Applied Category Theory Course – Videos

15 January, 2019

Yay! David Spivak and Brendan Fong are teaching a course on applied category theory based on their book, and the lectures are on YouTube! Here are the first two videos:

Their book is free here:

• Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

If you’re in Boston you can actually go to the course. It’s at MIT January 14 – Feb 1, Monday-Friday, 14:00-15:00 in room 4-237.

They taught it last year too, and last year’s YouTube videos are on the same YouTube channel.

Also, I taught a course based on the first 4 chapters of their book, and you can read my “lectures”, see discussions and do problems here:

Applied category theory course.

So, there’s no excuse not to start applying category theory in your everday life!

The Mathematics of the 21st Century

13 January, 2019

Abstract. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics—and be affected by it—just as dramatically as the agricultural revolution or industrial revolution. We should get ready!

Check out the video of my talk, the first in the Applied Category Theory Seminar here at U. C. Riverside. It was nicely edited by Paola Fernandez and uploaded by Joe Moeller.

The slides are rather hard to see, but you can read them here while you watch the talk:

The Mathematics of the 21st Century.

Click on links in green for more information!

Geometric Quantization (Part 7)

8 January, 2019


I’ve been falling in love with algebraic geometry these days, as I realize how many of its basic concepts and theorems have nice interpretations in terms of geometric quantization. I had trouble getting excited about them before. I’m talking about things like the Segre embedding, the Veronese embedding, the Kodaira embedding theorem, Chow’s theorem, projective normality, ample line bundles, and so on. In the old days, all these things used to make me nod and go “that’s nice”, without great enthusiasm. Now I see what they’re all good for!

Of course this is my own idiosyncratic take on the subject: obviously algebraic geometers have their own pefectly fine notion of what these things are good for. But I never got the hang of that.

Today I want to talk about how the Veronese embedding can be used to ‘clone’ a classical system. For any number k, you can take a classical system and build a new one; a state of this new system is k copies of the original system constrained to all be in the same state! This may not seem to do much, but it does something: for example, it multiplies the Kähler structure on the classical state space by k. And it has a quantum analogue, which has a much more notable effect!

Last time I looked at an example, where I built the spin-3/2 particle by cloning the spin-1/2 particle.

In brief, it went like this. The space of classical states of the spin-1/2 particle is the Riemann sphere, \mathbb{C}\mathrm{P}^1. This just happens to also be the space of quantum states of the spin-1/2 particle, since it’s the projectivization of \mathbb{C}^2. To get the 3/2 particle we look at the map

\text{cubing} \colon \mathbb{C}^2 \to S^3(\mathbb{C}^2)

You can think of this as the map that ‘triplicates’ a spin-1/2 particle, creating 3 of them in the same state. This gives rise to a map between the corresponding projective spaces, which we should probably call

P(\text{cubing}) \colon P(\mathbb{C}^2) \to P(S^3(\mathbb{C}^2))

It’s an embedding.

Algebraic geometers call the image of this embedding the twisted cubic, since it’s a curve in 3d projective space described by homogeneous cubic equations. But for us, it’s the embedding of the space of classical states of the spin-3/2 particle into the space of quantum states. (The fact that classical states give specially nice quantum states is familiar in physics, where these specially nice quantum states are called ‘coherent states’, or sometimes ‘generalized coherent states’.)

Now, you’ll have noted that the numbers 2 and 3 show up a bunch in what I just said. But there’s nothing special about these numbers! They could be arbitrary natural numbers… well, > 1 if we don’t enjoy thinking about degenerate cases.

Here’s how the generalization works. Let’s think of guys in \mathbb{C}^n as linear functions on the dual of this space. We can raise any one of them to the k power and get a homogeneous polynomial of degree k. The space of such polynomials is called S^k(\mathbb{C}^n), so raising to the kth power defines a map

\mathbb{C}^n \to S^k(\mathbb{C}^n)

This in turn gives rise to a map between the corresponding projective spaces:

P(\mathbb{C}^n) \to P(S^k(\mathbb{C}^n))

This map is an embedding, since different linear functions give different polynomials when you raise them to the k power, at least if k \ge 1. And this map is famous: it’s called the k Veronese embedding. I guess it’s often denoted

v_k \colon P(\mathbb{C}^n) \to P(S^k(\mathbb{C}^n))

An important special case occurs when we take n = 2, as we’d been doing before. The space of homogeneous polynomials of degree k in two variables has dimension k + 1, so we can think of the Veronese embedding as a map

v_k \colon \mathbb{C}\mathrm{P}^1 \to \mathbb{C}\mathrm{P}^k

embedding the projective line as a curve in \mathbb{C}\mathrm{P}^k. This sort of curve is called a rational normal curve. When d = 3 it’s our friend from last time, the twisted cubic.

