Symmetric Monoidal Categories: a Rosetta Stone

28 May, 2021

The Topos Institute is in business! I’m really excited about visiting there this summer and working on applied category theory.

They recently had a meeting with some people concerned about AI risks, called Finding the Right Abstractions, organized by Scott Garrabrant, David Spivak, and Andrew Critch. I gave a gentle introduction to the uses of symmetric monoidal categories:

• Symmetric monoidal categories: a Rosetta Stone.

To describe systems composed of interacting parts, scientists and engineers draw diagrams of networks: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams and the like. All these different diagrams fit into a common framework: the mathematics of symmetric monoidal categories. While originally the morphisms in such categories were mainly used to describe processes, we can also use them to describe open systems.

You can see the slides here, and watch a video here:

For a lot more detail on these ideas, see:

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone, in New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95—174.


Compositional Robotics (Part 2)

27 May, 2021

Very soon we’re having a workshop on applications of category theory to robotics:

2021 Workshop on Compositional Robotics: Mathematics and Tools, online, Monday 31 May 2021.

You’re invited! As of today it’s not too late to register and watch the talks online, and registration is free. Go here to register:

https://forms.gle/9v52EXgDFFGu3h9Q6

Here’s the schedule. All times are in UTC, so the show starts at 9:15 am Pacific Time:

Time (UTC) Speaker

Title

16:15-16:30   Intro and plan of the workshop

16:30-17:10

Jonathan Lorand

Category Theory Basics

17:20-18:00

John Baez Category Theory and Systems 

 

Breakout rooms

 

18:30-19:10

Andrea Censi
& Gioele Zardini

Categories for Co-Design

19:20-20:00

David Spivak

Dynamic Interaction Patterns

 

Breakout rooms

 

20:30-21:15

Aaron Ames

A Categorical Perspective on Robotics

21:30-22:15 Daniel Koditschek Toward a Grounded Type Theory for Robot Task Composition
22:30-00:30 Selected speakers Talks from open submissions

For more information go to the workshop website or my previous blog post on this workshop:

Compositional robotics (part 1).


Category Theory and Systems

27 May, 2021

I’m giving a talk on Monday the 31st of May, 2021 at 17:20 UTC, which happens to be 10:20 am Pacific Time for me. You can see my slides here:

Category theory and systems.

I’ll talk about how to describe open systems as morphisms in symmetric monoidal categories, and how to use ‘functorial semantics’ to describe the behavior of open systems.

It’s part of the 2021 Workshop on Compositional Robotics: Mathematics and Tools, and if you click the link you can see how to attend!  If you stick around for the rest of the workshop you’ll hear more concrete talks from people who really work on robotics. 


Electrostatics and the Gauss–Lucas Theorem

24 May, 2021

Say you know the roots of a polynomial P and you want to know the roots of its derivative. You can do it using physics! Namely, electrostatics in 2d space, viewed as the complex plane.

To keep things simple, let us assume P does not have repeated roots. Then the procedure works as follows.

Put equal point charges at each root of P, then see where the resulting electric field vanishes. Those are the roots of P’.

I’ll explain why this is true a bit later. But first, we use this trick to see something cool.

There’s no way the electric field can vanish outside the convex hull of your set of point charges. After all, if all the charges are positive, the electric field must point out of that region. So, the roots of P’ must lie in the convex hull of the roots of P!



This cool fact is called the Gauss–Lucas theorem. It always seemed mysterious to me. Now, thanks to this ‘physics proof’, it seems completely obvious!

Of course, it relies on my first claim: that if we put equal point
charges at the roots of P, the electric field they generate will vanish at the roots of P’. Why is this true?

By multiplying by a constant if necessary, we can assume

\displaystyle{   P(z) = \prod_{i = 1}^n  (z - a_i) }

Thus

\displaystyle{  \ln |P(z)| = \sum_{i = 1}^n \ln|z - a_i| }

This function is the electric potential created by equal point charges at the points ai in the complex plane. The corresponding electric field is minus the gradient of the potential, so it vanishes at the critical points of this function. Equivalently, it vanishes at the critical points of the exponential of this function, namely |P|. Apart from one possible exception, these points are the same as the critical points of P, namely the roots of P’. So, we’re almost done!

