17 February, 2019

This article describes some recent work on ‘direct air capture’ of carbon dioxide—essentially, sucking it out of the air:

• Jon Gerntner, The tiny Swiss company that thinks it can help stop climate change, New York Times Magazine, 12 February 2019.

There’s a Swiss company called Climeworks that’s built machines that do this—shown in the picture above. So far they are using these machines for purposes other than reducing atmospheric CO2 concentrations: namely, making carbonated water for soft drinks, and getting greenhouses to have lots of carbon dioxide in the air, for tastier vegetables. And they’re just experimental, not economically viable yet:

The company is not turning a profit. To build and install the 18 units at Hinwil, hand-assembled in a second-floor workshop in Zurich, cost between $3 million and $4 million, which is the primary reason it costs the firm between $500 and $600 to remove a metric ton of CO₂ from the air. Even as the company has attracted about $50 million in private investments and grants, it faces the same daunting task that confronted Carl Bosch a century ago: How much can it bring costs down? And how fast can it scale up?

If they ever make it in these markets, greenhouses and carbonation might want 6 megatonnes of CO₂ annually. This is nothing compared to the 37 gigatonnes of CO₂ that we put into the atmosphere in 2018. In principle the technology Climeworks is using could be massively scaled up. After all, Napoleon used aluminum silverware, back when aluminum was more precious than gold… and only later did the technology for making aluminum improve to the point where the metal gained a mass market.

But can Climeworks’ technology actually be scaled up? Some are dubious:

M.I.T.’s Howard Herzog, for instance, an engineer who has spent years looking at the potential for these machines, told me that he thinks the costs will remain between $600 and $1,000 per metric ton. Some of Herzog’s reasons for skepticism are highly technical and relate to the physics of separating gases. Some are more easily grasped. He points out that because direct-air-capture machines have to move tremendous amounts of air through a filter or solution to glean a ton of CO₂ — the gas, for all its global impact, makes up only about 0.04 percent of our atmosphere — the process necessitates large expenditures for energy and big equipment. What he has likewise observed, in analyzing similar industries that separate gases, suggests that translating spreadsheet projections for capturing CO₂ into real-world applications will reveal hidden costs. “I think there has been a lot of hype about this, and it’s not going to revolutionize anything,” he told me, adding that he thinks other negative-emissions technologies will prove cheaper. “At best it’s going to be a bit player.”

What actually is the technology Climeworks is using? And what other technologies are available for sucking carbon dioxide out of the air—or out of the exhaust from fossil-fuel-burning power plants, or out of water?

I’ll have a lot more to say about the latter question in future articles. As for Climeworks, they describe their technology rather briefly here:

• Climeworks, Our technology.

They write:

Our plants capture atmospheric carbon with a filter. Air is drawn into the plant and the CO2 within the air is chemically bound to the filter.

Once the filter is saturated with CO2 it is heated (using mainly low-grade heat as an energy source) to around 100 °C (212 °F). The CO2 is then released from the filter and collected as concentrated CO2 gas to supply to customers or for negative emissions technologies.

CO2-free air is released back into the atmosphere. This continuous cycle is then ready to start again. The filter is reused many times and lasts for several thousand cycles.

What is the filter material?

The filter material is made of porous granulates modified with amines, which bind the CO2 in conjunction with the moisture in the air. This bond is dissolved at temperatures of 100 °C.

So, it seems their technology is an example of ‘amine gas treating’:

• Wikipedia, Amine gas treating.

In future posts I’ll talk a bit more about amine gas treating, but also other methods for absorbing carbon dioxide from air or from solution in water. Maybe you can help me figure out what’s the best method!

Exploring New Technologies

13 February, 2019

I’ve got some good news! I’ve been hired by Bryan Johnson to help evaluate and explain the potential of various technologies to address the problem of climate change.

Johnson is an entrepreneur who sold his company Braintree for $800M and started the OS Fund in 2014, seeding it with $100M to invest in the hard sciences so that we can move closer towards becoming proficient system administrators of our planet: engineering atoms, molecules, organisms and complex systems. The fund has invested in many companies working on synthetic biology, genetics, new materials, and so on. Here are some writeups he’s done on these companies.

