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67th meeting Cascade Topology Seminar

November 12-13, 2022

Portland State University

All events except the conference dinner will be held in Room 230 of the Academic and Student Recreation Center, 1800 SW 6th Avenue on Portland State University's downtown campus.

Schedule

Saturday November 12, 2022

9:30 am -10:00 am Caffeine and carbohydrates

10:00 am - 11:00 am Kristen Hendricks, Rutgers University and SL Math Research Institute

Title: Rank inequalities for the Heegaard Floer homology of branched covers

Abstract: Heegaard Floer homology is a powerful invariant of three-manifolds and knots and links within them due to Oszvath and Szabo; among its other properties, the rank of the theory for rational homology spheres has deep connections to the fundamental group. Heegaard Floer homology is defined in terms of Lagrangian Floer homology in an auxiliary symplectic manifold. In the presence of a branched covering action on a three manifold, one naturally aspires to use equivariant Lagrangian Floer theory to extract additional information from Heegaard Floer homology. We discuss the history of applications of equivariant Lagrangian Floer theory to Heegaard Floer homology, and establish an inequality between the ranks of the Heegaard Floer homologies of any double cover between three-manifolds branched along a null-homologous link. This is joint work with R. Lipshitz and T. Lidman.

11:00 am – 11:15 am Break

11:15 am -12:15 pm XiaoLin Shi, University of Washington

Title: Equivariant methods in chromatic homotopy theory

Abstract: I will talk about equivariant homotopy theory and its role in the proof of the Segal conjecture and the Kervaire invariant one problem. Then, I will talk about chromatic homotopy theory and its role in studying the stable homotopy groups of spheres. These newly established techniques allow one to use equivariant machinery to attack chromatic computations that were long considered unapproachable.

12:15 pm - 2:15 pm Lunch

2:15 pm - 3:15 pm Bala Krishnamoorthy, Washington State University, Vancouver

Title: Scale Invariance of the Normalized Bottleneck Distance on Persistence Diagrams

Abstract: Persistent homology is a powerful tool in applied topology or topological data analysis (TDA). For point cloud data sampled from a manifold, persistence diagrams allows us to observe "holes" in data. Further, the bottleneck distance allows us to compare data sets by directly comparing their persistence diagrams.

One potential drawback of this TDA pipeline for comparing data sets is that point clouds sampled from homeomorphic manifolds can have arbitrarily large bottleneck distance when there is a large degree of scaling. We propose a solution to this problem via appropriate metric normalization and an associated bottleneck distance. We define a new distance between persistence diagrams termed scaled bottleneck distance and study its properties. We also explore specific examples from dimension reduction in which the scaled bottleneck may be a better choice for comparing the original and reduced data sets.

This is joint work with Nathan May.

3:15 pm – 3:45 pm Caffeine and conversation

3:45 pm – 4:45 pm Marissa Masden, University of Oregon

Title: Geometric Duality, Neural Networks, and Decision Boundaries

Abstract: One goal of the application of topological data analysis in machine learning is to understand the topology of a neural network's decision boundary, the level set of the function representing its performance on a classification task. While previous studies have focused on approximations to decision boundaries, we wish to obtain the topological invariants of exact decision boundaries of neural networks. Recently, E. Grigsby and K. Lindsey defined the canonical polyhedral complex of a "ReLU" network, which encodes the network's decomposition of input space into piecewise linear regions, and correspondingly its decision boundary. While the canonical polyhedral complex is composed of seemingly arbitrary combinatorial types, we use an algebraic structure from the theory of oriented matroids to prove that generically its geometric dual is a cubical complex, which we call the sign sequence cubical complex. Furthermore, the locations and sign sequences of the vertices of a network's polyhedral complex fully determine the complex combinatorially and topologically. We use these theoretical results to derive a novel algorithm which calculates the polyhedral complex of a ReLU network, including an encoding of all polyhedral adjacency relations, by computing the sign sequences of the vertices.

7 pm - ? Seminar dinner at the home of Steven Bleiler. Maps and directions available at meeting

Sunday, November 13, 2022

9:30 am – 10:00 am Caffeine and carbohydrates

10 am -11 am Steven Bleiler, Portland State University

Title: Topological foundations of quantum computation and algorithms

Abstract: Quantum computation is the idea, originally developed by physicists in the 1980s, that the counter-intuitive properties of quantum mechanics might allow for the construction of computers that could perform certain mathematical feats that no non-quantum machine would ever be capable of. That promise is now starting to become real.

Quantum computers will not beat their classical counterparts at everything, but those tasks that quantum computers will do better will be done significantly better. This is why many computing and technology companies large and small, from hardware manufacturers and chip makers such as IBM and Intel to software and tech giants such as Microsoft and Google, are building and refining quantum hardware and software. Some of these approaches, in particular, topological quantum computing, require the considerable topological background needed to define the quantum invariants of 3-manifolds, e.g. braid groups and their representations via the work of Jones, topological quantum field theory in the form of the Temperly-Lieb algebra, the Jones-Wenzl projectors in that algebra, in addition to the physical theories of certain subatomic particles, called fermions.

Not so for the abstract and now “standard” gate model of quantum computation, the foundations and demonstrations of computational superiority in various situations for which can be accessed by those with a modest topological background. This happy trick is performed through the use of what we call “elementary” quantum systems, systems that we will explore and apply in this talk.

This joint work with Marek Perkowski will soon be available as part of forthcoming text "Quantum Algorithms for Data Science and Machine Learning".

11 am -11:15 am Break

11:15 am – 12:15 pm Ryan Budney, University of Victoria

Title: The homotopy-types of diffeomorphism groups of small manifolds in dimensions four and up.

Abstract: An embedded circle in high dimensions can be unknotted. Moreover, often we can parametrize these unknotting operations in a multi-parameter family. This led David Gabai and I to the notion of a "barbell diffeomorphism", which is a (family of) diffeomorphism(s) of a boundary connect-sum of two trivial disc bundles over spheres. Using knotted embeddings of these barbell manifolds into small manifolds, such as S^1 x D^{n-1} we construct non-trivial diffeomorphism (families). These techniques are novel in dimensions n>=4, and are closely related to the diffeomorphisms constructed by Tadayuki Watanabe.

As mentioned in the schedule, in addition to usual talks and collaboration opportunities, the traditional seminar dinner will be offered at the home of Steven Bleiler the evening of November 12th. Directions available at the meeting. Portland State University is located in downtown Portland, Oregon, a major cosmopolitan center with many lodging, dining, and entertainment opportunities nearby.

For graduate student, early career, and participants from traditionally underrepresented groups without other grant support, a limited amount of dedicated support for lodging expenses incurred at meetings of the Cascade Topology Seminar is available, please contact Steve Bleiler, bleilers@pdx.edu, for details on how to apply.

The US operations of the Cascade Topology Seminar are supported in part by a grant from the National Science Foundation, Canadian operations of the seminar are supported in part by a grant from the Pacific Institute for the Mathematical Sciences, both of whose support is gratefully acknowledged.