The exception occurs when P has a critical point where P vanishes. |P| is not smooth where P vanishes, so in this case we cannot say the critical point of P is a critical point of |P|.

However, when P has a critical point where P vanishes, then this point is a repeated root of P, and I already said I’m assuming P has no repeated roots. So, we’re done—given this assumption.

Everything gets a bit more complicated when our polynomial has repeated roots. Greg Egan explored this, and also the case where its derivative has repeated roots.

However, the Gauss–Lucas theorem still applies to polynomials with repeated roots, and this proof explains why:

• Wikipedia, Gauss–Lucas theorem.

Alternatively, it should be possible to handle the case of a polynomial with repeated roots by thinking of it as a limit of polynomials without repeated roots.

By the way, in my physics proof of the Gauss–Lucas theorem I said the electric field generated by a bunch of positive point charges cannot vanish outside the convex hull of these point charges because the field ‘points out’ of this region. Let me clarify that.

It’s true even if the positive point charges aren’t all equal; they just need to have the same sign. The rough idea is that each charge creates an electric field that points radially outward, so these electric fields can’t cancel at a point that’s not ‘between’ several charges—in other words, at a point that’s not in the convex hull of the charges.

But let’s turn this idea into a rigorous argument.

Suppose z is some point outside the convex hull of the points ai. Then, by the hyperplane separation theorem, we can draw a line with z on one side and all the points ai on the other side. Let v be a vector normal to this line and pointing toward the z side. Then

v \cdot (z - a_i) > 0

for all i. Since the electric field created by the ith point charge is a positive multiple of z – ai at the point z, the total electric field at z has a positive dot product with v. So, it can’t be zero!

Credits

The picture of a convex hull is due to Robert Laurini.


Parallel Line Masses and Marden’s Theorem

22 May, 2021

Here’s an idea I got from Albert Chern on Twitter. He did all the hard work, and I think he also drew the picture I’m going to use. I’ll just express the idea in a different way.

Here’s a strange fact about Newtonian gravity.

Consider three parallel ‘line masses’ that have a constant mass per length—the same constant for each one. Choose a plane orthogonal to these lines. There will typically be two points on this plane, say a and b, where a mass can sit in equilibrium, with the gravitational pull from all three lines masses cancelling out. This will be an unstable equilibrium.

Put a mass at point a. Remove the three line masses—but keep in mind the triangle they formed where they pierced your plane!

You can now orbit a test particle in an elliptical orbit around the mass at a in such a way that:

• one focus of this ellipse is a,
• the other focus is b, and
• the ellipse fits inside the triangle, just touching the midpoint of each side of the triangle.

Even better, this ellipse has the largest possible area of any ellipse contained in the triangle!

Here is Chern’s picture:



 

The triangle’s corners are the three points where the line masses pierce your chosen plane. These line masses create a gravitational potential, and the contour lines are level curves of this potential.

You can see that the points a and b are at saddle points of the potential. Thus, a mass placed at either a and b will be in an unstable equilibrium.

You can see the ellipse with a and b as its foci, snugly fitting into the triangle.

You can sort of see that the ellipse touches the midpoints of the triangle’s edges.

What you can’t see is that this ellipse has the largest possible area for any ellipse fitting into the triangle!

Now let me explain the math. While the gravitational potential of a point mass in 3d space is proportional to 1/r, the gravitational potential of a line mass in 3d space is proportional to \log r, which is also the gravitational potential of a point mass in 2d space.

So, if we have three equal line masses, which are parallel and pierce an orthogonal plane at points p_1, p_2 and p_3, then their gravitational potential, as a function on this plane, will be proportional to

\phi(z) = \log|z - p_1| + \log|z - p_2| + \log|z - p_3|

Here I’m using z as our name for an arbitrary point on this plane, because the next trick is to think of this plane as the complex plane!

