Agenda: Who is going to give a talk, and on what days? How many physicists are we going to let attend?

If you are interested in giving a talk/demonstration or would like to invite a guest speaker, please contact the organizers: Danny Kim, Varun Scarlett, or Joe Wells.

Agenda: Who is going to give a talk, and on what days? How many physicists are we going to let attend?

In the 1970's, Thurston found hyperbolic 3-manifolds which fibered over the circle (\(S^1\)) and his results prompted a natural follow-up question: are there higher-dimensional hyperbolic manifolds which also fiber over \(S^1\)? In 2023, Italiano-Martelli-Migliorini used a combinatorial result about right-angled Coxeter groups (RACGs) to construct a hyperbolic 5-manifold which also fibered over the circle. Might we finally have a way forward in tackling the big question??

In the immortal words of Vizzini (The Princess Bride) "You'd like to think that, wouldn't you?" Alas, the crucial ingredient in the aformentioned construction was a nice right-angled hyperbolic Coxeter polytope, and few things are even known about the existence of such objects in higher dimensions, so answering this question with geometric methods seems impractical at this juncture. In this talk, we'll look at recent results to the analogous group theoretic question about fibering and virtual cohomological dimension (VCD) of these RACGs.

This work is joint with Lafont, Minemyer, Sorcar, and Stover.

Stratified spaces are a class of spaces which generalize smooth manifolds and real/complex algebraic varieties. In studying their geometry, the need for spaces which generalize vector bundles to allow the rank of the fibers to drop over lower dimensional strata naturally arises. In this talk, we will discuss a possible framework for a theory of stratified vector bundles and construct several classes of examples.

The curious question of "Which Blacksburg am I in?" invites spherical point-inclusion tests (PITs) on \(S^2\). We discuss BAE-gons, i.e., the class of spherical polygons with antipode-excluding boundaries. A subset of \(S^2\) is said to be *antipode-excluding* if it does not intersect its antipode. We discuss the main proposition that spherical PITs for BAE-gons are decidable using planar PITs for polygons, if both notions of interiorness are defined using the winding number. We present two transformation algorithms from spherical PITs to planar PITs, illustrate their practicality in the broader context of geographic information science (GIS) and numerical simulation, and highlight some open questions.

This talk will begin with an introduction to Witten-Reshetikhin-Turaev (WRT) invariants and a related q-series invariant which first appeared in the work of Lawrence and Zagier. This q-series was one of the first key examples of a quantum modular form, and unified the WRT invariants of the Poincaré homology sphere. I will then discuss results based on joint work with Eleanor McSpirit. We show that the series of Lawrence and Zagier is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3-manifolds whose radial limits toward roots of unity may be thought of as a deformation of the WRT invariants. We use a recently developed theory of Akhmechet, Johnson, and Krushkal (AJK) which extends lattice cohomology and BPS q-series of 3-manifolds. As part of this work, we provide the first calculation of the AJK series for an infinite family of 3-manifolds.

Transfer systems are mathematical objects that encode transfers within algebras over specific types of structures known as equivariant operads. These systems allow us to employ combinatorial tools to investigate equivariant homotopy theory. One significant aspect is the study of compatible pairs of transfer systems, which correspond to multiplicative structures that align with an underlying additive structure.

In this talk, I will introduce G-transfer systems, which are transfer systems defined for a given group G. I will discuss the fundamental concepts of saturation and pairs of compatible transfer systems. Additionally, I will present joint work with Kristen Mazur, Angelica Osorno, Constanze Roitzheim, Rekha Santhanam, and Danika Van Niel on the compatibility of \(C_{p^rq^s}\)-transfer systems, where \(C_{p^rq^s}\) denotes the cyclic group of order \(p^r q^s\). Specifically, we will outline a criterion for determining when transfer systems only form trivially compatible pairs.

Abstract TBD

An awkward fact about manifolds is that the intersection of two submanifolds isn't necessarily another submanifold. Various generalizations of manifolds remedy this problem. One of these generalizations, Smooth Sets, is defined from the key insight that the set of smooth functions from cartesian spaces into a manifold completely determines the manifold in an appropriate sense. The "appropriate sense" here requires understanding a sheaf condition, which I will discuss extensively.

Smooth sets fix the intersection problem, but many more issues are also fixed, such as Lie group quotients being much more well-behaved and the new existence of a natural way to do homotopy theory called a model structure. Along the way, we'll explore how some of these fixes give rise to interesting objects and ideas, such as the irrational torus and concordance.

This is a continuation of my work in making complex and symplectic geometry accessible to undergraduates. We review a method to visualize complex affine space as a means to visualize standard symplectic space, and discuss the standard construction of the Kodaira-Thurston manifold, the simplest non-Kahler symplectic manifold. We will then connect the two presentations.

This is a cultural talk. I’ll talk about a few areas I’ve worked in where it felt like one was exploring playgrounds. The first area will be growth series and growth functions for groups. The second will be constructing 3-manifolds by face pairings.

Abstract TBD

Abstract TBD

Past talk information can be found here.