This spring (2023) we will have a semi-active seminar.
Feb 10th
Speaker: Ryan Budney (U.Vic)
Title: Extending a theorem of Farrell and Jones
Abstract: Farrell and Jones proved that the automorphism groups of compact hyperbolic manifolds in dimensions n>=11 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.
Location: DSB C126, 10am--11am
Feb 24th
Speaker: Hans Boden (McMaster)
Title: Virtual knots and (algebraic) concordance
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.
Location: DSB C114, 11:30am--12:30pm
Mar 3rd
Speaker: Dayten Sheffar (U.Vic)
Title: Characterizing Statistical Noise in Persistent Homology
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.
Location: DSB C126, 10am--11am
Mar 17th
Speaker: Alan Mehlenbacher (U.Vic)
Title: Topology in Economics
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.
Location: DSB C126, 10am--11am
Apr. 7th
Speaker: Ryan Budney (U.Vic)
Title: The Schoenflies monoid
Abstract: Cerf gave a novel re-interpretation of Smale's proof that the diffeomorphism group of the 2-sphere has the homotopy-type of the isometry group. It fits into a family of proofs that allow for a partial description of the homotopy-fibres of the `scanning' maps Emb(D^{n-1}, S^1 x D^{n-1}) --> \Omega^j Emb(D^{n-1-j}, S^1 x D^{n-1}), as well as give a proof that the monoid of co-dimension one spheres \pi_0 Emb(S^{n-1}, S^n) is a group, under the connect-sum operation (for all n>=2). When n is not 4 this is a well-known result due to the resolution of the generalized Schoenflies problem in those dimensions, but this result would appear to be novel when n=4. I will outline these observations.
Location: DSB C130, 10:30am--11:30am. Given that Apr. 7th is a holiday, if the room in DSB is locked, we will have the seminar in the Math department, likely A514.
Apr. 14th
Speaker: Wenzhao Chen (UBC)
Title: The half-Alexander polynomial and Knot concordance
Abstract: Negative amphicheiral knots provide torsion elements in the knot concordance group, and torsion elements are less well understood than infinite-order elements. In this talk, I will introduce an equivariant version of the Alexander polynomial for strongly negative amphicheiral knots called the half-Alexander polynomial, focusing on its applications to knot concordance. In particular, we will show how understanding the geography behavior of the half-Alexander polynomial led to the construction of the first examples of non-slice amphichiral knots of determinant 1. This talk is based on joint work with Keegan Boyle.
Location: DSB C130, 11am--12pm
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