Where are the critical points (in fact saddle points) of this potential? They are just points where the gradient of \phi vanishes. To find these points, we can just take the exponential of \phi and see where the gradient of that vanishes. This is a nice idea because

e^{\phi(z)} = |(z-p_1)(z-p_2)(z-p_3)|

The gradient of this function will vanish whenever

P'(z) = 0

where

P(z) = (z-p_1)(z-p_2)(z-p_3)

Since P is a cubic polynomial, P' is a quadratic, hence proportional to

(z - a)(z - b)

for some a and b. Now we use

Marden’s theorem. Suppose the zeros p_1, p_2, p_3 of a cubic polynomial P are non-collinear. Then there is a unique ellipse inscribed in the triangle with vertices p_1, p_2, p_3 and tangent to the sides at their midpoints. The foci of this ellipse are the zeroes of the derivative of P.

For a short proof of this theorem go here:

Carlson’s proof of Marden’s theorem.

This ellipse is called the Steiner inellipse of the triangle:

• Wikipedia, Steiner inellipse.

The proof that it has the largest area of any ellipse inscribed in the triangle goes like this. Using a linear transformation of the plane you can map any triangle to an equilateral triangle. It’s obvious that there’s a circle inscribed in any equilateral triangle, touching each of the triangle’s midpoints. It’s at least very plausible that that this circle is the ellipse of largest area contained in the triangle. If we can prove this we’re done.

Why? Because linear transformations map circles to ellipses, and map midpoints of line segments to midpoints of line segments, and simply rescale areas by a constant fact. So applying the inverse linear transformation to the circle inscribed in the equilateral triangle, we get an ellipse inscribed in our original triangle, which will touch this triangle’s midpoints, and have the maximum possible area of any ellipse contained in this triangle!


Non-Equilibrium Thermodynamics in Biology (Part 1)

11 May, 2021

Together with William Cannon and Larry Li, I’m helping run a minisymposium as part of SMB2021, the annual meeting of the Society for Mathematical Biology:

• Non-equilibrium Thermodynamics in Biology: from Chemical Reaction Networks to Natural Selection, Monday June 14, 2021, beginning 9:30 am Pacific Time.

You can register for free here before May 31st, 11:59 pm Pacific Time. You need to register to watch the talks live on Zoom. I think the talks will be recorded.

Here’s the idea:

Abstract: Since Lotka, physical scientists have argued that living things belong to a class of complex and orderly systems that exist not despite the second law of thermodynamics, but because of it. Life and evolution, through natural selection of dissipative structures, are based on non-equilibrium thermodynamics. The challenge is to develop an understanding of what the respective physical laws can tell us about flows of energy and matter in living systems, and about growth, death and selection. This session will address current challenges including understanding emergence, regulation and control across scales, and entropy production, from metabolism in microbes to evolving ecosystems.

It’s exciting to me because I want to get back into work on thermodynamics and reaction networks, and we’ll have some excellent speakers on these topics. I think the talks will be in this order… later I will learn the exact schedule.

Christof Mast, Ludwig-Maximilians-Universität München

Coauthors: T. Matreux, K. LeVay, A. Schmid, P. Aikkila, L. Belohlavek, Z. Caliskanoglu, E. Salibi, A. Kühnlein, C. Springsklee, B. Scheu, D. B. Dingwell, D. Braun, H. Mutschler.

Title: Heat flows adjust local ion concentrations in favor of prebiotic chemistry

Abstract: Prebiotic reactions often require certain initial concentrations of ions. For example, the activity of RNA enzymes requires a lot of divalent magnesium salt, whereas too much monovalent sodium salt leads to a reduction in enzyme function. However, it is known from leaching experiments that prebiotically relevant geomaterial such as basalt releases mainly a lot of sodium and only little magnesium. A natural solution to this problem is heat fluxes through thin rock fractures, through which magnesium is actively enriched and sodium is depleted by thermogravitational convection and thermophoresis. This process establishes suitable conditions for ribozyme function from a basaltic leach. It can take place in a spatially distributed system of rock cracks and is therefore particularly stable to natural fluctuations and disturbances.

Supriya Krishnamurthy, Stockholm University

Coauthors: Eric Smith

Title: Stochastic chemical reaction networks

Abstract: The study of chemical reaction networks (CRNs) is a very active field. Earlier well-known results (Feinberg Chem. Enc. Sci. 42 2229 (1987), Anderson et al Bull. Math. Biol. 72 1947 (2010)) identify a topological quantity called deficiency, easy to compute for CRNs of any size, which, when exactly equal to zero, leads to a unique factorized (non-equilibrium) steady-state for these networks. No general results exist however for the steady states of non-zero-deficiency networks. In recent work, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments. Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for non-trivial examples, that in this manner we can predict any moment of interest, for CRNs with non-zero deficiency and non-factorizable steady states. It is however an open question how scalable these techniques are for large networks.

