Applied Category Theory Meeting at UCR (Part 3)

15 November, 2019

 

We had a special session on applied category theory here at UCR:

Applied category theory, Fall Western Sectional Meeting of the AMS, 9–10 November 2019, U.C. Riverside.

I was bowled over by the large number of cool ideas. I’ll have to blog about some of them. A bunch of people stayed for a few days afterwards, and we had lots of great conversations.

The biggest news was that Brendan Fong and David Spivak definitely want to set up an applied category theory in the San Francisco Bay Area, which they’re calling the Topos Institute. They are now in the process of raising funds for this institute! I plan to be involved, so I’ll be saying more about this later.

But back to the talks. We didn’t make videos, but here are the slides. Click on talk titles to see abstracts of the talks. For a multi-author talk, the person whose name is in boldface is the one who gave the talk. You also might enjoy comparing the 2017 talks.

Saturday November 9, 2019

8:00 a.m.
Fibrations as generalized lens categoriestalk slides.
David I. Spivak, Massachusetts Institute of Technology

9:00 a.m.
Supplying bells and whistles in symmetric monoidal categoriestalk slides.
Brendan Fong, Massachusetts Institute of Technology
David I. Spivak, Massachusetts Institute of Technology

9:30 a.m.
Right adjoints to operadic restriction functorstalk slides.
Philip Hackney, University of Louisiana at Lafayette
Gabriel C. Drummond-Cole, IBS Center for Geometry and Physics

10:00 a.m.
Duality of relationstalk slides.
Alexander Kurz, Chapman University

10:30 a.m.
A synthetic approach to stochastic maps, conditional independence, and theorems on sufficient statisticstalk slides.
Tobias Fritz, Perimeter Institute for Theoretical Physics

3:00 p.m.
Constructing symmetric monoidal bicategories functoriallytalk slides.
Michael Shulman, University of San Diego
Linde Wester Hansen, University of Oxford

3:30 p.m.
Structured cospanstalk slides.
Kenny Courser, University of California, Riverside
John C. Baez, University of California, Riverside

4:00 p.m.
Generalized Petri netstalk slides.
Jade Master, University of California, Riverside

4:30 p.m.
Formal composition of hybrid systemstalk slides and website.

Paul Gustafson, Wright State University
Jared Culbertson, Air Force Research Laboratory
Dan Koditschek, University of Pennsylvania
Peter Stiller, Texas A&M University

5:00 p.m.
Strings for cartesian bicategoriestalk slides.
M. Andrew Moshier, Chapman University

5:30 p.m.
Defining and programming generic compositions in symmetric monoidal categoriestalk slides.
Dmitry Vagner, Los Angeles, CA

Sunday November 10, 2019

8:00 a.m.
Mathematics for second quantum revolutiontalk slides.
Zhenghan Wang, UCSB and Microsoft Station Q

9:00 a.m.
A compositional and statistical approach to natural languagetalk slides.
Tai-Danae Bradley, CUNY Graduate Center

9:30 a.m.
Exploring invariant structure in neural activity with applied topology and category theorytalk slides.
Brad Theilman, UC San Diego
Krista Perks, UC San Diego
Timothy Q Gentner, UC San Diego

10:00 a.m.
Of monks, lawyers and villages: new insights in social network science — talk cancelled due to illness.
Nina Otter, Mathematics Department, UCLA
Mason A. Porter, Mathematics Department, UCLA

10:30 a.m.
Functorial cluster embeddingtalk slides.

Steve Huntsman, BAE Systems FAST Labs

2:00 p.m.
Quantitative equational logictalk slides.
Prakash Panangaden, School of Computer Science, McGill University
Radu Mardare, Strathclyde University
Gordon D. Plotkin, University of Edinburgh

3:00 p.m.
Brakes: an example of applied category theorytalk slides in PDF and Powerpoint.
Eswaran Subrahmanian, Carnegie Mellon University / National Institute of Standards and Technology

3:30 p.m.
Intuitive robotic programming using string diagramstalk slides.
Blake S. Pollard, National Institute of Standards and Technology

4:00 p.m.
Metrics on functor categoriestalk slides.
Vin de Silva, Department of Mathematics, Pomona College

4:30 p.m.
Hausdorff and Wasserstein metrics on graphs and other structured datatalk slides.
Evan Patterson, Stanford University


Why Is Category Theory a Trending Topic?