In general, we can think of \mathbb{C}\mathrm{P}^k as the space of quantum states of the spin-k/2 particle, since we got it from projectivizing the spin-k/2 representation of \mathrm{SU}(2), namely S^k(\mathbb{C}^n). Sitting inside here, the rational normal curve is the space of classical states of the spin-k/2 particle—or in other words, ‘coherent states’.

Maybe I should expand on this, since it flew by so fast! Pick any direction you want the angular momentum of your spin-k/2 particle to point. Think of this as a point on the Riemann sphere and think of that as coming from some vector \psi \in \mathbb{C}^2. That describes a quantum spin-1/2 particle whose angular momentum points in the desired direction. But now, form the tensor product

\underbrace{\psi \otimes \cdots \otimes \psi}_{k}

This is completely symmetric under permuting the factors, so we can think of it as a vector in S^k(\mathbb{C}^2). And indeed, it’s just what I was calling

v_k (\psi) \in S^k(\mathbb{C}^2)

This vector describes a collection of k indistinguishable quantum spin-1/2 particles with angular momenta all pointing in the same direction. But it also describes a single quantum spin-k/2 particle whose angular momentum points in that direction! Not all vectors in S^k(\mathbb{C}^2) are of this form, clearly. But those that are, are called ‘coherent states’.

Now, let’s do this all a bit more generally. We’ll work with \mathbb{C}^n, not just \mathbb{C}^2. And we’ll use a variety M \subseteq \mathbb{C}\mathrm{P}^{n-1} as our space of classical states, not necessarily all of \mathbb{C}\mathrm{P}^{n-1}.

Remember, we’ve got:

• a category \texttt{Class} where the objects are linearly normal subvarieties M \subseteq \mathbb{C}\mathrm{P}^{n-1} for arbitrary n,


• a category \texttt{Quant} where the objects are linear subspaces V \subseteq \mathbb{C}^n for arbitrary n.

The morphisms in each case are just inclusions. We’ve got a ‘quantization’ functor

\texttt{Q} \colon \texttt{Class} \to \texttt{Quant}

that maps M \subseteq \mathbb{C}\mathrm{P}^{n-1} to the smallest V \subseteq \mathbb{C}^n whose projectivization contains M. And we’ve got what you might call a ‘classicization’ functor going back:

\texttt{P} \colon \texttt{Quant} \to \texttt{Class}

We actually call this ‘projectization’, since it sends any linear subspace V \subseteq \mathbb{C}^n to its projective space sitting inside \mathbb{C}\mathrm{P}^{n-1}.

We would now like to get the Veronese embedding into the game, copying what we just did for the spin-k/2 particle. We’d like each Veronese embedding v_k to define a functor from \texttt{Class} to \texttt{Class} and also a functor \texttt{Quant} to \texttt{Quant}. For example, the first of these should send the space of classical states of the spin-1/2 particle to the space of classical states of the spin-k/2 particle. The second should do the same for the space of quantum states.

The quantum version works just fine. Here’s how it goes. An object in \texttt{Quant} is a linear subspace

V \subseteq \mathbb{C}^n

for some n. Our functor should send this to

S^k(V) \subseteq S^k(\mathbb{C}^n) \cong \mathbb{C}^{\left(\!\!{n\choose k}\!\!\right)}

Here \left(\!{n\choose k}\!\right) , pronounced ‘n multichoose k’ , is the number of ways to choose k not-necessarily-distinct items from a set of n, since this is the dimension of the space of degree-k homogeneous polynomials on \mathbb{C}^n. (We have to pick some sort of ordering on monomials to get the isomorphism above; this is one of the clunky aspects of our current framework, which I plan to fix someday.)

This process indeed defines functor, and the only reasonable name for it is

S^k \colon \texttt{Quant} \to \texttt{Quant}

Intuitively, it takes any state space of any quantum system and produces the state space for k indistinguishable copies that system. (If you’re a physicist, muttering the phrase ‘identical bosons’ may clarify things. There is also a fermionic version where we use exterior powers instead of symmetric powers, but let’s not go there now.)

