Collaborative Research Projects
The geometry/topology group is organizing several ongoing collaborative research projects. Our goal is to involve faculty, postdocs, and graduate students from mainly colleges and universities in the Virginia and mid-Atlantic region, in substantial front-line mathematical research in geometry and topology. Projects will be led by senior researchers; the first round of projects started off in summer of 2022, and an upcoming round will start in summer 2023.
COLLABORATIVE PROJECTS IN GEOMETRIC TOPOLOGY
The upcoming round of research projects will focus on three problems in different areas of geometric topology. After a period over the summer of individual background study, the collaborations will kick off with a retreat workshop August 6-12, 2023, in the Blue Ridge Mountains. Collaboration will then continue remotely through the fall, culminating—we hope!—in one or more research papers from each group. Potential participants are encouraged to watch this space for further details on application, or to contact one of the organizers (Sara Maloni and Thomas Mark). The planned projects are described briefly below.
Please click here to apply. Application deadline: June 10, 2023.
- Contact surgery, fillability, and quasipositivity
- Team Leaders: Thomas Mark (UVA) and Joshua Sabloff (Haverford College)
- Description: A recent result shows that a knot in the 3-sphere has a Legendrian representative for which some positive contact surgery yields a fillable contact manifold, if and only if that knot is isotopic to the closure of a quasipositive braid with a certain condition on the braid expression. This has allowed the question of existence of a fillable surgery to be settled for all knots of up to 10 crossings by examination of braid expressions. We will warm up by considering the situation for 11-crossing knots, for which some of the obstructions used in the simpler cases may not apply. The main project will be to extend the quasipositive characterization for fillable surgery to knots in certain other 3-manifolds, particularly lens spaces and S^1 x S^2, with exploration of potential applications in contact and symplectic topology.
- Parabolically geometrically finite subgroups of mapping class groups
- Team Leaders: Tarik Aougab (Haverford College), Spencer Dowdall (Vanderbilt), Sara Maloni (UVA)
- Description: The goal is to construct new examples of parabolically geometrically finite (PGF) subgroups of mapping class groups, as defined by Dowdall-Durham-Leininger-Sisto, by using right-angled Artin groups. The definition is motivated by the theory of geometrically finite Kleinian groups and is meant to generalize the theory of “convex cocompact” subgroups introduced by Farb and Mosher. More specifically, given a finite collection of multicurves that pairwise fill, the subgroup generated by sufficiently high powers of twists in the components of the multicurves should generate a non-free PGF subgroup. Beyond that, we also hope to look, more generally, for other sufficient conditions and also try to identify necessary conditions.
- Fibering and Coxeter groups
- Team Leaders: Jean-Francois Lafont (Ohio State), Matthew Stover (Temple)
- Description: Right-angled Coxeter groups have long been fundamental objects in geometric group theory. They famously played a central role in Agol’s proof of the virtual fiber conjecture for 3-manifolds, and this has inspired a great deal of work in recent years on connecting algebraic and geometric notions of fibering that were developed to study fibrations of 3-manifolds. This project will be centered around understanding recent breakthroughs by Martelli and collaborators that includes (1) the first hyperbolic 5-manifolds that fiber over the circle and (2) hyperbolic 4-manifolds with perfect circle-valued Morse functions. Understanding the tools used in these papers, this project will aim at answering questions left in the wake of these papers and constructing new and interesting examples, for example in the broader context of negatively-curved manifolds. Background in basic geometric group theory and hyperbolic geometry will be the only requirements.
COLLABORATIVE PROJECTS IN ALGEBRAIC TOPOLOGY
The projects below were started in summer of 2022. Results will be publicly available soon!
- Equivariant partition complexes and trees
- Team leaders: Julie Bergner (UVA) and Peter Bonventre (Georgetown)
- Team members: Maxine Elena Calle (Penn), David Chan (Vanderbilt), and Maru Sarazola (Johns Hopkins)
- Calculations with the equivariant sphere spectrum
- Team leaders: Bert Guillou and Nat Stapleton (Kentucky)
- Team members: Tanner Carawan (UVA), Rebecca Field (James Madison), and David Mehrle (Kentucky)
- Dyer-Lashof operations on Ext groups
- Team leaders: Nick Kuhn (UVA) and Don Larson (Catholic University)
- Team members: William Balderrama (UVA), Justin Barhite (Kentucky), and Andres Mejia (Penn)
Collaborative Workshop in Algebraic Topology
August 1-5, 2022
The goal of this workshop was to bring together researchers at various career stages in the Mid-Atlantic region working in the area of algebraic topology, with the aim of fostering collaborations. Each of three teams has two team leaders who develop a research idea, and three participants who learn background material in advance of the workshop. Each team will continue to work after the workshop itself and write a paper on their results.
The workshop took place at Good Place Farms in Lexington, Virginia, and was supported by the Geometry and Topology RTG grant.
