We’re going to have a seminar on applied category theory here at U. C. Riverside! My students have been thinking hard about category theory for a few years, but they’ve decided it’s time to get deeper into applications. Christian Williams, in particular, seems to have caught my zeal for trying to develop new math to help save the planet.

We’ll try to videotape the talks to make it easier for you to follow along. I’ll also start discussions here and/or on the Azimuth Forum. It’ll work best if you read the papers we’re talking about and then join these discussions. Ask questions, and answer any questions you can!

Here’s how the schedule of talks is shaping up so far. I’ll add more information as it becomes available, either here or on a webpage devoted to the task.

I’ll give an updated synthesized version of these earlier talks of mine, so check out these slides and the links:

• The mathematics of planet Earth.

Lorand is visiting U. C. Riverside to work with me on applications of symplectic geometry to chemistry. Here is the abstract of his talk:

In this talk we will look at various examples of classification problems in symplectic linear algebra: conjugacy classes in the symplectic group and its Lie algebra, linear lagrangian relations up to conjugation, tuples of (co)isotropic subspaces. I will explain how many such problems can be encoded using the theory of symplectic poset representations, and will discuss some general results of this theory. Finally, I will recast this discussion from a broader category-theoretic perspective.

Vasilakopoulou, a visiting professor here, previously worked with David Spivak. So, we really want to figure out how two frameworks for dealing with networks relate: Brendan Fong’s ‘decorated cospans’, and Spivak’s ‘monoidal category of wiring diagrams’. Since Fong is now working with Spivak they’ve probably figured it out already! But anyway, Vasilakopoulou will give a talk on systems as algebras for the wiring diagram monoidal category. It will be based on this paper:

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

but she will focus more on the algebraic description (and conditions for deterministic/total systems) rather than the sheaf theoretic aspect of the input types. This work builds on earlier papers such as these:

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

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

Cicala will discuss a topic from this paper:

• Mason A. Porter and James P. Gleeson, Dynamical systems on networks: a tutorial.

His leading choice is a model for social contagion (e.g. opinions) which is discussed in more detail here:

• Duncan J. Watts, A simple model of global cascades on random networks.

]]>• Second Symposium on Compositional Structures (SYCO2), 17-18 December 2018, University of Strathclyde, Glasgow.

http://events.cs.bham.ac.uk/syco/2/accepted.html

Please register asap so that catering can be arranged. Late registrants

might go hungry.

• Corina Cirstea, University of Southampton – Quantitative Coalgebras for

Optimal Synthesis

• Martha Lewis, University of Amsterdam – Compositionality in Semantic Spaces

The Symposium on Compositional Structures (SYCO) is an interdisciplinary series of meetings aiming to support the growing community of researchers interested in the phenomenon of compositionality, from both applied and abstract perspectives, and in particular where category theory serves as a unifying common language. The first SYCO was held at the School of Computer Science, University of Birmingham, 20-21 September, 2018, attracting 70 participants.

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

Submission is easy, with no format requirements or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere.

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

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

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

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

• abstract algebra and pure category theory, including monoidal category

theory, higher category theory, operads, polygraphs, and relationships to homotopy theory;

• quantum algebra, including quantum computation and representation theory;

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

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

descriptions.

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

SYCO will be a regular fixture in the academic calendar, running regularly throughout the year, and becoming over time a recognized venue for presentation and discussion of results in an informal and friendly atmosphere. To help create this community, and to avoid the need to make difficult choices between strong submissions, in the event that more good-quality submissions are received than can be accommodated in the timetable, the programme committee may choose to *defer* some submissions to a future meeting, rather than reject them. This would be done based largely on submission order, giving an incentive for early submission, but would also take into account other requirements, such as ensuring a broad scientific programme. Deferred submissions would be accepted for presentation at any future SYCO meeting without the need for peer review. This will allow us to ensure that speakers have enough time to present their ideas, without creating an unnecessarily competitive reviewing process. Meetings would be held sufficiently frequently to avoid a backlog of deferred papers.