8 November, 2019

I wrote something for the Spanish newspaper El País, which has a column on mathematics called “Café y Teoremas”. Ágata Timón helped me a lot with writing this, and she also translated it into Spanish:

• John Baez, Qué es la teoría de categorías y cómo se ha convertido en tendencia, El País, 8 November 2019.

Here’s the English-language version I wrote. It’s for a general audience so don’t expect hard-core math!

Why has “category theory” become a trending topic?

Recently, various scientific media have been paying attention to a branch of mathematics called “category theory” that has become pretty popular inside the mathematical community in recent years. Some mathematicians are even starting to complain on Twitter that more people are tweeting about category theory than their own specialties. But what is this branch of mathematics, and why is it becoming so fashionable?

Category theory was invented in 1945 as a general technique to transform problems in one field of pure mathematics into problems in another field, where they could be solved. For example, we know that at any moment there must be a location on the surface of the Earth there where the wind velocity is zero. This is a marvelous result—but to prove this result, we must translate it into a fact about algebra, and a bit of category theory is very helpful here. More difficult results often require more category theory. The proof of Fermat’s Last Theorem, for example, builds on a vast amount of 20th-century mathematics, in which category theory plays a crucial role.

Category theory is sometimes called “the mathematics of mathematics”, since it stands above many other fields of mathematics, connecting and uniting them. Unfortunately even mathematicians have a limited tolerance for this high level of abstraction. So, for a long time many mathematicians called category theory “abstract nonsense”—using it reluctantly when it was necessary for their work, but not really loving it.

On the other hand, other mathematicians embraced the beauty and power of category theory. Thus, its influence has gradually been spreading. Since the 1990s, it has been infiltrating computer science: for example, new programming languages like Haskell and Scala use ideas from this subject. But now we are starting to see people apply category theory to chemistry, electrical engineering, and even the design of brakes in cars! “Applied category theory”, once an oxymoron, is becoming a real subject.

To understand this we need a little taste of the ideas. A category consists of a set of “objects” together with “morphisms”—some kind of processes, or paths—going between these objects. For example, we could take the objects to be cities, and the morphisms to be routes from one city to another. The key requirement is that if we have a morphism from an object x to an object y and a morphism from y to an object z, we can “compose” them and get a morphism from x to z. For example, if you have a way to drive from Madrid to Seville and a way to drive from Seville to Faro, that gives a way to drive from Madrid to Faro. Thus there is a category of cities and routes between them.

In mathematics, this focus on morphisms represented a radical shift of viewpoint. Starting around 1900, logicians tried to build the whole of mathematics on solid foundations. This turned out to be a difficult and elusive task, but their best attempt at the time involved “set theory”. A set is simply a collection of elements. In set theory as commonly practiced by mathematicians, these elements are also just sets. In this worldview, everything is just a set. It is a static worldview, as if we had objects but no morphisms. On the other hand, category theory builds on set theory by emphasizing morphisms—ways of transforming things—as equal partners to things themselves. It is not incompatible with set theory, but it offers new ways of thinking.

The idea of a category is simple. Exploiting it is harder. A loose group of researchers are starting to apply category theory to subjects beyond pure mathematics. The key step is to focus a bit less on things and a bit more on morphisms, which are ways to go between things, or ways to transform one thing into another. This is attitude is well suited to computer programming: a program is a way to transform input data into output data, and composing programs is the easiest way to build complicated programs from simpler ones. But personally, I am most excited by applications to engineering and the natural sciences, because these are newer and more surprising.

