Low Dimensional Topology

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.

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 (^_^)

March 6, 2016

Sexual harassment in academia

Filed under: Uncategorized — dmoskovich @ 11:18 am

An interesting piece has come out in the New York Times about sexual harassment in an academic setting:

A. Hope Jahren, She Wanted to Do Her Research. He Wanted to Talk ‘Feelings.’, New York Times, March 4, 2016.

What makes this piece especially interesting for me is that it’s written so that one understands the harasser, and is made to realize that “it could be me”. The pattern she describes sounds more common than one might like to admit- and the person writing the e-mail would almost certainly not be cogniscent of it being harassment. A male TA, professor, or supervisor, using the excuse of an altered state of mind (haven’t slept, drank too much) e-mails a love confession to a female student or colleague in a way that blames her, is a total power play, and is creepy and maybe a bit threatening (although of course he doesn’t see it that way). A wrong response to this first e-mail might mean that the victim gets harassed for a long time.

The author says that this first e-mail must be answered by firmly telling him (not asking him) to stop. But, Jahren laments, it never, never stops. While surely Jahren’s suggestion is sensible, a firm, “Dude, I have zero romantic interest in you. In addition you might want to read this piece by Jahren,” might, I think, be even more effective.

What do you all think? How prevalent is this type of sexual harassment in mathematics, and what can be done to effectively nip such harassment patterns in the bud?

February 28, 2016

Nonlocality and statistical inference

Filed under: Uncategorized — dmoskovich @ 2:02 pm

It doesn’t have much to do with topology, but I’d like to share with you something Avishy Carmi and I have been thinking about quite a bit lately, that is the EPR paradox and the meaning of (non)locality. Avishy and I have a preprint about this:

A.Y. Carmi and D.M., Statistics Limits Nonlocality, arXiv:1507.07514.

It offers a statistical explanation for a Physics inequality called Tsirelson’s bound (perhaps to be compared to a known explanation called Information Causality). Behind the fold I will sketch how it works. (more…)

September 29, 2015

Dispatches from the Dark Side, part 2

Filed under: Uncategorized — Jesse Johnson @ 7:10 pm

July 28, 2015

Tangle diagram crossings and quantum entanglement

Filed under: Uncategorized — dmoskovich @ 1:55 pm

In low dimensional topology we speak of tangles, while quantum physics speaks of entanglement. Similar words, but is there a deeper connection? Kauffman conjectured that the answer is yes (and I think he’s right, although maybe for other reasons). Glancing through arXiv this morning, I came across the following recent preprint:

Alagic, G., Jarret M., and Jordan S.P. Yang-Baxter operators need quantum entanglement to distinguish knots
arXiv:1507.05979

Their result is what it says in the title. Namely, we comb the knot into a braid, and assign R-matrices to crossings. An R-matrix underlies a linear operator $V\otimes V \rightarrow V \otimes V$. The authors prove that if the this operator maps product states to product states, then it gives rise (via a certain “taking the normalized trace of the operator the braid gives” procedure) to a trivial quantum knot invariant. Thus, entanglement is an essential part of being a nontrivial quantum invariant. Very cool!

There’s a suggestive picture in my head. Entanglement is all about nonlocality, where two non-interacting objects (an overstrand and an understrand?) cannot be described as separate systems (crossing?), but are inseparably intertwined in that they share some sort of coordination. It’s the entanglement which allows the overstrand to “communicate” to the understrand that it is there, making it possible to construct a nontrivial quantum invariant.

I suspect there’s a lot more to this story. Well done Alagic, Jarret, and Jordan!

July 16, 2015

Boolean logic with braids

Filed under: Uncategorized — dmoskovich @ 3:03 am

First-off, I’m fairly chuffed that Tangle Machines (arXiv version HERE) was published in Proc. R Soc. A, and they even chose our figure for the cover! Computing with Coloured Tangles has also been accepted for publication. This is good.

One of the constructions of Tangle Machines, which I previously discussed HERE and HERE, is a universal set of logic gates using coloured tangles (and in fact we cheated, because our colouring wasn’t by a quandle but by a more general algebraic structure). It turns out that this idea isn’t new, and actually it’s been done better a long time ago in a different setting, and in a very nice way (thanks anonymous referee!). Boolean logic can be realized using coloured braids! And it’s even potentially useful in quantum computing! So today I’ll discuss this paper and the papers it references:

Alagic, G., Jeffery, S., and Jordan, S. Circuit Obfuscation Using Braids. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014) (eds. S.T. Flammia and A.W. Harrow), Vol. 27, pp. 141–160.

June 11, 2015

Slice-ribbon progress

Filed under: Uncategorized — dmoskovich @ 10:29 am

There has been some recent interesting progress around the Slice Ribbon Conjecture. In particular, Yasui is giving talks on an infinite family counterexamples to the Akbulut-Kirby Conjecture (1978) that he has constructed:

Akbulut-Kirby Conjecture: If 0-surgeries on two knots give the same 3-manifold, then the knots with relevant orientations are concordant.

Note that some knots are not concordant to their reverses (Livingston), but the 0-surgery of a knot and its reverse are homeomorphic, so Akbulut-Kirby had to revise their original formalism to allow for arbitrary orientations. Abe and Tagami recently showed that if the Slice-Ribbon Conjecture is true then the Akbulut-Kirby Conjecture is false. Thus Yasui has eliminated an avenue to falsify the Slice-Ribbon Conjecture.

June 7, 2015

Eynard-Orantin Theory enters Quantum Topology

Filed under: Uncategorized — dmoskovich @ 9:18 am

I’m now reading the following paper:

G. Borot and B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, Quantum Topol. 6 (2015), 39-138.

In it, the authors apply the Eynard-Orantin topological recursion to conjecture an all-order asymptotic expansion of the coloured Jones polynomial of the complement of a hyperbolic knot, extending the volume conjecture.

To get an overview of Eynard-Orantin Theory, I’m looking at:

1. The original paper.
2. Eynard’s own overview– an expanded version of an ICM talk.
3. Some superb slides on the topic by Mulase.
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