The classical version of this functor suffers from a small glitch, which however is easy to fix. An object in \texttt{Class} is a linearly normal subvariety

M \subseteq \mathbb{C}\mathrm{P}^{n-1}

for some n. Applying the k Veronese embedding we get a subvariety

v_k(M) \subseteq \mathbb{C}\mathrm{P}^{\left(\!\!{n\choose k}\!\!\right)-1}

However, I don’t think this is linearly normal, in general. I think it’s linearly normal iff M is k-normal. You can take this as a definition of k-normality, if you like, though there are other equivalent ways to say it.

Luckily, a projectively normal subvariety of projective space is k-normal for all k \ge 1. And even better, projectively normal varieties are fairly common! In particular, any projective space is a projectively normal subvariety of itself.

So, we can redefine the category \texttt{Class} by letting objects be projectively normal subvarieties M \subseteq \mathbb{C}\mathrm{P}^{n-1} for arbitrary n \ge 1. I’m using the same notation for this new category, which is ordinarily a very dangerous thing to do, because all our results about the original version are still true for this one! In particular, we still have adjoint functors

\texttt{Q} \colon \texttt{Class} \to \texttt{Quant}, \qquad \texttt{P} \colon \texttt{Quant} \to \texttt{Class}

defined exactly as before. But now the kth Veronese embedding gives a functor

v_k \colon \texttt{Class} \to \texttt{Class}

Intuitively, this takes any state space of any classical system and produces the state space for k indistinguishable copies that system that are all in the same state. It has no effect on the classical state space M as an abstract variety, just its embedding into projective space—which in turn affects its Kähler structure and the line bundle it inherits from projective space. In particular, its symplectic structure gets multiplied by k, and the line bundle over it gets replaced by its kth tensor power. (These are well-known facts about the Veronese embedding.)

I believe that this functor obeys

\texttt{Q} \circ v_k = S^k \circ \texttt{Q}

and it’s just a matter of unraveling the definitions to see that

\texttt{P} \circ S^k = v_k \circ \texttt{P}

So, very loosely, the functors

v_k \colon \texttt{Class} \to \texttt{Class}, \qquad S^k \colon \texttt{Quant} \to \texttt{Quant}

should be thought of as replacing a classical or quantum system by a new ‘cloned’ version of that system. And they get along perfectly with quantization and its adjoint, projectivization!

Applied Category Theory 2019 School

5 January, 2019

Dear scientists, mathematicians, linguists, philosophers, and hackers:

We are writing to let you know about a fantastic opportunity to learn about the emerging interdisciplinary field of applied category theory from some of its leading researchers at the ACT2019 School. It will begin February 18, 2019 and culminate in a meeting in Oxford, July 22–26. Applications are due January 30th; see below for details.

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 community’s members is as varied as the systems being studied.

The goal of the ACT2019 School is to help grow this community by pairing ambitious young researchers together with established researchers in order to work on questions, problems, and conjectures in applied category theory.

Who should apply

Anyone from anywhere who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language—the definition of monoidal category for example—is encouraged.

We will consider advanced undergraduates, PhD students, and post-docs. We ask that you commit to the full program as laid out below.

Instructions for how to apply can be found below the research topic descriptions.

Senior research mentors and their topics

Below is a list of the senior researchers, each of whom describes a research project that their team will pursue, as well as the background reading that will be studied between now and July 2019.

Miriam Backens

Title: Simplifying quantum circuits using the ZX-calculus

Description: The ZX-calculus is a graphical calculus based on the category-theoretical formulation of quantum mechanics. A complete set of graphical rewrite rules is known for the ZX-calculus, but not for quantum circuits over any universal gate set. In this project, we aim to develop new strategies for using the ZX-calculus to simplify quantum circuits.

Background reading:

  1. Matthes Amy, Jianxin Chen, Neil Ross. A finite presentation of CNOT-Dihedral operators.
  2. Miriam Backens. The ZX-calculus is complete for stabiliser quantum mechanics.

Tobias Fritz

Title: Partial evaluations, the bar construction, and second-order stochastic dominance

Description: We all know that 2+2+1+1 evaluates to 6. A less familiar notion is that it can partially evaluate to 5+1. In this project, we aim to study the compositional structure of partial evaluation in terms of monads and the bar construction and see what this has to do with financial risk via second-order stochastic dominance.

Background reading:

  1. Tobias Fritz and Paolo Perrone. Monads, partial evaluations, and rewriting.
  2. Maria Manuel Clementino, Dirk Hofmann, George Janelidze. The monads of classical algebra are seldom weakly cartesian.
  3. Todd Trimble. On the bar construction.