I was very pleased when two of my students got internships at the engineering firm Siemens, applying category theory to industrial processes. The first, Blake Pollard, now has a postdoctoral position at the National Institute of Standards and Technology in the USA. Among other things, he has used a programming method based on category theory to help design a “smart grid”—an electrical power network that is flexible enough to handle the ever-changing power generated by thousands of homes equipped with solar panels.

Rumors say that soon there may even be an institute of applied category theory, connecting mathematicians to programmers and businesses who need this way of thinking. It is too early to tell if this is the beginning of a trend, but my friends and colleagues on Twitter are very excited.


Diversity Workshop at UCR

14 October, 2019

We’re having a workshop to promote diversity in math here at UCR:

Riverside Mathematics Workshop for Excellence and Diversity, Friday 8 November 2019, U. C. Riverside. Organized by John Baez, Weitao Chen, Edray Goins, Ami Radunskaya, and Fred Wilhelm.

If you want to come, please register here.

It’s happening right before the applied category theory meeting, so I hope some of you can make both… especially since the category theorist Eugenia Cheng will be giving a talk!

Three talks will take place in Skye Hall—home of the math department—starting at 1 pm. After this we’ll have refreshments and an hour for students to talk to the speakers. Starting at 6 pm there will be a reception across the road at the UCR Alumni Center, with food and a panel discussion on the challenges we face in promoting diversity at U.C. Riverside.

All the talks will be in Skye 284:

• 1:00–1:50 p.m. Abba Gumel, Arizona State University.

Some models for enhancing diversity and capacity-building in STEM education in under-represented minority communities.

STEM (science, technology, engineering and mathematics) education is undoubtedly the necessary bedrock for the development and sustenance of the vitally-needed knowledge-based economy that fuels and sustains the development of modern nations. Central to STEM education are, of course, the mathematical science … which are the rock-solid foundation of all the natural and engineering sciences. Hence, it is vital that all diverse populations are not left behind in the quest to build and sustain capacity in the mathematical sciences. This talk focuses on discussion around a number of pedagogic and mentorship models that have been (and are being) used to help increase diversity and capacity-building in STEM education in general, and in the mathematical sciences in particular, in under-represented minority populations. Some examples from Africa, Canada and the U.S. will be presented.

• 2:00–2:50. Marissa Loving, Georgia Tech.

Where do I belong? Creating space in the math community.

I will tell the story of my mathematical journey with a focus on my time in grad school. I will be blunt about the ups and downs I have experienced and touch on some of the barriers (both structural and internalized) I have encountered. I will also discuss some of the programs and spaces I have helped create in my quest to make the mathematics community into a place where folks from historically under-represented groups (particularly women of color) can feel safe, seen, and free to devote their energy to their work. If you have ever felt like you don’t belong or worried that you have made others feel that way, this talk is for you.

• 3:00–3:50 p.m. Eugenia Cheng, School of the Art Institute of Chicago.

Inclusion–exclusion in mathematics and beyond: who stays in, who falls out, why it happens, and what we could do about it.

The question of why women and minorities are under-represented in mathematics is complex and there are no simple answers, only many contributing factors. I will focus on character traits, and argue that if we focus on this rather than gender we can have a more productive and less divisive conversation. To try and focus on characters rather than genders I will introduce gender-neutral character adjectives “ingressive” and “congressive” as a new dimension to shift our focus away from masculine and feminine. I will share my experience of teaching congressive abstract mathematics to art students, in a congressive way, and the possible effects this could have for everyone in mathematics, not just women. Moreover I will show that abstract mathematics is applicable to working towards a more inclusive, congressive society in this politically divisive era. This goes against the assumption that abstract math can only be taught to high level undergraduates and graduate students, and the accusation that it is removed from real life.

• 4:00–4:30 p.m. Refreshments in Skye 284.

• 4:30–5:30 p.m. Conversations Between Speakers & Students, Not Faculty, in Skye 284.