Pierre Gaspard, Université libre de Bruxelles

Title: Nonequilibrium biomolecular information processes

Abstract: Nearly 70 years have passed since the discovery of DNA structure and its role in coding genetic information. Yet, the kinetics and thermodynamics of genetic information processing in DNA replication, transcription, and translation remain poorly understood. These template-directed copolymerization processes are running away from equilibrium, being powered by extracellular energy sources. Recent advances show that their kinetic equations can be exactly solved in terms of so-called iterated function systems. Remarkably, iterated function systems can determine the effects of genome sequence on replication errors, up to a million times faster than kinetic Monte Carlo algorithms. With these new methods, fundamental links can be established between molecular information processing and the second law of thermodynamics, shedding a new light on genetic drift, mutations, and evolution.

Carsten Wiuf, University of Copenhagen

Coauthors: Elisenda Feliu, Sebastian Walcher, Meritxell Sáez

Title: Reduction and the Quasi-Steady State Approximation

Abstract: Chemical reactions often occur at different time-scales. In applications of chemical reaction network theory it is often desirable to reduce a reaction network to a smaller reaction network by elimination of fast species or fast reactions. There exist various techniques for doing so, e.g. the Quasi-Steady-State Approximation or the Rapid Equilibrium Approximation. However, these methods are not always mathematically justifiable. Here, a method is presented for which (so-called) non-interacting species are eliminated by means of QSSA. It is argued that this method is mathematically sound. Various examples are given (Michaelis-Menten mechanism, two-substrate mechanism, …) and older related techniques from the 50-60ies are briefly discussed.

Matteo Polettini, University of Luxembourg

Coauthor: Tobias Fishback

Title: Deficiency of chemical reaction networks and thermodynamics

Abstract: Deficiency is a topological property of a Chemical Reaction Network linked to important dynamical features, in particular of deterministic fixed points and of stochastic stationary states. Here we link it to thermodynamics: in particular we discuss the validity of a strong vs. weak zeroth law, the existence of time-reversed mass-action kinetics, and the possibility to formulate marginal fluctuation relations. Finally we illustrate some subtleties of the Python module we created for MCMC stochastic simulation of CRNs, soon to be made public.

Ken Dill, Stony Brook University

Title: The principle of maximum caliber of nonequilibria

Abstract: Maximum Caliber is a principle for inferring pathways and rate distributions of kinetic processes. The structure and foundations of MaxCal are much like those of Maximum Entropy for static distributions. We have explored how MaxCal may serve as a general variational principle for nonequilibrium statistical physics – giving well-known results, such as the Green-Kubo relations, Onsager’s reciprocal relations and Prigogine’s Minimum Entropy Production principle near equilibrium, but is also applicable far from equilibrium. I will also discuss some applications, such as finding reaction coordinates in molecular simulations non-linear dynamics in gene circuits, power-law-tail distributions in “social-physics” networks, and others.

Joseph Vallino, Marine Biological Laboratory, Woods Hole

Coauthors: Ioannis Tsakalakis, Julie A. Huber

Title: Using the maximum entropy production principle to understand and predict microbial biogeochemistry