Pieter Hofstra

Title: Complexity classes, computation, and Turing categories

Description: Turing categories form a categorical setting for studying computability without bias towards any particular model of computation. It is not currently clear, however, that Turing categories are useful to study practical aspects of computation such as complexity. This project revolves around the systematic study of step-based computation in the form of stack-machines, the resulting Turing categories, and complexity classes. This will involve a study of the interplay between traced monoidal structure and computation. We will explore the idea of stack machines qua programming languages, investigate the expressive power, and tie this to complexity theory. We will also consider questions such as the following: can we characterize Turing categories arising from stack machines? Is there an initial such category? How does this structure relate to other categorical structures associated with computability?

Background reading:

  1. J.R.B. Cockett and P.J.W. Hofstra. Introduction to Turing categories. APAL, Vol 156, pp. 183-209, 2008.
  2. J.R.B. Cockett, P.J.W. Hofstra and P. Hrubes. Total maps of Turing categories. ENTCS (Proc. of MFPS XXX), pp. 129-146, 2014.
  3. A. Joyal, R. Street and D. Verity. Traced monoidal categories. Mat. Proc. Cam. Phil. Soc. 3, pp. 447-468, 1996.

Bartosz Milewski

Title: Traversal optics and profunctors

Description: In functional programming, optics are ways to zoom into a specific part of a given data type and mutate it. Optics come in many flavors such as lenses and prisms and there is a well-studied categorical viewpoint, known as profunctor optics. Of all the optic types, only the traversal has resisted a derivation from first principles into a profunctor description. This project aims to do just this.

Background reading:

  1. Bartosz Milewski. Profunctor optics, categorical view.
  2. Craig Pastro, Ross Street. Doubles for monoidal categories.

Mehrnoosh Sadrzadeh

Title: Formal and experimental methods to reason about dialogue and discourse using categorical models of vector spaces

Description: Distributional semantics argues that meanings of words can be represented by the frequency of their co-occurrences in context. A model extending distributional semantics from words to sentences has a categorical interpretation via Lambek’s syntactic calculus or pregroups. In this project, we intend to further extend this model to reason about dialogue and discourse utterances where people interrupt each other, there are references that need to be resolved, disfluencies, pauses, and corrections. Additionally, we would like to design experiments and run toy models to verify predictions of the developed models.

Background reading:

  1. Gerhard Jager (1998): A multi-modal analysis of anaphora and ellipsis. University of Pennsylvania Working Papers in Linguistics 5(2), p. 2.
  2. Matthew Purver, Ronnie Cann, and Ruth Kempson. Grammars as parsers: meeting the dialogue challenge. Research on Language and Computation, 4(2-3):289–326, 2006.

David Spivak

Title: Toward a mathematical foundation for autopoiesis

Description: An autopoietic organization—anything from a living animal to a political party to a football team—is a system that is responsible for adapting and changing itself, so as to persist as events unfold. We want to develop mathematical abstractions that are suitable to found a scientific study of autopoietic organizations. To do this, we’ll begin by using behavioral mereology and graphical logic to frame a discussion of autopoeisis, most of all what it is and how it can be best conceived. We do not expect to complete this ambitious objective; we hope only to make progress toward it.

Background reading:

  1. Brendan Fong, David Jaz Myers, David Spivak. Behavioral mereology.
  2. Brendan Fong, David Spivak. Graphical regular logic.
  3. Luhmann. Organization and Decision, CUP. (Preface)

School structure

All of the participants will be divided up into groups corresponding to the projects. A group will consist of several students, a senior researcher, and a TA. Between January and June, we will have a reading course devoted to building the background necessary to meaningfully participate in the projects. Specifically, two weeks are devoted to each paper from the reading list. During this two week period, everybody will read the paper and contribute to discussion in a private online chat forum. There will be a TA serving as a domain expert and moderating this discussion. In the middle of the two week period, the group corresponding to the paper will give a presentation via video conference. At the end of the two week period, this group will compose a blog entry on this background reading that will be posted to the n-category cafe.

After all of the papers have been presented, there will be a two-week visit to Oxford University, 15–26 July 2019. The second week is solely for participants of the ACT2019 School. Groups will work together on research projects, led by the senior researchers.

The first week of this visit is the ACT2019 Conference, where the wider applied category theory community will arrive to share new ideas and results. It is not part of the school, but there is a great deal of overlap and participation is very much encouraged. The school should prepare students to be able to follow the conference presentations to a reasonable degree.

To apply

To apply please send the following to by January 30th, 2019:

  • Your CV
  • A document with:
    • An explanation of any relevant background you have in category theory or any of the specific projects areas
    • The date you completed or expect to complete your Ph.D and a one-sentence summary of its subject matter.
  • Order of project preference
  • To what extent can you commit to coming to Oxford (availability of funding is uncertain at this time)
  • A brief statement (~300 words) on why you are interested in the ACT2019 School. Some prompts:
    • how can this school contribute to your research goals?
    • how can this school help in your career?