Ross Duncan, University of Strathclyde

Fabrizio Romano Genovese, Statebox and University of Oxford

Jules Hedges, University of Oxford

Chris Heunen, University of Edinburgh

Dominic Horsman, University of Grenoble

Aleks Kissinger, Radboud University Nijmegen

Eliana Lorch, University of Oxford

Guy McCusker, University of Bath

Samuel Mimram, École Polytechnique

Koko Muroya, RIMS, Kyoto University & University of Birmingham

Paulo Oliva, Queen Mary

Nina Otter, UCLA

Simona Paoli, University of Leicester

Robin Piedeleu, University of Oxford and UCL

Julian Rathke, University of Southampton

Bernhard Reus, Univeristy of Sussex

David Reutter, University of Oxford

Mehrnoosh Sadrzadeh, Queen Mary

Pawel Sobocinski, University of Southampton (chair)

Jamie Vicary, University of Birmingham and University of Oxford (co-chair)

The first one is this. As beginners, we start by thinking of geometric quantization as a procedure for taking a symplectic manifold and constructing a Hilbert space: that is, taking a space of *classical* states and contructing the corresponding space of *quantum* states. We soon learn that this procedure requires additional data as its input: a symplectic manifold is not enough. We learn that it works much better to start with a Kähler manifold equipped with a holomorphic hermitian line bundle with a connection whose curvature is the imaginary part of the Kähler structure. Then the space of holomorphic sections of that line bundle gives the Hilbert space we seek.

That’s quite a mouthful—but it makes for such a nice story that I’d love to write a bunch of blog articles explaining it with lots of examples. Unfortunately I don’t have time, so try these:

• Matthias Blau, Symplectic geometry and geometric quantization.

• A. Echeverria-Enriquez, M.C. Munoz-Lecanda, N. Roman-Roy, C. Victoria-Monge, Mathematical foundations of geometric quantization.

But there’s a flip side to this story which indicates that something big and mysterious is going on. Geometric quantization is not just a procedure for converting a space of classical states into a space of quantum states. It also reveals that *a space of quantum states can be seen as a space of classical states!*

To reach this realization, we must admit that quantum states are not really vectors in a Hilbert space ; from a certain point of view they are really *1-dimensonal subspaces* of a Hilbert space, so the set of quantum states I’m talking about is the projective space But this projective space, at least when it’s finite-dimensional, turns out to be the simplest example of that complicated thing I mentioned: a Kähler manifold equipped with a holomorphic hermitian line bundle whose curvature is the imaginary part of the Kähler structure!

So a space of quantum states is an *example* of a space of classical states—equipped with precisely all the complicated extra structure that lets us geometrically quantize it!

At this point, if you don’t already know the answer, you should be asking: *and what do we get when we geometrically quantize it?*

The answer is exciting only in that it’s surprisingly dull: when we geometrically quantize we get back the Hilbert space

You may have heard of ‘second quantization’, where we take a quantum system, treat it as classical, and quantize it again. In the usual story of second quantization, the new quantum system we get is more complicated than the original one… and we can repeat this procedure again and again, and keep getting more interesting things:

• John Baez, Nth quantization.

The story I’m telling now is different. I’m saying that when we take a quantum system with Hilbert space we can think of it as a classical system whose symplectic manifold of states is but then we can geometrically quantize this and get back.

The two stories are not in contradiction, because they rely on two different notions of what it means to ‘think of a quantum system as classical’. In today’s story that means getting a symplectic manifold from a Hilbert space In the other story we use the fact that *itself* is a symplectic manifold!

I should explain the relation of these two stories, but that would be a big digression from today’s intended blog article: indeed I’m already regretting having drifted off course. I only brought up this other story to heighten the mystery I’m talking about now: namely, that when we geometrically quantize the space we get back.

The math is not mysterious here; it’s the *physical meaning* of the math that’s mysterious. The math seems to be telling us that contrary to what they say in school, *quantum systems are special classical systems*, with the special property that when you quantize them nothing new happens!

This idea is not mine; it goes back at least to Kibble, the guy who with Higgs invented the method whereby the Higgs boson does its work:

• Tom W. B. Kibble, Geometrization of quantum mechanics, *Comm. Math. Phys.* **65** (1979), 189–201.

This led to a slow, quiet line of research that continues to this day. I find this particular paper especially clear and helpful:

• Abhay Ashtekar, Troy A. Schilling, Geometrical formulation of quantum mechanics, in *On Einstein’s Path*, Springer, Berlin, 1999, pp. 23–65.

so if you’re wondering what the hell I’m talking about, this is probably the best place to start. To whet your appetite, here’s the abstract:

Abstract.States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a Kähler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schrödinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features—such as uncertainties and state vector reductions—which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric—a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.

Personally I’m not interested in the generalizations of quantum mechanics: I’m more interested in what this circle of ideas *means* for quantum mechanics.

One rather cynical thought is this: when we start our studies with geometric quantization, we naively hope to extract a space of quantum states from a space of classical states, e.g. a symplectic manifold. But we then discover that to do this in a systematic way, we need to equip our symplectic manifold with lots of bells and whistles. Should it really be a surprise that when we’re done, the bells and whistles we need are exactly what a space of quantum states *has*?