As part of my research I’ll be blogging about some new technologies, asking questions and hoping experts can help me out. Stay tuned!

Applied Category Theory 2019

7 February, 2019

I hope to see you at this conference, which will occur right before the associated school meets in Oxford:

Applied Category Theory 2019, July 15-19, 2019, Oxford, UK.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying systems of all sorts using category-theoretic tools. These systems are found in the natural sciences and social sciences, as well as in computer science, linguistics, and engineering. The background and experience of our members is as varied as the systems being studied. The goal of the ACT2019 Conference is to bring the majority of researchers in the field together and provide a platform for exposing the progress in the area. Both original research papers as well as extended abstracts of work submitted/accepted/published elsewhere will be considered.

There will be best paper award(s) and selected contributions will be awarded extended keynote slots.

The conference will include a business showcase and tutorials, and there also will be an adjoint school, the following week (see webpage).

Important dates

Submission of contributed papers: 3 May
Acceptance/Rejection notification: 7 June


Prospective speakers are invited to submit one (or more) of the following:

• Original contributions of high quality work consisting of a 5-12 page extended abstract that provides sufficient evidence of results of genuine interest and enough detail to allow the program committee to assess the merits of the work. Submissions of works in progress are encouraged but must be more substantial than a research proposal.

• Extended abstracts describing high quality work submitted/published elsewhere will also be considered, provided the work is recent and relevant to the conference. These consist of a maximum 3 page description and should include a link to a separate published paper or preprint.

The conference proceedings will be published in a dedicated Proceedings issue of the new Compositionality journal:

Only original contributions are eligible to be published in the proceedings.

Submissions should be prepared using LaTeX, and must be submitted in PDF format. Use of the Compositionality style is encouraged. Submission is done via EasyChair:

Program chairs

John Baez (U.C. Riverside)
Bob Coecke (University of Oxford)

Program committee

Bob Coecke (chair)
John Baez (chair)
Christina Vasilakopoulou
David Moore
Josh Tan
Stefano Gogioso
Brendan Fong
Steve Lack
Simona Paoli
Joachim Kock
Kathryn Hess Bellwald
Tobias Fritz
David I. Spivak
Ross Duncan
Dan Ghica
Valeria de Paiva
Jeremy Gibbons
Samuel Mimram
Aleks Kissinger
Jamie Vicary
Martha Lewis
Nick Gurski
Dusko Pavlovic
Chris Heunen
Corina Cirstea
Helle Hvid Hansen
Dan Marsden
Simon Willerton
Pawel Sobocinski
Dominic Horsman
Nina Otter
Miriam Backens

Steering committee

John Baez (U.C. Riverside)
Bob Coecke (University of Oxford)
David Spivak (M.I.T.)
Christina Vasilakopoulou (U.C. Riverside)

Fermat Primes and Pascal’s Triangle

5 February, 2019

If you take the entries Pascal’s triangle mod 2 and draw black for 1 and white for 0, you get a pleasing pattern:

The 2^nth row consists of all 1’s. If you look at the triangle consisting of the first 2^n rows, and take the limit as n \to \infty, you get a fractal called the Sierpinski gasket. This can also be formed by repeatedly cutting triangular holes out of an equilateral triangle:

Something nice happens if you interpret the rows of Pascal’s triangle mod 2 as numbers written in binary:

1 = 1
11 = 3
101 = 5
1111 = 15
10001 = 17
110011 = 51
1010101 = 85
11111111 = 255
100000001 = 257

Notice that some of these rows consist of two 1’s separated by a row of 0’s. These give the famous ‘Fermat numbers‘:

11 = 3 = 2^{2^0} + 1
101 = 5 = 2^{2^1} + 1
10001 = 17 = 2^{2^2} + 1
10000001 = 257 = 2^{2^3} + 1
1000000000000001 = 65537 = 2^{2^4} + 1

The numbers listed above are all prime. Based on this evidence Fermat conjectured that all numbers of the form 2^{2^n} + 1 are prime. But Euler crushed this dream by showing that the next Fermat number, 2^{2^5} + 1, is not prime.