• 6:00–6:45 p.m. Reception with Food at the Alumni Center.

• 6:45 – 7:45 p.m. Panel Discussion at Alumni Center with Alissa Crans, Jose Gonzalez and Paige Helms, moderated by Edray Goins.


Climate Technology Primer (Part 2)

13 October, 2019

Here’s the second of a series of blog articles:

• Adam Marblestone, Climate technology primer (2/3): CO2 removal.

The first covered the basics of climate science as related to global warming. This one moves on to consider technologies for removing carbon dioxide from the air.

I hope you keep the following warning in mind as you read on:

I’m focused here on trying to understand the narrowly technical aspects, not on the political aspects, despite those being crucial. This is meant to be a review of the technical literature, not a political statement. I worried that writing a blog purely on the topic of technological intervention in the climate, without attempting or claiming to do justice to the social issues raised, would implicitly suggest that I advocate a narrowly technocratic or unilateral approach, which is not my intention. By focusing on technology, I don’t mean to detract from the importance of the social and policy aspects.

The technological issues are worth studying on their own, since they constrain what’s possible. For example: to draw down as much CO2 as human civilization is emitting now, with trees their peak growth phase and their carbon stored permanently, could be done by covering the whole USA with such trees.


Foundations of Math and Physics One Century After Hilbert

10 October, 2019

I wrote a review of this book with chapters by Penrose, Witten, Connes, Atiyah, Smolin and others:

• John Baez, review of Foundations of Mathematics and Physics One Century After Hilbert: New Perspectives, edited by Joseph Kouneiher, Notices of the American Mathematical Society 66 no. 11 (November 2019), 1690–1692.

It gave me a chance to say a bit—just a tiny bit—about the current state of fundamental physics and the foundations of mathematics.


Quantales from Petri Nets

6 October, 2019

A referee pointed out this paper to me:

• Uffe Engberg and Glynn Winskel, Petri nets as models of linear logic, in Colloquium on Trees in Algebra and Programming, Springer, Berlin, 1990, pp. 147–161.

It contains a nice observation: we can get a commutative quantale from any Petri net.

I’ll explain how in a minute. But first, what does have to do with linear logic?

In linear logic, propositions form a category where the morphisms are proofs and we have two kinds of ‘and’: \& , which is a cartesian product on this category, and \otimes, which is a symmetric monoidal structure. There’s much more to linear logic than this (since there are other connectives), and maybe also less (since we may want our category to be a mere poset), but never mind. I want to focus on the weird business of having two kinds of ‘and’.

Since \& is cartesian we have P \Rightarrow P \& P as usual in logic.

But since \otimes is not cartesian we usually don’t have P \Rightarrow P \otimes P. This other kind of ‘and’ is about resources: from one copy of a thing P you can’t get two copies.

Here’s one way to think about it: if P is “I have a sandwich”, P \& P is like “I have a sandwich and I have a sandwich”, while P \otimes P is like “I have two sandwiches”.

A commutative quantale captures these two forms of ‘and’, and more. A commutative quantale is a commutative monoid object in the category of cocomplete posets: that is, posets where every subset has a least upper bound. But it’s a fact that any cocomplete poset is also complete: every subset has a greatest lower bound!

If we think of the elements of our commutative quantale as propositions, we interpret x \le y as “x implies y”. The least upper bound of any subset of proposition is their ‘or’. Their greatest lower bound is their ‘and’. But we also have the commutative monoid operation, which we call \otimes. This operation distributes over least upper bounds.

So, a commutative quantale has both the logical \& (not just for pairs of propositions, but arbitrary sets of them) and the \otimes operation that describes combining resources.

To get from a Petri net to a commutative quantale, we can compose three functors.

First, any Petri net gives a commutative monoidal category—that is, a commutative monoid object in \mathsf{Cat}. Indeed, my student Jade has analyzed this in detail and shown the resulting functor from the category of Petri nets to the category of commutative monoidal categories is a left adjoint:

• Jade Master, Generalized Petri nets, Section 4.

