Fridays from 2:30-3:30pm in LGRT 1681 at UMass Amherst
Organizers: İnanç Baykur, Patricia Cahn, Miriam Kuzbary and Riccardo Pedrotti
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Speaker: Friedrich Bauermeister
Title: Gordian split links in the Gehring ropelength problem
Abstract: A link in R^3 is thick if different components are at least unit distance apart. The Gehring ropelength problem asks about the shortest thick links in a given link homotopy class. Little is known about the “energy landscape” of Gehring ropelength. Given two thick links in the same homotopy class, can they be trapped in distinct local wells of low energy? In this talk I will present two new constructions:
- A 4-component link that is thickly embedded (the normal radius-$\tfrac12$ disk bundle embeds), but which cannot be cannot be split without increasing length even when curvature and self-distance constraints are discarded and length trading between components is permitted.
- A local minimizer of Gehring ropelength for the 2-component unlink. This is the first known example of any local minimizer for Gehring ropelength.
These constructions make substantial progress toward proving things about the energy landscape of ropelength energies. Some of the discussed techniques may yield progress on the open problem of whether a Gordian unknot exists.
Institution: Dartmouth College
Website: https://sites.google.com/view/friedrich-bauermeister/home
Speaker: Mark Hughes
Title: Branched covers of twist roll spun 2-knots and CP^2
Abstract: In 2023 Miyazawa produced a family of potentially exotic complex projective planes (i.e. 4-manifolds which were known to be homeomorphic to CP^{2}, but not necessarily diffeomorphic to it). These manifolds were constructed as the double branched coverings of a certain family of surface knots obtained via roll-spinning classical knots. In this talk I will show how the branched coverings of these knots can instead be obtained via torus surgeries, which can in turn be used to show that the resulting manifolds are indeed diffeomorphic to CP^{2}. We also use these results to show that certain homotopy 4-spheres created by Juhász-Powell are also standard. This is joint work with Seungwon Kim and Maggie Miller.
Institution: BYU
Website: https://mathdept.byu.edu/~hughes/
Speaker: Sze Hong Kwong
Title: Singular upward flow in the moduli space of Higgs bundles
Abstract: The moduli space of Higgs bundles admits a Białynicki-Birula-type decomposition. The upward flow through a stable fixed point can be parametrized by complex vector space and is a Lagrangian. In this talk, using the gauge theoretic construction of the moduli space, we look at the case where the fixed point is polystable and in special cases, discuss the existence of their conformal limits. It is based on my previous work (arXiv: 2504.14472) and an ongoing project.
Institution: UMass Amherst
Website: https://www.umass.edu/mathematics-statistics/about/directory/sze-hong-kwong
Speaker: Sierra Knavel
Title: New bounds on the first Betti number of Lefschetz fibrations.
Abstract: Results of Donaldson and Gompf together show that Lefschetz pencils are in one-to-one correspondence with symplectic 4 manifolds. In the study of symplectic 4 manifolds, it is often nice to consider Lefschetz pencils as they can be blown up to give a Lefschetz fibration. These fibrations have the structure of a fiber bundle with finitely many tractable singularities that are well understood due to their correlation to identity factorizations in the mapping class group. The possible fundamental groups of Lefschetz fibrations have been studied for quite some time, and it is conjectured that genus 2 Lefschetz fibrations have Abelian fundamental group on at most 2 generators. In this talk, I will discuss my progress which supports this conjecture and present a result on a new bound for the first Betti number for a nontrivial genus-g Lefschetz fibration.
Institution: Georgia Insitute of Technology
Website: https://sites.gatech.edu/sknavel3/
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Speaker: Hannah Hoganson
Title: Groups acting on Graphs and Homeomorphisms of Stone Spaces
Abstract: We will discuss the expansion of the theory of Cayley graphs and Cayley-Abels graphs to the setting of Polish groups due to Rosendal. An analog of the Milnor-Schwarz lemma applies, giving a quasi-isometry between a group with a coarsely bounded generating set and its associated Cayley-Abels-Rosendal graph. We will then apply these concepts to homeomorphism groups of Stone spaces, including a complete classification of countable Stone spaces whose homeomorphism groups are coarsely bounded, are locally bounded, and admit a Cayley-Abels-Rosendal graph, producing a coarsely bounded generating set.
Institution: University of Maryland
Website: https://www.math.umd.edu/~hoganson/
Baillieul Distinguished Lecture Series
Speaker: Sofía Martínez Alberga
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Institution: Stonehill College
Website: https://sites.google.com/view/sofiamartinezalberga/home
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Speaker: Ryan Stees
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Institution: University of Virginia
Website: https://sites.google.com/view/ryanstees
Speaker: Seraphina Eun Bi Lee
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Institution: Harvard
Website: https://people.math.harvard.edu/~slee
Thanksgiving
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Speaker: Beibei Liu
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Institution: The Ohio State University