Indeed, even today, no other Fermat numbers are known to be prime! People have checked all of them up to 2^{2^{32}} + 1. They’ve even checked a few bigger ones, the largest being

2^{2^{3329780}} + 1

which turns out to be divisible by

193 \times 2^{3329782} + 1

Here are some much easier challenges:

Puzzle 1. Show that every row of Pascal’s triangle mod 2 corresponds to a product of distinct Fermat numbers:

1 = 1
11 = 3
101 = 5
1111 = 15 = 3 × 5
10001 = 17
110011 = 51 = 3 × 17
1010101 = 85 = 5 × 17
11111111 = 255 = 3 × 5 × 17
100000001 = 257

and so on. Also show that every product of distinct Fermat numbers corresponds to a row of Pascal’s triangle mod 2. What is the pattern?

By the way: the first row, 1, corresponds to the empty product.

Puzzle 2. Show that the product of the first n Fermat numbers is 2 less than the next Fermat number:

3 + 2 = 5
3 × 5 + 2 = 17
3 × 5 × 17 + 2 = 257
3 × 5 × 17 × 257 + 2 = 65537

and so on.

Now, Gauss showed that we can construct a regular n-gon using straight-edge and compass if n is a prime Fermat number. Wantzel went further and showed that if n is odd, we can construct a regular n-gon using straight-edge and compass if and only if n is a product of distinct Fermat primes.

We can construct other regular polygons from these by repeatedly bisecting the angles. And it turns out that’s all:

Gauss–Wantzel Theorem. We can construct a regular n-gon using straight-edge and compass if and only if n is a power of 2 times a product of distinct Fermat primes.

There are only 5 known Fermat primes: 3, 5, 17, 257 and 65537. So, our options for constructing regular polygons with an odd number of sides are extremely limited! There are only 2^5 = 32 options, if we include the regular 1-gon.

Puzzle 3. What is a regular 1-gon? What is a regular 2-gon?

And, as noted in The Book of Numbers by Conway and Guy, the 32 constructible regular polygons with an odd number of sides correspond to the first 32 rows of Pascal’s triangle!

1 = 1
11 = 3
101 = 5
1111 = 15 = 3 × 5
10001 = 17
110011 = 51 = 3 × 17
1010101 = 85 = 5 × 17
11111111 = 255 = 3 × 5 × 17
100000001 = 257
1100000011 = 771 = 3 × 257
10100000101 = 1285 = 5 × 257
101010010101 = 3855 = 3 × 5 × 257

and so on. Here are all 32 rows, borrowed from the Online Encylopedia of Integer Sequences:

Click to enlarge! And here are all 32 odd numbers n for which we know that a regular n-gon is constructible by straight-edge and compass:

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

So, the largest known odd n for which a regular n-gon is constructible is 4294967295. This is the product of all 5 known Fermat primes:

4294967295 = 3 × 5 × 17 × 257 × 65537

Thanks to Puzzle 2, this is 2 less than the next Fermat number:

4294967295 = 2^{2^5} - 1

We can construct a regular polygon with one more side, namely

4294967296 = 2^{2^5}

sides, because this is a power of 2. But we can’t construct a regular polygon with one more side than that, namely

4294967297 = 2^{2^5} + 1

because Euler showed this Fermat number is not prime.

So, we’ve hit the end of the road… unless someone discovers another Fermat prime.

From Classical to Quantum and Back

30 January, 2019

Damien Calaque has invited me to speak at FGSI 2019, a conference on the Foundations of Geometric Structures of Information. It will focus on scientific legacy of Cartan, Koszul and Souriau. Since Souriau helped invent geometric quantization, I decided to talk about this. That’s part of why I’ve been writing about it lately!

I’m looking forward to speaking to various people at this conference, including Mikhail Gromov, who has become interested in using category theory to understand biology and the brain.

Here’s my talk:

From classical to quantum and back.

