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Maru Sarazola (University of Minnesota) - A homotopical framework for path homology of directed graphs

Traditionally, the way we compare mathematical structures is by using the notion of equality, or even of isomorphism. However, there are many settings where this is no longer the natural notion of "sameness". A notable example occurs when our objects admit a homology construction: then, we want two objects to be "the same" if they have identical homology.

Homotopy theory provides the framework required to work in these settings. In this talk, we will describe an algebraic invariant of directed graphs called "path homology", and introduce a new homotopical framework that encodes "sameness up to path homology". We will also show how this new framework allows us to make homology-invariant constructions.

Joshua Sabloff (Haverford College) - Cobordism Relations in Legendrian Knot Theory

Cobordisms between links, and the classical concordance group in particular, are a central framework in knot theory. In this talk, I will compare and contrast structures arising from smooth cobordism between smooth knots to those that appear in the world of contact and symplectic topology, namely Lagrangian cobordism between Legendrian knots. What type of relations are these? Is there any algebraic structure? What about metric structure? The bias will be towards open questions and sample results rather than a deep dive into any particular aspect of the theory. I will discuss joint work with Shea Vela-Vick, Mike Wong, and Angela Wu, as well as with Lisa Traynor.

Algebraic Topology session (Clark Hall 101)

1. Tanner Carawan (UVA) - Homotopy G-Mackey functors of $L_{KU_G} S_G$
2. David Chan (Michigan State) - Equivariant trees and partition complexes
3. Justin Barhite (Colorado) - Constructing type 2 complexes out of Brown-Gitler spectra

Geometric Topology session (Clark Hall 102)

1. Joe Wells (Virginia Tech) - High dimensional hyperbolic Coxeter groups that virtually fiber.
   Jankiewicz, Norin, and Wise provided a combinatorial criterion to determine whether right-angled Coxeter groups virtually fiber (that is, a finite-index subgroup admits a surjective homomorphism onto $\mathbb{Z}$ with finitely-generated kernel). Utilizing their result and generalizing a procedure of Osajda, we'll describe an iterative method for producing hyperbolic right-angled Coxeter groups (of arbitrarily-high virtual cohomological dimension) which virtually fiber. This is joint with Jean-François Lafont, Barry Minemyer, Gangotryi Sorcar, and Matthew Stover.
2. Harry Bray (George Mason University) and Hannah Hoganson (University of Maryland) - Geometrically finite subgroups of mapping class groups
   In Kleinian groups, geometrically finite groups are natural generalizations of lattices and convex cocompact groups, and they are well-understood. For instance, a geometrically finite Kleinian group is hyperbolic relative to finitely many virtually abelian parabolic subgroups. An analogy of convex cocompactness has been extended to subgroups of the mapping class group of a surface, but all examples are virtually free. In this talk, we will discuss a more general notion of geometrical finiteness for subgroups of mapping class groups. We present recent progress towards developing examples. This is joint work with Tarik Aougab, Spencer Dowdall, Sara Maloni, and Brandis Whitfield

We gratefully acknowledge financial support from the National Science Foundation (RTG grant DMS-1839968).