Also have sent on your behalf to a brief letter of recommendation confirming any of the following:

  • your background
  • ACT2019 School’s relevance to your research/career
  • your research experience


For more information, contact either
– Daniel Cicala. cicala (at) math (dot) ucr (dot) edu
– Jules Hedges. julian (dot) hedges (at) cs (dot) ox (dot) ac (dot) uk

Unsolved Mysteries of Fundamental Physics

2 January, 2019

In this century, progress in fundamental physics has been slow. The Large Hadron Collider hasn’t yet found any surprises, attempts to directly detect dark matter have been unsuccessful, string theory hasn’t made any successful predictions, and nobody really knows what to do about any of this. But there is no shortage of problems, and clues. Watch the talk I gave at the Cambridge University Physics Society for some ideas on this! Warning: this is for ordinary folks, not experts.

There are some squeaky sounds on the video at first, but they seem to go away pretty quick, so hang in there! You can also see my talk slides here:

Unsolved mysteries of theoretical physics.

and click on the links for extra information.

Geometric Quantization (Part 6)

1 January, 2019


Now let’s do some more interesting examples of geometric quantization using the functor described in Part 4. Let’s look at the spin-j particle with j > 1/2.

To be specific, let’s consider the spin-3/2 particle. There’s nothing special about the number 3 here: everything I’ll say can be generalized. But the number 3 will give me a nice excuse to show you a picture of a curve called the ‘twisted cubic’.

We can build a spin-3/2 particle from three spin-1/2 particles, all having angular momenta pointing in the same direction. Classically this procedure amounts to tripling a vector, but quantum-mechanically it’s related to cubing. This is a bit mysterious to me, but let me explain.

Classically, if we take a vector of length 1/2 in \mathbb{R}^3 and triple it we get a vector of length 3/2. This gives a map from classical states of the spin-1/2 particle to classical states of the spin-3/2 particle. In other words: a map from the sphere of radius 1/2 to the sphere of radius 3/2. Simple! But this is not a symplectic map, since as we saw last time, we are giving the latter sphere a symplectic structure that’s 3 times as big. So, it’s not a valid classical process. Still, it’s a perfectly fine way to get our hands on lots of states of the classical spin-3/2 particle! All of them, in fact.

Quantum-mechanically the state space for the spin-1/2 particle is \mathbb{C}^2. We can think of a guy in here as a linear functional on the dual {\mathbb{C}^2}^\ast. We can cube this and get a homogeneous polynomial of degree 3 on {\mathbb{C}^2}^\ast. The space of such polynomials is called S^3(\mathbb{C}^2), and this is the space of states of the quantum spin-3/2 particle. So, we get a map

\text{cubing} \colon \mathbb{C}^2 \to S^3(\mathbb{C}^2)

Simple! This describes the process of ‘triplicating’ the state of spin-1/2 particle and getting a spin-3/2 particle. But this is not a linear map, so it’s not a valid quantum process. As the saying goes, “you can’t clone a quantum”. Still, it’s a perfectly fine way to get our hands on lots of states of the quantum spin-3/2 particle! But not all of them, as we’ll soon see.

Geometric quantization should reconcile and combine classical and quantum mechanics. How can we do this here?

It’s pretty simple. We projectivize the map

\text{cubing} \colon \mathbb{C}^2 \to S^3(\mathbb{C}^2)

Cubing sends any line through the origin in \mathbb{C}^2 into a unique line through the origin in S^3(\mathbb{C}^2). So, we get a map from \mathbb{C}\mathrm{P}^1 to the projective space of S^3(\mathbb{C}^2). And we define the image of this map to be the space of classical states of the spin-3/2 particle!

Let’s see what it looks like. I’m thinking of an element (a,b) \in \mathbb{C}^2 as a linear functional

f(x,y) = ax + by

If we cube it we get this homogeneous polynomial of degree 3:

f(x,y)^3 = a^3 \, x^3 + 3a^2b \, x^2y + 3ab^2 \, xy^2 + b^3 \, y^3

If we take x^3, x^2y, xy^2 and y^3 as our basis of the homogeneous polynomials of degree 3, we can identify S^3(\mathbb{C}^2) with \mathbb{C}^4. So we get

\begin{array}{clll}  \text{cubing} \colon & \mathbb{C}^2 &\to& \mathbb{C}^4 \\                      & (a,b) & \mapsto & (a^3, 3a^2b, 3ab^2, b^3)  \end{array}

If we projectivize this map we get a map between projective varieties, which I’ll call

P(\text{cubing}) \colon  \mathbb{C}\mathrm{P}^1 \to \mathbb{C}\mathrm{P}^3

The image of this map is the space of states of the classical spin 3/2-particle! It’s a copy of \mathbb{C}\mathrm{P}^1 sitting inside projective 3-space. But because cubing a nonlinear map, this copy will be twisted: not a ‘line’ but a ‘curve’. People call it the twisted cubic.