Second, any category gives a poset where we say x \le y if there is a morphism from x to y. Moreover, the resulting functor \mathsf{Cat} \to \mathsf{Poset} preserves products. As a result, every commutative monoidal category gives a commutative monoidal poset: that is, a commutative monoid object in the category of Posets.

Composing these two functors, every Petri net gives a commutative monoidal poset. Elements are of this poset are markings of the Petri net, the partial order is “reachability”, and the commutative monoid structure is addition markings.

Third, any poset P gives another poset \widehat{P} whose elements are downsets of P: that is, subsets S \subseteq P such that

x \in S, y \le x \; \implies \; y \in S

The partial order on downsets is inclusion. This new poset \widehat{P} is ‘better’ than P because it’s cocomplete. That is, any union of downsets is again a downset. Moreover, \widehat{P} contains P as a sub-poset. The reason is that each x \in P gives a downset

\downarrow x = \{y \in P : \; y \le x \}

and clearly

x \le y \; \iff \;  \downarrow x \subseteq \downarrow y

Composing this third functor with the previous two, every Petri net gives a commutative monoid object in the category of cocomplete posets. But this is just a commutative quantale!

What is this commutative quantale like? Its elements are downsets of markings of our Petri net: sets of markings such that if x is in the set and x is reachable from y then y is also in the set.

It’s good to contemplate this a bit more. A marking can be seen as a ‘resource’. For example, if our Petri net has a place in it called sandwich there is a marking 2sandwich, which means you have two sandwiches. Downsets of markings are sets of markings such that if x is in the set and x is reachable from y then y is also in the set! An example of a downset would be “a sandwich, or anything that can give you a sandwich”. Another is “two sandwiches, or anything that can give you two sandwiches”.

The tensor product \otimes comes from addition of markings, extended in the obvious way to downsets of markings. For example, “a sandwich, or anything that can give you a sandwich” tensored with “a sandwich, or anything that can give you a sandwich” equals “two sandwiches, or anything that can give you two sandwiches”.

On the other hand, the cartesian product \& is the logical ‘and’:
if you have “a sandwich, or anything that can give you a sandwich” and you have “a sandwich, or anything that can give you a sandwich”, then you just have “a sandwich, or anything that can give you a sandwich”.

So that’s the basic idea.


Climate Technology Primer (Part 1)

5 October, 2019

Here’s the first of a series of blog articles on how technology can help address climate change:

• Adam Marblestone, Climate technology primer (1/3): basics.

Adam Marblestone is a research scientist at Google DeepMind studying connections between neuroscience and artificial intelligence. Previously, he was Chief Strategy Officer of the brain-computer interface company Kernel, and a research scientist in Ed Boyden’s Synthetic Neurobiology Group at MIT working to develop new technologies for brain circuit mapping. He also helped to start companies like BioBright, and advised foundations such as the Open Philanthropy Project.

Now, like many of us, he’s thinking about climate change, and what to do about it. He writes:

In this first of three posts, I attempt an outsider’s summary of the basic physics/chemistry/biology of the climate system, focused on back of the envelope calculations where possible. At the end, I comment a bit about technological approaches for emissions reductions. Future posts will include a review of the science behind negative emissions technologies, as well as the science (with plenty of caveats, don’t worry) behind more controversial potential solar radiation management approaches. This first post should be very basic for anyone “in the know” about energy, but I wanted to cover the basics before jumping into carbon sequestration technologies.

Check it out! I like the focus on “back of the envelope” calculations because they serve as useful sanity checks for more complicated models… and also provide a useful vaccination against the common denialist argument “all the predictions rely on complicated computer models that could be completely wrong, so why should I believe them?” It’s a sad fact that one of the things we need to do is make sure most technically literate people have a basic understanding of climate science, to help provide ‘herd immunity’ to everyone else.

The ultimate goal here, though, is to think about “what can technology do about climate change?”