Abstract: Natural microbial communities contain billions of individuals per liter and can exceed a trillion cells per liter in sediments, as well as harbor thousands of species in the same volume. The high species diversity contributes to extensive metabolic functional capabilities to extract chemical energy from the environment, such as methanogenesis, sulfate reduction, anaerobic photosynthesis, chemoautotrophy, and many others, most of which are only expressed by bacteria and archaea. Reductionist modeling of natural communities is problematic, as we lack knowledge on growth kinetics for most organisms and have even less understanding on the mechanisms governing predation, viral lysis, and predator avoidance in these systems. As a result, existing models that describe microbial communities contain dozens to hundreds of parameters, and state variables are extensively aggregated. Overall, the models are little more than non-linear parameter fitting exercises that have limited, to no, extrapolation potential, as there are few principles governing organization and function of complex self-assembling systems. Over the last decade, we have been developing a systems approach that models microbial communities as a distributed metabolic network that focuses on metabolic function rather than describing individuals or species. We use an optimization approach to determine which metabolic functions in the network should be up regulated versus those that should be down regulated based on the non-equilibrium thermodynamics principle of maximum entropy production (MEP). Derived from statistical mechanics, MEP proposes that steady state systems will likely organize to maximize free energy dissipation rate. We have extended this conjecture to apply to non-steady state systems and have proposed that living systems maximize entropy production integrated over time and space, while non-living systems maximize instantaneous entropy production. Our presentation will provide a brief overview of the theory and approach, as well as present several examples of applying MEP to describe the biogeochemistry of microbial systems in laboratory experiments and natural ecosystems.

Gheorge Craciun, University of Wisconsin-Madison

Title: Persistence, permanence, and global stability in reaction network models: some results inspired by thermodynamic principles

Abstract: The standard mathematical model for the dynamics of concentrations in biochemical networks is called mass-action kinetics. We describe mass-action kinetics and discuss the connection between special classes of mass-action systems (such as detailed balanced and complex balanced systems) and the Boltzmann equation. We also discuss the connection between the “global attractor conjecture” for complex balanced mass-action systems and Boltzmann’s H-theorem. We also describe some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

Hong Qian, University of Washington

Title: Large deviations theory and emergent landscapes in biological dynamics

Abstract: The mathematical theory of large deviations provides a nonequilibrium thermodynamic description of complex biological systems that consist of heterogeneous individuals. In terms of the notions of stochastic elementary reactions and pure kinetic species, the continuous-time, integer-valued Markov process dictates a thermodynamic structure that generalizes (i) Gibbs’ macroscopic chemical thermodynamics of equilibrium matters to nonequilibrium small systems such as living cells and tissues; and (ii) Gibbs’ potential function to the landscapes for biological dynamics, such as that of C. H. Waddington’s and S. Wright’s.

John Harte, University of Berkeley

Coauthors: Micah Brush, Kaito Umemura

Title: Nonequilibrium dynamics of disturbed ecosystems

Abstract: The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories, and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time-varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic, by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth, or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time-varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.


Applied Category Theory 2021 — Call for Papers

16 April, 2021


The deadline for submitting papers is coming up soon: May 12th.

Fourth Annual International Conference on Applied Category Theory (ACT 2021), July 12–16, 2021, online and at the Computer Laboratory of the University of Cambridge.

Plans to run ACT 2021 as one of the first physical conferences post-lockdown are progressing well. Consider going to Cambridge! Financial support is available for students and junior researchers.

Applied category theory is a topic of interest for a growing community of researchers, interested in studying many different kinds of systems using category-theoretic tools. These systems are found across computer science, mathematics, and physics, as well as in social science, linguistics, cognition, and neuroscience. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, disseminate the latest results, and facilitate further development of the field.

We accept submissions of both original research papers, and work accepted/submitted/ published elsewhere. Accepted original research papers will be invited for publication in a proceedings volume. The keynote addresses will be drawn from the best accepted papers. The conference will include an industry showcase event.

We hope to run the conference as a hybrid event, with physical attendees present in Cambridge, and other participants taking part online. However, due to the state of the pandemic, the possibility of in-person attendance is not yet confirmed. Please do not book your travel or hotel accommodation yet.

Financial support

We are able to offer financial support to PhD students and junior researchers. Full guidance is on the webpage.

Important dates (all in 2021)

• Submission Deadline: Wednesday 12 May
• Author Notification: Monday 7 June
• Financial Support Application Deadline: Monday 7 June
• Financial Support Notification: Tuesday 8 June
• Priority Physical Registration Opens: Wednesday 9 June
• Ordinary Physical Registration Opens: Monday 13 June
• Reserved Accommodation Booking Deadline: Monday 13 June
• Adjoint School: Monday 5 to Friday 9 July
• Main Conference: Monday 12 to Friday 16 July

Submissions

The following two types of submissions are accepted:

Proceedings Track. Original contributions of high-quality work consisting of an extended abstract, up to 12 pages, that provides evidence of results of genuine interest, and with enough detail to allow the program committee to assess the merits of the work. Submission of work-in-progress is encouraged, but it must be more substantial than a research proposal.

