Seminars and Colloquia by Series

Joint GT-UGA Seminar at GT - On the topological expressiveness of neural networks

Series
Geometry Topology Seminar
Time
Monday, April 22, 2019 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eli Grigsby – Boston College

One can regard a (trained) feedforward neural network as a particular type of function , where  is a (typically high-dimensional) Euclidean space parameterizing some data set, and the value  of the function on a data point  is the probability that the answer to a particular yes/no question is "yes." It is a classical result in the subject that a sufficiently complex neural network can approximate any function on a bounded set. Last year, J. Johnson proved that universality results of this kind depend on the architecture of the neural network (the number and dimensions of its hidden layers). His argument was novel in that it provided an explicit topological obstruction to representability of a function by a neural network, subject to certain simple constraints on its architecture. I will tell you just enough about neural networks to understand how Johnson's result follows from some very simple ideas in piecewise linear geometry. Time permitting, I will also describe some joint work in progress with K. Lindsey aimed at developing a general theory of how the architecture of a neural network constrains its topological expressiveness.

Joint GT-UGA Seminar at GT - Simply-connected, spineless 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, April 22, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Adam Levine – Duke University
Given an m-dimensional manifold M that is homotopy equivalent to an n-dimensional manifold N (where n4, Cappell and Shaneson showed that if M is simply-connected or if m is odd, then it contains a spine. In contrast, I will show that there exist smooth, compact, simply-connected 4-manifolds which are homotopy equivalent to the 2-sphere but do not contain a spine (joint work with Tye Lidman). I will also discuss some related results about PL concordance of knots in homology spheres (joint with Lidman and Jen Hom).

Doubly slice knots and L^2 signatures by Patrick Orson

Series
Geometry Topology Seminar
Time
Monday, April 15, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Patrick Orson – Boston College –

The question of which high-dimensional knots are slice was entirely solved by Kervaire and Levine. Compared to this, the question of which knots are doubly slice in high-dimensions is still a largely open problem. Ruberman proved that in every dimension, some version of the Casson-Gordon invariants can be applied to obtain algebraically doubly slice knots that are not doubly slice. I will show how to use L^2 signatures to recover the result of Ruberman for (4k-3)-dimensional knots, and discuss how the derived series of the knot group might be used to organise the high-dimensional doubly slice problem.

Heegaard Floer homology and non-zero degree maps

Series
Geometry Topology Seminar
Time
Monday, April 8, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tye Lidman – NCSU

We will use Heegaard Floer homology to analyze maps between a certain family of three-manifolds akin to the Gromov norm/hyperbolic volume.  Along the way, we will study the Heegaard Floer homology of splices.  This is joint work with Cagri Karakurt and Eamonn Tweedy.

Moebius bands in S^1xB^3 and the square peg problem by Peter Feller

Series
Geometry Topology Seminar
Time
Wednesday, April 3, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Feller – ETH Zurich –

Following an idea of Hugelmeyer, we give a knot theory reproof of a theorem of Schnirelman: Every smooth Jordan curve in the Euclidian plane has an inscribed square. We will comment on possible generalizations to more general Jordan curves.

Our main knot theory result is that the torus knot T(2n,1) in S^1xS^2 does not arise as the boundary of a locally-flat Moebius band in S^1xB^3 for square-free integers n>1. For context, we note that for n>2 and the smooth setting, this result follows from a result of Batson about the non-orientable 4-genus of certain torus knots. However, we show that Batson's result does not hold in the locally flat category: the smooth and topological non-orientable 4-genus differ for the T(9,10) torus knot in S^3.

Based on joint work with Marco Golla.

Embedding Seifert fibered spaces in the 4-sphere

Series
Geometry Topology Seminar
Time
Monday, April 1, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ahmad Issa – University of Texas, Austin

Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards a couple of interesting conjectures which I'll discuss. This is joint work with Duncan McCoy.

Joint GT-UGA Seminar at UGA - A spectral sequence from Khovanov homology to knot Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 25, 2019 - 14:30 for 1 hour (actually 50 minutes)
Location
Boyd
Speaker
Nathan Dowlin – Dartmouth
Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.

Using 2-torsion to obstruct topological isotopy

Series
Geometry Topology Seminar
Time
Monday, March 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Hannah Schwartz – Bryn Mawr
It is well known that two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but are not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres; this shows that Gabai's recent ``4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.

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