Just for kicks, let’s take a closer look at it. Let’s use homogeneous coordinates and write [a,b] for the point in \mathbb{C}\mathrm{P}^1 corresponding to a nonzero vector (a,b) \in \mathbb{C}^2. Similarly, any point in \mathbb{C}\mathrm{P}^3 can be written as a nonzero 4-tuple of complex numbers with a bracket around it. We get

\begin{array}{cccl} P(\text{cubing}) \colon & \mathbb{C}\mathrm{P}^1 &\to& \mathbb{C}\mathrm{P}^3 \\                                     & [a,b] & \mapsto & [a^3, 3a^2b, 3ab^2, b^3]  \end{array}

so the twisted cubic is

\{ [a^3, 3a^2b, 3ab^2, b^3] : \, (0,0) \ne (a,b) \in \mathbb{C}^2 \} \subset \mathbb{C}\mathrm{P}^3

This is hard to visualize, so we can work with a copy of \mathbb{C}^3 that’s dense inside \mathbb{C}\mathrm{P}^3, namely the set where a = 1. The portion of the twisted cubic sitting in this set is

\{ (b, 3b^2, b^3) : \, b \in \mathbb{C} \} \subset \mathbb{C}^3

This is still hard to visualize, so we can restrict to the reals and think about the curve

\{ (b, 3b^2, b^3) : \, b \in \mathbb{R} \} \subset \mathbb{R}^3

This is also called the twisted cubic! People often rescale the y axis to get rid of the number 3 here.

This version of the twisted cubic is still a bit hard to visualize, but it’s the intersection of two very nice surfaces: the surface y = x^2 and the surface z = x^3. So, the twisted cubic is the black curve:

in this nice picture uploaded to ResearchGate by Alexander M. Kasprzyk.

Okay, back to serious business! Let’s call the twisted cubic C_3 \subseteq \mathbb{C}\mathrm{P}^3. It’s exactly the sort of thing we can geometrically quantize using our functor

\texttt{Q} \colon \texttt{Class} \to \texttt{Quant}

The reason is that it’s a projective variety and it’s linearly normal. Moreover when we quantize C_3 we get \mathbb{C}^4, which is just what we want for the spin-3/2 particle!

\texttt{Q}(C_3) = \mathbb{C}^4

Why? Remember, \texttt{Q}(C_3) is defined to be the smallest linear subspace V \subseteq \mathbb{C}^4 such that C_3 \subseteq P(V). But C_3 twists around so much that the smallest V that works is all of \mathbb{C}^4.

Now let’s take stock of where we are and draw some general conclusions from what we’ve seen in this example. We now have a firm grip on the space of quantum states of the spin-3/2 particle:

S^3(\mathbb{C}^2) \cong \mathbb{C}^4

and also the space of classical states of the spin-3/2 particle, the twisted cubic:

C_3 = \{ [a^3, 3a^2b, 3ab^2, b^3] : \, (0,0) \ne (a,b) \in \mathbb{C}^2 \}

Now for something cool: the latter sits inside the projectivization of the former! This is obvious, but it has a very nice physical meaning. While I’ve been calling \mathbb{C}^4 the space of quantum states of the spin-3/2 particle, it is very reasonable to argue that quantum states are actually points of its projectivization, \mathbb{C}\mathrm{P}^3. I will skip the argument, which is old, famous and convincing:

• Wikipedia, Projective Hilbert space.

What matters for us here is that classical states of the spin-3/2 particle give some of these quantum states—far from all, but some of the nicest ones! They are the states obtained by ‘cubing’ a state of a spin-1/2 particle. In other words, they are the states where we can think of our spin-3/2 particle as made of three spin-1/2 particles with their spins perfectly aligned.

It’s nice to say this with a bit more physics jargon. These quantum states coming from classical ones are called ‘coherent states’ . A general quantum state of the spin-3/2 particle is a ‘quantum superposition’ of these coherent states. By this, I mean that the smallest linear subspace V \subseteq \mathbb{C}^4 for which P(V) contains the twisted cubic is all of \mathbb{C}^4.