Non-Proceedings Track. Descriptions of high-quality work submitted or published elsewhere will also be considered, provided the work is recent and relevant to the conference. The work may be of any length, but the program committee members may only look at the first 3 pages of the submission, so you should ensure that these pages contain sufficient evidence of the quality and rigour of your work.

Papers in the two tracks will be reviewed against the same standards of quality. Since ACT is an interdisciplinary conference, we use two tracks to accommodate the publishing conventions of different disciplines. For example, those from a Computer Science background may prefer the Proceedings Track, while those from a Mathematics, Physics or other background may prefer the Non-Proceedings Track. However, authors from any background are free to choose the track that they prefer, and submissions may be moved from the Proceedings Track to the Non-Proceedings Track at any time at the request of the authors.

Contributions must be submitted in PDF format. Submissions to the Proceedings Track must be prepared with LaTeX, using the EPTCS style files available at http://style.eptcs.org.

The submission link will soon be available on the ACT2021 web page: https://www.cl.cam.ac.uk/events/act2021

Program Committee

Chair:

• Kohei Kishida, University of Illinois, Urbana-Champaign

Members:

• Richard Blute, University of Ottawa
• Spencer Breiner, NIST
• Daniel Cicala, University of New Haven
• Robin Cockett, University of Calgary
• Bob Coecke, Cambridge Quantum Computing
• Geoffrey Cruttwell, Mount Allison University
• Valeria de Paiva, Samsung Research America and University of Birmingham
• Brendan Fong, Massachusetts Institute of Technology
• Jonas Frey, Carnegie Mellon University
• Tobias Fritz, Perimeter Institute for Theoretical Physics
• Fabrizio Romano Genovese, Statebox
• Helle Hvid Hansen, University of Groningen
• Jules Hedges, University of Strathclyde
• Chris Heunen, University of Edinburgh
• Alex Hoffnung, Bridgewater
• Martti Karvonen, University of Ottawa
• Kohei Kishida, University of Illinois, Urbana -Champaign (chair)
• Martha Lewis, University of Bristol
• Bert Lindenhovius, Johannes Kepler University Linz
• Ben MacAdam, University of Calgary
• Dan Marsden, University of Oxford
• Jade Master, University of California, Riverside
• Joe Moeller, NIST
• Koko Muroya, Kyoto University
• Simona Paoli, University of Leicester
• Daniela Petrisan, Université de Paris, IRIF
• Mehrnoosh Sadrzadeh, University College London
• Peter Selinger, Dalhousie University
• Michael Shulman, University of San Diego
• David Spivak, MIT and Topos Institute
• Joshua Tan, University of Oxford
• Dmitry Vagner
• Jamie Vicary, University of Cambridge
• John van de Wetering, Radboud University Nijmegen
• Vladimir Zamdzhiev, Inria, LORIA, Université de Lorraine
• Maaike Zwart


Mathematics in the 21st Century

16 March, 2021

I’m giving a talk in the Topos Institute Colloquium on Thursday March 25, 2021 at 18:00 UTC. That’s 11:00 am Pacific Time.

I’ll say a bit about the developments we might expect if mathematicians could live happily in an ivory tower and never come down for the rest of the century. But my real focus will be on how math will interact with the world outside mathematics.

Mathematics in the 21st Century

Abstract. The climate crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that no physical quantity can grow exponentially forever. This transformation may affect mathematics—and be affected by it—just as dramatically as the agricultural and industrial revolutions. After a review of the problems, we discuss how mathematicians can help make this transformation a bit easier, and some ways in which mathematics may change.

You can see my slides here, and click on links in dark brown for more information. You can watch the talk on YouTube here, either live or recorded later:

You can also watch the talk live on Zoom. Only Zoom lets you ask questions. The password for Zoom can be found on the Topos Institute Colloquium website.


Emerging Researchers in Category Theory

11 March, 2021

 

Eugenia Cheng is an expert on giving clear, fun math talks.

