Seminars and Colloquia by Series

SU(2) representations for toroidal homology spheres

Series
Geometry Topology Seminar
Time
Monday, August 17, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/803706608
Speaker
Tye Lidman – NCSU

The three-dimensional Poincare conjecture shows that any closed three-manifold other than the three-sphere has non-trivial fundamental group. A natural question is how to measure the non-triviality of such a group, and conjecturally this can be concretely realized by a non-trivial representation to SU(2). We will show that the fundamental groups of three-manifolds with incompressible tori admit non-trivial SU(2) representations. This is joint work with Juanita Pinzon-Caicedo and Raphael Zentner.

The speaker will hold online office hours from 3:15-4:15 pm for interested graduate students and postdocs.

Satellite operations and knot genera

Series
Geometry Topology Seminar
Time
Monday, March 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Allison Miller – Rice University –

The satellite construction, which associates to a pattern knot P in a solid torus and a companion knot K in the 3-sphere the so-called satellite knot P(K), features prominently in knot theory and low-dimensional topology.  Besides the intuition that P(K) is “more complicated” than either P or K, one can attempt to quantify how the complexity of a knot changes under the satellite operation. In this talk, I’ll discuss how several notions of complexity based on the minimal genus of an embedded surface change under satelliting. In the case of the classical Seifert genus of a knot, Schubert gives an exact formula. In the 4-dimensional context the situation is more complicated, and depends on whether we work in the smooth or topological category: the smooth category is sometimes asymptotically similar to the classical setting, but our main results show that the topological category is much weirder.  This talk is based on joint work with Peter Feller and Juanita Pinzón-Caicedo. 

Joint UGA-GT Topology Seminar at UGA: The dihedral genus of a knot

Series
Geometry Topology Seminar
Time
Monday, March 2, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Patricia Cahn – Smith College

We consider dihedral branched covers of $S^4$, branched along an embedded surface with one non-locally flat point, modelled on the cone on a knot $K\subset S^3$. Kjuchukova proved that the signature of this cover is an invariant $\Xi_p(K)$ of the $p$-colorable knot $K$. We prove that the values of $\Xi_p(K)$ fall in a bounded range for homotopy-ribbon knots. We also construct a family of (non-slice) knots for which the values of $\Xi_p$ are unbounded. More generally, we introduce the notion of the dihedral 4-genus of a knot, and derive a lower bound on the dihedral 4-genus of $K$ in terms of $\Xi_p(K)$. This work is joint with A. Kjuchukova.

Joint UGA-GT Topology Seminar at UGA: Stein domains in complex two space with prescribed boundary

Series
Geometry Topology Seminar
Time
Monday, March 2, 2020 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Bolent Tosun – University of Alabama

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.

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