And while we’ve seen this in a particular example, it’s a completely general feature of our setup! Remember, projectivization

\texttt{P} \colon \texttt{Quant} \to \texttt{Class}

is the left adjoint of quantization

\texttt{Q} \colon \texttt{Class} \to \texttt{Quant}

This means that for any M \in \texttt{Class}, V \in \texttt{Quant} we have

\texttt{Q}(M) \subseteq V \quad \iff \quad M \subseteq \texttt{P}(V)

We get something interesting if we take V = \texttt{Q}(M) here. Since \texttt{Q}(M) \subseteq \texttt{Q}(M), we get

M \subseteq \texttt{P} (\texttt{Q}(M))

Category theorists call this inclusion of M in \texttt{P} (\texttt{Q}(M)) the ‘unit’ of our pair of adjoint functors. It says how classical states sit inside the projectivization of \texttt{Q}(M). If we call points of \texttt{P}(\texttt{Q}(M)) quantum states, those in M are called the coherent states.

Moreover every quantum state is a ‘quantum superposition’ of coherent states! In other words, the smallest linear subspace V \subseteq \texttt{Q}(M) for which M sits inside \texttt{P}(V) is all of \texttt{Q}(M).

Why is this true? It’s just the definition of \texttt{Q}(M)!

So, I hope you see how much physics is packed into these adjoint functors \texttt{Q} and \texttt{P}. I’ll do more examples next time. In the meantime, you can read more about this approach to the spin-3/2 particle here:

• Dorje C. Brody and Lane P. Hughston, Geometric quantum mechanics.

I was greatly inspired by this article!

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.


1 January, 2019

Happy New Year! People like to ponder grand themes each time the Earth completes another orbit around the Sun, so let’s give that a try.

Maria Mannone is a musician who studies the relation between mathematics, music and the visual arts. We met at a conference on The Philosophy and Physics of Noether’s Theorems. Later she decided to interview me for the blog Math is in the Air. There’s a version in English and one in Italian.

She let me reprint the interview here… so with no further ado, here it is!

MM: You are one of the pioneers in using the internet and blogs for scientific education, with ‘This Week’s Finds.’ Which words would you use to feed the enthusiasm of young minds towards abstract mathematics?

JB: It seems only certain people are drawn to mathematics, and that’s fine: there are many wonderful things in life and there’s no need for everyone explore all of them. Mathematics seems to attract people who enjoy patterns, who enjoy precision, and who don’t want to remember lists of arbitrary facts, like the names of all 206 bones in the human body. In math everything has a reason and you can understand it, so you don’t really need to remember much. At first it may seem like there’s a lot to remember – for examples, lists of trig identities. But as you go deeper into math, and understand more, everything becomes simpler. These days I don’t bother to remember more than a couple of trigonometric identities; if I ever need them I can figure them out.

But the really surprising thing is that as you go deeper and deeper into mathematics, it keeps revealing more beauty, and more mysteries. You enter new worlds full of profound questions that are quite hard to explain to nonmathematicians. As the Fields medalist Maryam Mirzakhani said, “The beauty of mathematics only shows itself to more patient followers.”

MM: I love the reference to patterns, and the beauty to find. Thus, we can say that mathematical beauty is not ‘all out there’ as the beauty of a flower can be. Or, that some beautiful geometry present in nature can give a hint or can embody some mathematical beauty, but people have to work hard to find more of it—at least they have to learn how to look at things, and thus, how to mathematically think of them.

In the common opinion, a rose, or a water lily is beautiful (and it is!), but a bone is not ‘beautiful’ per se. Personally, each time I find patterns, regularities, hierarchical structures, I get excited and things seem to be at least mathematically interesting. I would like to ask you how would you relate the beauty in the natural world, both visible and ‘to discover,’ and the beauty of math. I’m wondering if they should be considered as two separate sets with occasional, random intersections, or as two displays of a generalized ‘beauty,’ as two different perspectives. Or, maybe, if the first can guide our search into math, or if math can teach us ‘how to look at things and finding beauty.’

JB: I think all forms of beauty are closely connected, and I think almost anything can be beautiful if it’s not the result of someone being heedless to their environment or deliberately hurtful.

It’s not surprising that flowers are very easy to find beautiful, since they evolved precisely to be attractive. Not to humans, at first, but to pollinators like birds and bees. It’s imaginable that what attracts those animals would not be attractive to us. But in fact there’s enough commonality that we enjoy flowers too! And then we bred them to please us even more; many of them are now symbiotic with us.