Now you can take a free class from her on how to give clear, fun math talks!

You need to be a grad student in category theory—and priority will be given to those who aren’t at fancy schools, etc.

Her course is called the Emerging Researchers in Category Theory Virtual Seminar, or Em-Cats for short. You can apply for it here:

https://topos.site/em-cats/

The first round of applications is due April 30th. It looks pretty cool, and knowing Eugenia, you’ll get a lot of help on giving talks.

Aims

The aims are, broadly:

• Help the next generation of category theorists become wonderful speakers.
• Make use of the virtual possibilities, and give opportunities to graduate students in places where there is not a category theory group or local seminar they can usefully speak in.
• Give an opportunity to graduate students to have a global audience, especially giving more visibility to students from less famous/large groups.
• Make a general opportunity for community among category theorists who are more isolated than those with local groups.
• Make a series of truly intelligible talks, which we hope students and researchers around the world will enjoy and appreciate.

Talk Preparation and Guidelines

Eugenia Cheng has experience with training graduate students in giving talks, from when she ran a similar seminar for graduate students at the University of Sheffield. Everyone did indeed give an excellent talk.

We ask that all Em-Cats speakers are willing to work with Eugenia and follow her advice. The guidelines document outlines what she believes constitutes a good talk. We acknowledge that this is to some extent a matter of opinion, but these are the guidelines for this particular seminar. Eugenia is confident that with her assistance everyone who wishes to do so will be able to give an excellent, accessible talk, and that this will benefit both the speaker and the community.


Magic Numbers

9 March, 2021

Working in the Manhattan Project, Maria Goeppert Mayer discovered in 1948 that nuclei with certain numbers of protons and/or neutrons are more stable than others. In 1963 she won the Nobel prize for explaining this discovery with her ‘nuclear shell model’.

Nuclei with 2, 8, 20, 28, 50, or 82 protons are especially stable, and also nuclei with 2, 8, 20, 28, 50, 82 or 126 neutrons. Eugene Wigner called these magic numbers, and it’s a fun challenge to explain them.

For starters one can imagine a bunch of identical fermions in a harmonic oscillator potential. In one-dimensional space we have evenly spaced energy levels, each of which holds one state if we ignore spin. I’ll write this as

1, 1, 1, 1, ….

But if we have spin-1/2 fermions, each of these energy levels can hold two spin states, so the numbers double:

2, 2, 2, 2, ….

In two-dimensional space, ignoring spin, the pattern changes to

1, 1+1, 1+1+1, 1+1+1+1, ….

or in other words

1, 2, 3, 4, ….

That is: there’s one state of the lowest possible energy, 2 states of the next energy, and so on. Including spin the numbers double:

2, 4, 6, 8, ….

In three-dimensional space the pattern changes to this if we ignore spin:

1, 1+2, 1+2+3, 1+2+3+4, ….

or

1, 3, 6, 10, ….

So, we’re getting triangular numbers! Here’s a nice picture of these states, drawn by J. G. Moxness:


Including spin the numbers double:

2, 6, 12, 20, ….

So, there are 2 states of the lowest energy, 2+6 = 8 states of the first two energies, 2+6+12 = 20 states of the first three energies, and so on. We’ve got the first 3 magic numbers right! But then things break down: next we get 2+6+12+20 = 40, while the next magic number is just 28.

Wikipedia has a nice explanation of what goes wrong and how to fix it to get the next few magic numbers right:

Nuclear shell model.

We need to take two more effects into account. First, ‘spin-orbit interactions’ decrease the energy of a state when some spins point in the opposite direction from the orbital angular momentum. Second, the harmonic oscillator potential gets flattened out at large distances, so states of high angular momentum have less energy than you’d expect. I won’t attempt to explain the details, since Wikipedia does a pretty good job and I’m going to want breakfast soon. Here’s a picture that cryptically summarizes the analysis:

The notation is old-fashioned, from spectroscopy—you may know it if you’ve studied atomic physics, or chemistry. If you don’t know it, don’t worry about it! The main point is that the energy levels in the simple story I just told change a bit. They don’t change much until we hit the fourth magic number; then 8 of the next 20 energy levels get lowered so much that this magic number is 2+6+12+8 = 28 instead of 2+6+12+20 = 40. Things go on from there.

