Topology seminar
Title: Topology in Economics
Speaker: Alan Mehlenbacher, University of Victoria
Date and time:
17 Mar 2023,
10:00am -
11:00am
Location: DSB C126
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Abstract: I highlight what I consider to be the two major contributions of topology to economic theory: the application of algebraic topology to social choice theory and the application of differential topology to general equilibrium theory. Interesting, but not surprisingly, both of these were directly influenced by the Bourbaki group.
Title: Characterizing Statistical Noise in Persistent Homology
Speaker: Dayten Sheffar, University of Victoria
Date and time:
03 Mar 2023,
10:00am -
11:00am
Location: DSB C126
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Abstract: Persistent Homology (PH) is an active research area in Algebraic Topology, especially in Topological Data Analysis, that concerns capturing structural information of data. It arises in critical applications such as machine learning, neuroscience, cosmology, medicine and biology among others. One of the primary challenges in TDA is distinguishing between signal (meaningful structures) and noise (arising from local randomness or other inaccuracies). A key tool of PH is the persistence diagram, which captures the birth and death of various features in a filtration of a dataset. Characterizing the distribution of these diagrams is an open problem in the field. This talk will introduce the open problem, address some fundamental results, and discuss our results so far on a given approach to the problem.
Title: Virtual knots and (algebraic) concordance
Speaker: Hans Boden, McMasters
Date and time:
24 Feb 2023,
11:30am -
12:30pm
Location: DSB C114
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Abstract: The concordance group of virtual knots is known to be nonabelian but still somewhat mysterious. Using the Gordon-Litherland form, we introduce new invariants of virtual knots defined in terms of non-orientable spanning surfaces for the knot. The associated mock Seifert matrices lead to a new algebraic concordance group, which is abelian but not finitely generated. This talk is based on joint work with Homayun Karimi.
Title: Extending a theorem of Farrell and Jones
Speaker: Ryan Budney, University of Victoria
Date and time:
10 Feb 2023,
10:00am -
11:00am
Location: DSB C126
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Abstract: Farrell and Jones proved that the automorphism groups of compact hyperbolic manifolds in dimensions n≥1 do not have the homotopy-type of finite CW complexes. Here "automorphism group" means groups of diffeomorphisms, PL-homeomorphisms, and homeomorphisms respectively, i.e. we are not discussing homotopy-equivalences or isometries. Farrell and Jones primarily used the machinery of "Higher Simple Homotopy Theory", which reduces the problem of computing homotopy groups of automorphism groups (in a range) to that of the group of automorphisms of S^1 x D^{n-1}, which was worked-out by Hatcher and Wagoner, plus a good amount of K-theory. With David Gabai we found an alternate route to such theorems, that avoids Higher Simple Homotopy Theory entirely. For us the results occur in dimensions n≥4. I will describe how these kinds of arguments work.