Something like a bone only becomes beautiful if you examine it carefully and think about how complex it is and how admirably it carries out its function.


Bones are initially scary or ‘disgusting’ because when they’re doing their job they are hidden: we usually see them only when an animal is seriously injured or dead. So, you have to go past that instinctive reaction—which by the way serves a useful purpose—to see the beauty in a bone.

Mathematics is somewhere between a rose and a bone. Underlying all of nature there are mathematical patterns – but normally they are hidden from view, like bones in a body. Perhaps to some people they seem harsh or even disgusting when first revealed, but in fact they are extremely elegant. Even those who love mathematics find its patterns austere at first—but as we explore it more deeply, we see they connect in complicated delicate patterns that put the petals of a rose to shame.

MM: Thus, there seems to be an intimate dialogue between nature, both visible and hidden, and mathematical thinking. About nature and environment: in your Twitter image, there is a sketch of you as a superhero saving the planet, with the mathematical symbol ‘There is one and only one’ applied to our planet Earth. Can you tell the readers something about the way you combine your research in mathematics with your engagement for the environment?

Also, it is often said that beauty will save the world. Do you think that mathematical beauty can save the world?

JB: I mainly think of beauty—in all its forms—-as a reason why the world is worth saving. But we are very primitive when it comes to the economics of beauty. Paintings can sell for hundreds of millions of dollars, and we have a market for them. But nobody attaches any value to this critically endangered frog, Atelopus varius:

Atelopus varius,

To my mind it’s more beautiful and precious than any painting. Not the individual, of course, but the species, which has taken millions of years to evolve. We are busy destroying species like this as if they were worthless trash. Our descendants, if we have any, will probably think we were barbaric idiots.

But I digress! I switched from pure mathematics and highly theoretical physics to more practical concerns around 2010, when I spent two years at the Centre for Quantum Technologies, in Singapore. I was very lucky that the director encouraged me to think about whatever I wanted. I was wanting a change in direction, and I soon realized that mathematicians, like everyone else, need to think about global warming and what we can do about it: it’s the crisis of our time. I spent some time learning the basics of climate science and working on some projects connected to that. It became clear that to do anything about global warming we need new ideas in politics and economics. Unfortunately, I’m not especially good at those things. So I decided to do something I can actually do, namely to get mathematicians to turn their attention from math inspired by the physics of the microworld—for example string theory—toward math inspired by the visible world around us: biology, ecology, engineering, economics and the like. I’m hoping that mathematicians can solve some problems by thinking more abstractly than anyone else can.

So to finally answer your last question: I’m not sure the beauty of mathematics can save the world, but its beauty is closely connected to clear thinking, and we really need clear thinking.

MM: Yes, in a certain sense, despite culture, technology, and thousands of years of human history, people are quite primitive when it comes to evaluating beauty as detached from the economy.

You brought up an important point: the research focus of mathematicians. This is a tricky point because young researchers are kind of split between following new ideas and projects, and the search for funds, that often leads them to join existing projects or just well-funded areas and to put aside their more ‘visionary’ ideas. What would be your suggestion to find a balance?

JB: I don’t know if I can give advice here: I’ve never needed to search for funds, I get paid to teach calculus and other courses, so I always just do the best research I can. That’s already quite hard—I could talk all day about that!

I suppose if you’re struggling for funds you have to fight to remember your dreams, and try to work your way into a situation where you can pursue these dreams. I imagine this is also true for any entrepreneur with a visionary idea. Academics struggling to get grants really aren’t all that different from executives in a large corporation trying to get funding for their projects.

MM: My last question is about the theme of peace, very important to the Baez family. Many innovations are related to the military. Do you think that the needed clear thinking you mentioned, can first of all come from times, themes, and ideas of peace?

JB: We are currently in a struggle that’s much bigger, and more inspiring, than any war between human tribes. We’re struggling to come to terms with the Anthropocene: the epoch where the Earth’s ecosystems and even geology are being transformed by humans. We are used to treating our impact on nature as negligible. This is no longer true! The Arctic is rapidly melting:

And since 1970, the abundance of many vertebrate species worldwide has dropped 60%. You can see it in this chart prepared by the Worldwide Wildlife Fund:

If this were a war, and these were humans dying, this would be the worst war the world has ever seen! But these changes will not merely affect other species; they are starting to hit us too. We need to wake up. We will either deliberately change our civilization, quite quickly, or we will watch as our cities burn and drown. Isn’t it better to use that intelligence we humans love to boast about, and take action?

MM: Thank you Professor, I hope these words will enlighten many people.