Abstract. Edward Nelson famously claimed that quantization is a mystery, not a functor. In other words, starting from the phase space of a classical system (a symplectic manifold) there is no functorial way of constructing the correct Hilbert space for the corresponding quantum system. In geometric quantization one gets around this problem by equipping the classical phase space with extra structure: for example, a Kähler manifold equipped with a suitable line bundle. Then quantization becomes a functor. But there is also a functor going the other way, sending any Hilbert space to its projectivization. This makes quantum systems into specially well-behaved classical systems! In this talk we explore the interplay between classical mechanics and quantum mechanics revealed by these functors going both ways.

For more details, read these articles:

  • Part 1: the mystery of geometric quantization: how a quantum state space is a special sort of classical state space.
  • Part 2: the structures besides a mere symplectic manifold that are used in geometric quantization.
  • Part 3: geometric quantization as a functor with a right adjoint, ‘projectivization’, making quantum state spaces into a reflective subcategory of classical ones.
  • Part 4: making geometric quantization into a monoidal functor.
  • Part 5: the simplest example of geometric quantization: the spin-1/2 particle.
  • Part 6: quantizing the spin-3/2 particle using the twisted cubic; coherent states via the adjunction between quantization and projectivization.
  • Part 7: the Veronese embedding as a method of ‘cloning’ a classical system, and taking the symmetric tensor powers of a Hilbert space as the corresponding method of cloning a quantum system.
  • Part 8: cloning a system as changing the value of Planck’s constant.

  • Systems as Wiring Diagram Algebras

    28 January, 2019


    Check out the video of Christina Vasilakopoulou’s talk, the third in the Applied Category Theory Seminar here at U. C. Riverside! It was nicely edited by Paola Fernandez and uploaded by Joe Moeller.

    Abstract. We will start by describing the monoidal category of labeled boxes and wiring diagrams and its induced operad. Various kinds of systems such as discrete and continuous dynamical systems have been expressed as algebras for that operad, namely lax monoidal functors into the category of categories. A major advantage of this approach is that systems can be composed to form a system of the same kind, completely determined by the specific way the composite systems are interconnected (‘wired’ together). We will then introduce a generalized system, called a machine, again as a wiring diagram algebra. On the one hand, this abstract concept is all-inclusive in the sense that discrete and continuous dynamical systems are sub-algebras; on the other hand, we can specify succinct categorical conditions for totality and/or determinism of systems that also adhere to the algebraic description.

    Reading material:

    • Patrick Schultz, David I. Spivak and Christina Vasilakopoulou, Dynamical systems and sheaves.

    • David I. Spivak, The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits.

    • Dmitry Vagner, David I. Spivak and Eugene Lerman, Algebras of open dynamical systems on the operad of wiring diagrams.

    Symposium on Compositional Structures 3

    28 January, 2019

    One of the most lively series of conferences on applied category theory is ‘SYCO’: the Symposium on Compositional Structures. And the next one is coming soon!

    Symposium on Compositional Structures 3, University of Oxford, 27-28 March, 2019.

    The Symposium on Compositional Structures (SYCO) is an interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. The first SYCO was in September 2018 at the University of Birmingham. The second SYCO was in December 2019, at the University of Strathclyde, each attracting about 70 people.

    We welcome submissions from researchers across computer science, mathematics, physics, philosophy, and beyond, with the aim of fostering friendly discussion, disseminating new ideas, and spreading knowledge between fields. Submission is encouraged for both mature research and work in progress, and by both established academics and junior researchers, including students.

    Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. Think creatively—you could submit a recent paper, or notes on work in progress, or even a recent Masters or PhD thesis.

    While no list of topics could be exhaustive, SYCO welcomes submissions with a compositional focus related to any of the following areas, in particular from the perspective of category theory:

    • logical methods in computer science, including classical and quantum programming, type theory, concurrency, natural language processing and machine learning;

    • graphical calculi, including string diagrams, Petri nets and
    reaction networks;

    • languages and frameworks, including process algebras, proof nets, type theory and game semantics;

    • abstract algebra and pure category theory, including monoidal
    category theory, higher category theory, operads, polygraphs, and
    relationships to homotopy theory;

    • quantum algebra, including quantum computation and representation theory;

    • tools and techniques, including rewriting, formal proofs and proof assistants, and game theory;

    • industrial applications, including case studies and real-world
    problem descriptions.

