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 were held in summer 2022 and 2023. Another round of projects will be held in summer 2025 and 2026.

COLLABORATIVE PROJECTS IN GEOMETRIC TOPOLOGY (Summer 2026)

Organizers: Harry Bray, Anton Lukyanenko, Sara Maloni and Allison Moore

The upcoming round of research projects will focus on three problems in geometric topology. Research teams will begin meeting virtually during the end of spring 2026 before participating in an in-person collaborative workshop August 10-14, 2026, in the Blue Ridge Mountains. The planned projects are described briefly below. Please click here to apply. Application deadline: April 13, 2026.

• Unknotting numbers of trivalent spatial graphs
  • Team Leaders: Allison Moore (VCU) and Danielle O’Donnol (Marymount).
  • Description: This project is focused on unknotting numbers of spatial graphs. Generalizing a knot, a spatial graph is an embedding of a graph in the three-sphere. Planar spatial graphs are unknotted when they are isotopic to planar embeddings. As with the classic setting of knots and links, a wide variety of invariants (geometric, polynomial, homological) may be used to obstruct unknotting pathways and bound or calculate unknotting numbers. Although unknotting numbers of spatial graphs are less understood than knots and links, yet natural applications arise in both low-dimensional topology and molecular biology. We will focus on calculating and bounding unknotting numbers of trivalent spatial graphs via existing methods and by developing new ones. Familiarity with geometric topology and knot theory is helpful, but not required.
• Distinguishing links using surface intersection patterns
  • Team Leaders: Nir Gadish (UPenn) and Ryan Stees (UVA).
  • Description: Starting with the classical linking number of Gauss and the triple linking number which detects the linking of the Borromean rings, Milnor’s mu-invariants are among the most fundamental tools for distinguishing links in the 3-sphere. A recent series of works centers the Bar construction of cochain algebras as a generalization of these invariants, leading simultaneously to a geometric interpretation involving intersections of surfaces in the link exterior as well as to subtle new invariants. This project seeks to systematize and rigidify the choices of surfaces, thus resolving the problem of self-intersection. This should ultimately facilitate explicit geometric formulas for Milnor’s invariants, and possibly computer implementation that takes link diagrams as input.
• Entropy graph for the moduli space of convex real projective structures
  • Team Leaders: Harrison Bray (GMU), Lien-Yung (Nyima) Kao (GWU), and Anton Lukyanenko (GMU).
  • Description: The moduli space of convex real projective structures (CRPS) on a surface of negative Euler characteristic is the set of representations of the fundamental group of the surface into PSL(3,R) with certain dynamical properties. It is one moduli space of interest in the growing and active area of higher Teichmuller spaces. It contains the Teichmuller space of representations into PSL(2,\mathbb R), called the Fuchsian locus. For the moduli space of CRPS, the representation has geometric connections to the Hilbert metric. It also admits many other notions of lengths of closed curves such as the spectral radius length or simple root length. The entropy of a length function is the exponential growth rate in T of the number of closed curves of length at most T. The entropy can be thought of as a measure of chaos; it is a real-valued function which captures the dynamical complexity of a point in the moduli space with a fixed length function. In this project, we study the entropy graph on this moduli space of CRPS on a surface. It is known that the entropy function is analytic and achieves a strict global maximum precisely along the Fuchsian locus. However, the existence of other critical points or other geometric properties such as the curvature, are unknown. We will study this entropy graph analytically and computationally, with explicit examples as a starting point. This project combines a variety of tools from projective geometry, topology of surfaces, dynamical systems theory, ergodic theory, and Lie theory. We are seeking participants with experience in some of these components and/or computational skills, with enthusiasm to expand their expertise into these other areas.

COLLABORATIVE PROJECTS IN ALGEBRAIC TOPOLOGY (Summer 2025)

Organizers: Rebecca Field and J. D. Quigley

The upcoming round of research projects will focus on three problems in algebraic topology. Research teams will begin meeting virtually during spring 2025 before participating in an in-person collaborative workshop June 16–20, 2025, in the Blue Ridge Mountains. The planned projects are described briefly below. Please click here to apply. Application deadline: February 28, 2025.

• Equivariant algebras and equivariant operads
  • Team Leaders: Anna Marie Bohmann (Vanderbilt), Bert Guillou (Kentucky), and David Mehrle (Kentucky)
  • Description: There is a long history of using the theory of operads to describe the algebraic structures on naturally-occurring topological spaces, for example loop spaces. In the stable context, operads are used to describe multiplicative structures on spectra. In recent years, the use of operads in equivariant spaces and spectra has greatly increased. The interaction between the algebraic structure on an equivariant space or spectrum and its underlying object and fixed points provides a rich structure. This project will investigate specific types of equivariant algebraic structures governed by equivariant operads. Familiarity with other concepts from algebraic topology, such as spectra, operads, or equivariance, is welcome but not required.
• Hermitian K-theory of motivic classifying spaces
  • Team Leaders: Rebecca Field (JMU) and J.D. Quigley (UVA)
  • Description: About a decade ago, motivated by connections to character theory and chromatic homotopy theory, Bruner and Greenlees used the motivic Adams spectral sequence to compute the topological K-theory of the classifying spaces of many finite groups. In this project, we will investigate the motivic analogues of their computations, i.e., we will investigate the hermitian K-theory of the geometric classifying spaces of various finite groups using the motivic Adams spectral sequence. Prior experience with Adams spectral sequences and/or motivic homotopy theory will be helpful, but is not strictly necessary.
• Cohomology Rings of Dani-Mainkar Nilmanifolds
  • Team Leaders: Marco Aldi (VCU)
  • Description: Introduced two decades ago, the Dani-Mainkar construction attaches a nilmanifold to each finite simple graph. In this project we aim at explicitly calculating the Betti numbers as well as generators and relations for the cohomology ring of certain families of Dani-Mainkar nilmanifolds including those attached to trees. No background beyond standard graduate-level homological algebra is needed.

COLLABORATIVE PROJECTS IN GEOMETRIC TOPOLOGY

Organizers: Sara Maloni and Tom Mark

The projects below were started in summer of 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)
  • Arxiv: https://arxiv.org/abs/2501.13234
  • 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)
  • Arxiv: https://arxiv.org/abs/2502.12906
  • 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 (Summer 2022)

Organizers: Julie Bergner and Nick Kuhn

The projects below were started in summer of 2022.

• 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 continued 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.

Geometry, Topology