Low Dimensional Topology

April 4, 2018

Two widely-believed conjectures. One is false.

Filed under: Uncategorized — dmoskovich @ 12:10 pm

I haven’t blogged for a long time- for the last few years, my research has been leading me away from low dimensional topology and more towards foundations of quantum physics. You can read my latest paper on the topic HERE.

Today I’d like to tell you about a preprint by Malyutin, that shows that two widely believed knot theory conjectures are mutually exclusive!

A Malyutin, On the Question of Genericity of Hyperbolic Knots, https://arxiv.org/abs/1612.03368.

Conjecture 1: Almost all prime knots are hyperbolic. More precisely, the proportion of hyperbolic knots amongst all prime knots of n or fewer crossings approaches 1 as n approaches \infty.

Conjecture 2: The crossing number (the minimal number of crossings of a knot diagram of that knot) of a composite knot is not less than that of each of its factors. (more…)

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December 19, 2017

Computation in geometric topology

Complete lecture videos for last week’s workshop Computation in Geometric Topology at Warwick are now posted on YouTube. The complete list of talks with abstracts and video links is here.

September 14, 2017

What is the complexity of the homeomorphism problem?

Filed under: Uncategorized — dmoskovich @ 4:19 am

Homeomorphism Problems are the guiding problems of low-dimensional topology: Given two topological objects, determine whether or not they are homeomorphic to one another. A lot of topology is tangential to this problem- we may define invariants, investigate their properties, and prove relationships between them, but we rarely directly touch on the Homeomorphism Problem. Direct progress on the Homeomorphism Problem is a big deal!

I’m excited by a number of new and semi-new papers by Greg Kuperberg and collaborators. From my point of view, the most interesting of all is:

G. Kuperberg, Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization, http://front.math.ucdavis.edu/1508.06720

This paper contains two results:

1) That Geometrization implies that there exists a recursive algorithm to determine whether two closed oriented 3–manifolds are homeomorphic.

2) Result (1), except with the words “elementary recursive” replacing the words “recursive”.

Result (1) is sort-of a well-known folklore theorem, and is essentially due to Riley and Thurston (with lots of subsections of it obtaining newer fancier proofs in the interim), but no full self-contained proof had appeared for it in one place until now. It’s great to have one- moreover, a proof which uses only the tools that were available in the 1970’s.

Knowing that we have a recursive algorithm, the immediate and important question is the complexity class of the best algorithm. Kuperberg has provided a worst-case bound, but “elementary recursive” is a generous computational class. The real question I think, and one that is asked at the end of the paper, is where exactly the homeomorphism problem falls on the heirarchy of complexity classes:

P \subseteq PSPACE \subseteq EXP \subseteq EXPSPACE \subseteq EEXP \subseteq \cdots

And whether the corresponding result holds for compact 3–manifolds with boundary, and for non-orientable 3–manifolds.

March 18, 2017

A new “Mathematician’s Apology”

Filed under: Misc. — Jesse Johnson @ 8:18 am

In the two and a half years (or so) since I left academia for industry, I’ve worked with a number of math majors and math PhDs outside of academia and talked to a number of current grad students who were considering going into industry. As a result, my perspective on the role of the math research community within the larger world has changed quite a bit from what it was in the early days of may academic career. In the post below, I explore this new perspective.

(more…)

March 3, 2017

Office House with a Geometric Group Theorist

Filed under: Geometric Group Theory — dmoskovich @ 9:49 am

It’s hard enough for one author to write a coherent work; for many authors, even if it’s one and the same topic one might end up with Rashoumon (in the sense of different and contradicting narratives). But, as the Princeton Companion to Mathematics shows, it is possible to have a coherent book with each author writing a chapter, and now geometric group theory has one too.

A new book has just come out, and it’s very good.

Office Hours with a Geometric Group Theorist, Edited by Matt Clay & Dan Margalit, 2017.

An undergraduate student walks into the office of a geometric group theorist, curious about the subject and perhaps looking for a senior thesis topic. The researcher pitches their favourite sub-topic to the student in a single “office hour”.

The book collects together 16 independent such “office hours”, plus two introductory office hours by the editors (Matt Clay and Dan Margalit) to get the student off the ground. (more…)

October 7, 2016

A bird’s eye view of topological recursion

Filed under: Quantum topology — dmoskovich @ 8:53 am

Eynard-Orantin Theory (topological recursion) has got to be one of the biggest ideas in quantum topology in recent years (see also HERE). Today I’d like to attempt a bird’s eye explanation of what all the excitement is about from the perspective of low-dimensional topology. (more…)

October 2, 2016

A gorgeous but incomplete proof of “The Smale Conjecture”

Filed under: 3-manifolds,Algebraic topology,Smooth Topology,Uncategorized — Ryan Budney @ 10:33 pm

In 1959 Stephen Smale gave a proof that the group of diffeomorphisms of the 2-sphere has the homotopy-type of the subgroup of linear diffeomorphisms, i.e. the Lie Group O_3.  His proof went in two steps: (more…)

July 15, 2016

Postdoctoral positions- call for applications

Filed under: Uncategorized — dmoskovich @ 9:12 am

I’m pleased to announce that we have been awarded Israel Science Foundation (ISF) to support 2-year postdoctoral positions at Ben-Gurion University of the Negev, to work on Tangle Machines and low-dimensional topological approaches to information theory. Interested applicants may contact me or Avishy Carmi at avcarmi@bgu.ac.il .

We’re looking toward developing applications, so we’re primarily searching for people who can program and maybe who have some signal processing knowledge. So primarily for computer science postdocs, I suppose.

An official announcement will be posted at relevant places in due time- but you heard it here first (^_^)

May 24, 2016

SnapPy 2.4 released

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Knot theory — Nathan Dunfield @ 6:11 pm

A new version of SnapPy, a program for studying the topology and geometry of 3-manifolds, is available.  Added features include a census of Platonic manifolds, rigorous computation of cusp translations, and substantial improvements to its link diagram component.

April 9, 2016

Arithmetic Chern-Simons

Filed under: Knot theory,Number theory,Quantum topology — dmoskovich @ 1:48 pm

The most compelling aspect of Quantum Topology for me is its connection to analytic number theory. Today I’d like to draw your attention to recent work of Minhyong Kim on Arithmetic Chern-Simons Theory (see his paper for more details). I was fortunate to hear him give a talk about this last Wednesday and a colloquium talk on related subjects the day before. People have been talking about such “quantum topological number theory” for a long time- e.g. this 2010 MO question – but we haven’t seen much of an uptake so far. This isn’t an easy direction to pursue because one needs to know both quantum topology and analytic number theory, but I was left with the strong feeling that “There’s gold in them thar hills”, both for topologists and for number theorists. (more…)

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