(GT)^2: Graduate Talks in Geometry and Topology

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Adele Jackson, Triangulation complexity of mapping tori: One measure of the complexity of a 3-manifold M is the minimal number of tetrahedra in a simplicial complex homeomorphic to M, called the triangulation complexity of M. A natural question is whether we can relate this with more geometric measures of the complexity of a manifold, especially understanding these relationships as combinatorial complexity grows.In the case when the manifold fibres over the circle, a recent theorem of Marc Lackenby and Jessica Purcell gives both an upper and lower bound on the triangulation complexity in terms of a geometric invariant of the gluing map (its translation length in the curve graph). We will discuss this result, as well as, if time permits, a new result concerning what happens when we alter the gluing map by a Dehn twist.

Campbell Wheeler, Quantum modularity of invariants of 3-manifolds: We will discuss newly observed modularity properties of q-hypergeometric functions. These functions often arise when studying quantum invariants of knots and 3-manifolds. This quantum modularity unifies and generalises various conjectures on the asymptotic behaviour of WRT invariants of closed manifolds for example.

Emily Thompson, Simplifying A-polynomial calculations for knots related by Dehn filling: The A-polynomial is a knot invariant that captures information about the topology of the knot complement and is conjectured to relate to the coloured Jones polynomial. However, it is difficult to compute in general. Recent work by Howie, Mathews and Purcell uncovers rich algebraic structure in the equations defining the A-polynomial in the case of knots that arise from Dehn fillings. This structure is similar to that of a cluster algebra. In this talk, we show how this cluster algebra-like structure leads to simplified calculations of the A-polynomial for infinitely many families of knots related by Dehn filling.
Rohin Berichon, The Alekseevskii Conjecture in 9 and 10 Dimensions: The study of Einstein Riemannian manifolds is a broad, yet rich field of study. In the case of homogeneous manifolds, Alekseevskii famously conjectured in 1975 that every connected homogeneous Einstein manifold with negative Ricci curvature is diffeomorphic to Euclidean space. Until now, the conjecture is only known up to dimension 8, besides 5 possible exceptions, and in some cases in dimension 10. Our thesis extends the results of Arroyo and Lafuente (2016) to show that the conjecture holds up to dimension 10, with the addition of 3 new possible exceptions.

Songqi Han, Parametrised h-cobordism theorem for smooth manifolds with tangential structures: The Whitehead space connects the Waldhausen A-theory space to the CAT h-cobordism space, where CAT means either TOP, PL, or DIFF. In this talk, I will outline the proof of Waldhausen's parametrised h-cobordism theorem that the loop space of the Whitehead space is homotopy equivalent to the stabilised space of h-cobordisms. Then I will discuss how to generalise this result to the case of smooth manifolds with tangential structures. In particular, this generalisation identifies the loop space of the A-theory space with the stabilised h-cobordism space of smooth manifolds with framings.

Alex He, On the hardness of finding normal surfaces: For fundamental topological problems like unknot recognition and 3-sphere recognition, polynomial-time algorithms remain elusive. Nevertheless, these problems can often be solved in practice using algorithms based on normal surface theory; such algorithms often rely on finding a normal surface of a certain type. Thus, it would be useful to work out which types of normal surface are hard to find. In this talk, we focus on two closely related types of normal surface; we will see that finding one of these types can be done in polynomial time, whereas finding the other type is an NP-complete problem.
Tamara Hogan, A basic introduction to planar algebras: This talk will introduce the notion of a planar algebra as defined by Jones (ie. as an algebra over the operad of planar tangles), as well as provide some discussion of the motivation for their existence. Some concrete examples will be presented, including both classical and virtual tangles.

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