But here’s something cute: our simplified calculation of the magic numbers actually matches the count of states in each energy level for a four-dimensional harmonic oscillator! In four dimensions, if we ignore spin, the number of states in each energy level goes like this:

1, 1+3, 1+3+6, 1+3+6+10, …

These are the tetrahedral numbers:

Doubling them to take spin into account, we get the first three magic numbers right! Then, alas, we get 40 instead of 28.

But we can understand some interesting features of the world using just the first three magic numbers: 2, 8, and 20.

For example, helium-4 has 2 protons and 2 neutrons, so it’s ‘doubly magic’ and very stable. It’s the second most common substance in the universe! And in radioactive decays, often a helium nucleus gets shot out. Before people knew what it was, people called it an ‘alpha particle’… and the name stuck.

Oxygen-16, with 8 protons and 8 neutrons, is also doubly magic. So is calcium-40, with 20 protons and 20 neutrons. This is the heaviest stable element with the same number of protons and neutrons! After that, the repulsive electric charge of the protons needs to be counteracted by a greater number of neutrons.

A wilder example is helium-10, with 2 protons and 8 neutrons. It’s doubly magic, but not stable. It just barely clings to existence, helped by all that magic.

Here’s one thing I didn’t explain yet, which is actually pretty easy. Why is it true that—ignoring the spin—the number of states of the harmonic oscillator in the nth energy level follows this pattern in one-dimensional space:

1, 1, 1, 1, ….

and this pattern in two-dimensional space:

1, 1+1 = 2, 1+1+1 = 3, 1+1+1+1 = 4, …

and this pattern in three-dimensional space:

1, 1+2 = 3, 1+2+3 = 6, 1+2+3+4 = 10, ….

and this pattern in four-dimensional space:

1, 1+3 = 4, 1+3+6 = 10, 1+3+6+10 = 20, ….

and so on?

To see this we need to know two things. First, the allowed energies for a harmonic oscillator in one-dimensional space are equally spaced. So, if we say the lowest energy allowed is 0, by convention, and choose units where the next allowed energy is 1, then the allowed energies are the natural numbers:

0, 1, 2, 3, 4, ….

Second, a harmonic oscillator in n-dimensional space is just like n independent harmonic oscillators in one-dimensional space. In particular, its energy is just the sum of their energies.

So, the number of states of energy E for an n-dimensional oscillator is just the number of ways of writing E as a sum of a list of n natural numbers! The order of the list matters here: writing 3 as 1+2 counts as different than writing it as 2+1.

This leads to the patterns we’ve seen. For example, consider a harmonic oscillator in two-dimensional space. It has 1 state of energy 0, namely

0+0

It has 2 states of energy 1, namely

1+0 and 0+1

It has 3 states of energy 2, namely

2+0 and 1+1 and 0+2

and so on.

Next, consider a harmonic oscillator in three-dimensional space. This has 1 state of energy 0, namely

0+0+0

It has 3 states of energy 1, namely

1+0+0 and 0+1+0 and 0+0+1

It has 6 states of energy 2, namely

2+0+0 and 1+1+0 and 1+0+1 and 0+2+0 and 0+1+1 and 0+0+2

and so on. You can check that we’re getting triangular numbers: 1, 3, 6, etc. The easiest way is to note that to get a state of energy E, the first of the three independent oscillators can have any natural number j from 0 to E as its energy, and then there are E – j ways to choose the energies of the other two oscillators so that they sum to E – j. This gives a total of

E + (E-1) + (E-2) + \cdots + 1

states, and this is a triangular number.

The pattern continues in a recursive way: in four-dimensional space the same sort of argument gives us tetrahedral numbers because these are sums of triangular numbers, and so on. We’re getting the diagonals of Pascal’s triangle, otherwise known as binomial coefficients.



We often think of the binomial coefficient

\displaystyle{\binom{n}{k} }

as the number of ways of choosing a k-element subset of an n-element set. But here we are seeing it’s the number of ways of choosing an ordered (k+1)-tuple of natural numbers that sum to n. You may enjoy finding a quick proof that these two things are equal!