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/
No seminar
Speaker: Valentina Zapata Castro
Title: Model categories in a grid
Abstract: Model categories provide a powerful framework for abstract homotopy theory, but their complexity often makes them difficult to classify. By focusing on finite categories, especially grids, we gain a combinatorial setting where the problem becomes explicit. In this talk, we explore model structures through weak factorization systems (WFS) on posets, which are in one-to-one correspondence with transfer systems and their duals, introduced here. This perspective leads to a method for constructing model structures and a characterization theorem for valid weak equivalences in posets. Our approach offers a pathway towards classifying model structures in a controlled setting. No prior knowledge of homotopy theory is assumed.
This is joint work with Kristen Mazur, Angelica Osorno, Constanze Roitzheim, Rekha Santhanam and Danika Van Niel.
Institution: UMass Amherst
Website: https://sites.google.com/virginia.edu/valentinazapatacastro
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
Title: Left and Right Bousfield Localization on Lattices
Abstract: The key information of a model category structure on a poset is encoded in a transfer system, which is a combinatorial gadget, originally introduced to investigate homotopy coherence structures in equivariant homotopy theory. In this talk I will describe how a transfer system associated with in a model structure on a lattice is affected by left and right Bousfield localization and provide a minimal generating system of morphisms which are responsible for the change in model structure. This leads to new concrete insights into the behavior of model categories on posets in general. This is joint work with Andres Carnero Bravo, Shuchita Goyal, Constanze Roitzheim, and Daniel Tolosa.
Institution: Stonehill College
Website: https://sites.google.com/view/sofiamartinezalberga/home
Speaker: Álvaro Pelayo
Title: Classical and Quantum Integrable Systems
Abstract: I will give a general audience introduction to symplectic geometry of classical integrable systems, with an emphasis on systems with many symmetries. Then I will briefly discuss a new p-adic approach to the subject, as well as the inverse problem for quantum integrable systems.
Institution: Royal Spanish Academy of Sciences, Universidad Complutense de Madrid
Website: https://blogs.mat.ucm.es/alvpel01/
Speaker: Ryan Stees
Title: Triple linking and rational homology cobordism
Abstract: In this talk, we will study the recently-defined Freedman-Krushkal triple torsion linking form in the context of 3-dimensional topological rational homology cobordism. After providing some background, we will give a new condition under which this form vanishes. Time permitting, we will discuss some related open questions.
Institution: University of Virginia
Website: https://sites.google.com/view/ryanstees
Speaker: Seraphina Eun Bi Lee
Title: Infinitely many Lefschetz pencils on ruled surfaces
Abstract: Works of Donaldson and Gompf show that a closed, oriented 4-manifold admits a symplectic structure if and only if it admits the structure of a Lefschetz pencil. However, the question of how many Lefschetz pencils (or fibrations) a given symplectic 4-manifold admits remains open. Works of Park--Yun and Baykur construct 4-manifolds admitting arbitrarily large (but finite) numbers of Lefschetz pencils or fibrations of the same genus. In this talk, we will construct infinitely many non-isomorphic Lefschetz pencils of the same genus on ruled surfaces of negative Euler characteristic. In fact, our construction gives the first example of infinitely many non-isomorphic but diffeomorphic Lefschetz pencils and fibrations of the same genus. This is joint work in progress with Carlos A. Serván.
Institution: Harvard
Website: https://people.math.harvard.edu/~slee
Thanksgiving
Speaker: Arya Yae
Title: Metric Degeneration in Moduli Spaces of Parabolic Higgs Bundles
Abstract: Moduli spaces of parabolic Higgs bundles on the four-punctured sphere are examples of ALG gravitational instantons, a kind of complete noncompact hyperkähler 4-manifold with r² volume growth. There are also ALE, ALF, and ALH instantons with r⁴, r³, and r¹ volume growth respectively. Gravitational instantons arise as bubbling limits in moduli spaces of K3 surfaces, and degenerations between instantons lend insight into the boundary faces of the compactified K3 moduli spaces. In this talk, I will discuss a fine-tuned degeneration of the Higgs bundle data which results in the degeneration of the full ALG hyperkähler metric to that of an ALE instanton modeled by a Nakajima quiver variety. This talk is based on joint work with my advisor Laura Fredrickson.
Institution: University of Oregon
Website: https://aryayae.com