    This new series aims to bring together the communities behind many previous successful events which have taken place over the last decade, including “Categories, Logic and Physics”, “Categories, Logicand Physics (Scotland)”, “Higher-Dimensional Rewriting and Applications”, “String Diagrams in Computation, Logic and Physics”, “Applied Category Theory”, the “Simons Workshop on Compositionality”, and the “Peripatetic Seminar in Sheaves and Logic”.

    SYCO will be a regular fixture in the academic calendar, running
    regularly throughout the year, and becoming over time a recognized venue for presentation and discussion of results in an informal and friendly atmosphere. To help create this community, and to avoid the need to make difficult choices between strong submissions, in the event that more good-quality submissions are received than can be accommodated in the timetable, the programme committee may choose to defer some submissions to a future meeting, rather than reject them. This would be done based largely on submission order, giving an incentive for early submission, but would also take into account other requirements, such as ensuring a broad scientific programme. Deferred submissions can be re-submitted to any future SYCO meeting, where they would not need peer review, and where they would be prioritised for inclusion in the programme. This will allow us to ensure that speakers have enough time to present their ideas, without creating an unnecessarily competitive reviewing process. Meetings will be held sufficiently frequently to avoid a backlog of deferred papers.

    Invited speakers

    • Marie Kerjean, INRIA Bretagne Atlantique

    • Alessandra Palmigiano, Delft University of Technology and University of Johannesburg

    Important dates

    All times are anywhere-on-earth.

    • Submission deadline: Friday 15 February 2019
    • Author notification: Wednesday 27 February 2019
    • Registration deadline: Wednesday 20 March 2019
    • Symposium dates: Wednesday 27 and Thursday 28 March 2019


    Submission is by EasyChair, via the following link:

    Submissions should present research results in sufficient detail to allow them to be properly considered by members of the programme committee, who will assess papers with regards to significance, clarity, correctness, and scope. We encourage the submission of work in progress, as well as mature results. There are no proceedings, so work can be submitted even if it has been previously published, or has been submitted for consideration elsewhere. There is no specific formatting requirement, and no page limit, although for long submissions authors should understand that reviewers may not be able to read the entire document in detail.

    Financial support

    Some funding is available to cover travel and subsistence costs, with a priority for PhD students and junior researchers. To apply for this funding, please contact the local organizers Antonin Delpeuch ( or Ben Musto ( with subject line “SYCO 3 funding request” by March 6, with a short statement of your current status, travel costs and funding required.

    Programme committee

    Fatimah Ahmadi, University of Oxford
    Corina Cirstea, University of Southampton
    Bob Coecke, University of Oxford
    Carmen Maria Constantin, University of Oxford
    Antonin Delpeuch, University of Oxford
    Brendan Fong, Massachusetts Institute of Technology
    Dan Ghica, University of Birmingham
    Giuseppe Greco, Utrecht University
    Helle Hvid Hansen, Delft University
    Jules Hedges, University of Oxford
    Chris Heunen, University of Edinburgh
    Dominic Horsman, University of Grenoble
    Dimitri Kartsaklis, Apple
    Aleks Kissinger, Radboud University Nijmegen
    Alexander Kurz, Chapman University
    Jean-Simon Lemay, University of Oxford
    Martha Lewis, University of Amsterdam
    Dan Marsden, University of Oxford
    Samuel Mimram, École Polytechnique
    Nina Otter, UCLA
    Simona Paoli, University of Leicester
    Robin Piedeleu, University of Oxford
    David Reutter, University of Oxford
    Christine Tasson, Paris Diderot University
    Jamie Vicary, University of Birmingham
    Tamara von Glehn, University of Cambridge
    Quanlong Wang, University of Oxford
    Gijs Wijnholds, Queen Mary University of London
    Philipp Zahn, University of St.Gallen