In Part 1 of this series, I gave an abstract description of one of the main problems in Machine Learning, the Generalization Problem, in which one uses the values of a function at a finite number of points to infer the entire function. The typical approach to this problem is to choose a finite-dimensional subset of the space of all possible functions, then choose the function from this family that minimizes something called *cost function*, defined by how accurate each function is on the sampled points. In this post, I will describe how the regression example from the last post generalizes to a family of models called Neural Networks, then describe how I recently used some fairly basic topology to demonstrate restrictions on the types of functions certain neural networks can produce.

## March 7, 2019

### The Topology of Neural Networks, Part 2: Compositions and Dimensions

## October 21, 2018

### The Topology of Neural Networks, Part 1: The Generalization Problem

I gave a talk a few months ago at the Thompson-Scharlemann-Kirby conference about a theorem I recently proved about topological limitations on certain families of neural networks. Most of the talk was a description of how neural networks work in terms of more abstract mathematics than they’re typically described in, and I thought this was probably a good thing to write up in a blog post. I decided to split the post into two parts because it was getting quite long. So the first post will describe the general approach to defining Machine Learning models, and the second post will cover Neural Networks in particular.

## April 4, 2018

### Two widely-believed conjectures. One is false.

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 or fewer crossings approaches 1 as approaches .

**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…)

## 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?

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 -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 –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:

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

## October 2, 2016

### A gorgeous but incomplete proof of “The Smale Conjecture”

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

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

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

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

Back in January, I wrote a post about my experience with the differences and trade-offs between academic careers and private sector careers. In this post, I want to present some practical advice for anyone with an academic math background who might be seeking a non-academic job. This advice is based on both my own experience and the advice I found while trying to make the transition. Much of it is very similar to the advice you’ll find (in more detail) in a book called What Are You Going To Do With That? which I read early on and found very helpful. (The book is about finding a non-academic job with any type of PhD.)