Border Services Agency (2006-2010). He has published 33 research papers across numerous fields, including graph theory, marine container risk-scoring, biometric identification, and sports tournament scheduling.

Richard has consulted for a billion-dollar professional baseball league, as well as three Canadian TV game shows (Qubit, Splatalot, Spin-Off), and has worked with his undergraduate students to implement automated employee scheduling systems for small businesses in British Columbia. He is heavily involved in high school outreach, and continues to visit dozens of schools each year to share his passion and love for mathematics. Richard is also the author of "The Math Olympian", a novel that is currently ranked #2 by GoodReads in Best Young Adult Books that Empower.

Richard holds a B.Math. from the University of Waterloo, a B.Ed. from Queen's University, and an M.Sc. and Ph.D. from Dalhousie University.

- Speaker: Prof. Gang Tian, Chair Professor and Vice President of Peking University
- Title: Poincare Conjecture and Geometrization
- Abstract: For more than one hundred years, the Poincare conjecture was a driving force for topologists and its study led to many important advances in the area. It was finally solved by Perelman using differential geometric methods. In this lecture, Dr. Gang Tian will introduce the Poincare conjecture and give a brief history of its mathematical pursuit. He will explain some of the geometric ideas involved in solving the conjecture, in particular, the geometrization of 3-spaces. The talk will end with some speculations on future developments in geometry.

Short Bio

- Speaker: Antoine Henrot (Universite de Lorraine, France/CRM)
- Title: Isoperimetric inequalities for eigenvalues
- Abstract: In this talk, really in the spirit of a colloquium, we will review classical and more recent isoperimetric inequalities involving the eigenvalues of the Laplacian (mainly with Dirichlet boundary conditions) with various geometric constraints (on the area, the perimeter, the diameter...). These questions will lead us to study the links between the eigenvalues and the geometry of the domain. It will also be the occasion to introduce some classical tools of modern analysis like Schwarz and Steiner rearrangement, variational methods, Gamma-convergence, derivative with respect to the domain. We will also give several open problems showing the vitality of this field.