I think this indeed dissolves some of the mystery. It’s a bit like the parable of ‘stone soup’: you can make a tasty soup out of just a stone… *if* you season it with some vegetables, some herbs, some salt and such.

However, perhaps because by nature I’m an optimist, I also think there are interesting things to be learned from the tight relation between quantum and classical mechanics that appears in geometric quantization. And I hope to talk more about those in future articles.

]]>The danger then is that we rush headlong into something untested that we’ll regret.

For a while I’ve been advocating research in geoengineering, to prevent a big mistake like this. Those who consider it “unthinkable” often object to such research, but I think preventing research is not a good long-term policy. I think it actually makes it more likely that at some point, when enough people become really desperate about climate change, we will do something rash without enough information about the possible effects.

Anyway, one can argue about this all day: I can see the arguments for both sides. But here is some news: scientists will soon study how calcium carbonate disperses when you dump a little into the atmosphere:

• First sun-dimming experiment will test a way to cool Earth, *Nature*, 27 November 2018.

It’s a good article—read it! Here’s the key idea:

If all goes as planned, the Harvard team will be the first in the world to move solar geoengineering out of the lab and into the stratosphere, with a project called the Stratospheric Controlled Perturbation Experiment (SCoPEx). The first phase — a US$3-million test involving two flights of a steerable balloon 20 kilometres above the southwest United States — could launch as early as the first half of 2019. Once in place, the experiment would release small plumes of calcium carbonate, each of around 100 grams, roughly equivalent to the amount found in an average bottle of off-the-shelf antacid. The balloon would then turn around to observe how the particles disperse.

The test itself is extremely modest. Dai, whose doctoral work over the past four years has involved building a tabletop device to simulate and measure chemical reactions in the stratosphere in advance of the experiment, does not stress about concerns over such research. “I’m studying a chemical substance,” she says. “It’s not like it’s a nuclear bomb.”

Nevertheless, the experiment will be the first to fly under the banner of solar geoengineering. And so it is under intense scrutiny, including from some environmental groups, who say such efforts are a dangerous distraction from addressing the only permanent solution to climate change: reducing greenhouse-gas emissions. The scientific outcome of SCoPEx doesn’t really matter, says Jim Thomas, co-executive director of the ETC Group, an environmental advocacy organization in Val-David, near Montreal, Canada, that opposes geoengineering: “This is as much an experiment in changing social norms and crossing a line as it is a science experiment.”

Aware of this attention, the team is moving slowly and is working to set up clear oversight for the experiment, in the form of an external advisory committee to review the project. Some say that such a framework, which could pave the way for future experiments, is even more important than the results of this one test. “SCoPEx is the first out of the gate, and it is triggering an important conversation about what independent guidance, advice and oversight should look like,” says Peter Frumhoff, chief climate scientist at the Union of Concerned Scientists in Cambridge, Massachusetts, and a member of an independent panel that has been charged with selecting the head of the advisory committee. “Getting it done right is far more important than getting it done quickly.”

For more on SCoPEx, including a FAQ, go here:

• Stratospheric Controlled Perturbation Experiment (SCoPEx), Keutsch Group, Harvard.

The hand-written notes here are by Christian Williams. They are probably best seen as a reminder to myself as to what I’d like to include in a short book someday.

• Lecture 1: What is pure mathematics all about? The importance of free structures.

• Lecture 2: The natural numbers as a free structure. Adjoint functors.

• Lecture 3: Adjoint functors in terms of unit and counit.

• Lecture 4: 2-Categories. Adjunctions.

• Lecture 5: 2-Categories and string diagrams. Composing adjunctions.

• Lecture 6: The ‘main spine’ of mathematics. Getting a monad from an adjunction.

• Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

• Lecture 8: The walking monad, the augmented simplex category and the simplex category.

• Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

• Lecture 10: The Dold-Thom theorem: the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. The homology of a chain complex.

• Lecture 7: Definition of a monad. Getting a monad from an adjunction. The augmented simplex category.

• Lecture 8: The walking monad, the

augmented simplex category and the simplex category.

• Lecture 9: Simplicial abelian groups from simplicial sets. Chain complexes from simplicial abelian groups.

• Lecture 10: Chain complexes from simplicial abelian groups. The homology of a chain complex.

• Lecture 12: The bar construction: getting a simplicial objects from an adjunction. The bar construction for G-sets, previewed.

• Lecture 13: The adjunction between G-sets and sets.

• Lecture 14: The bar construction for groups.

• Lecture 15: The simplicial set obtained by applying the bar construction to the one-point -set, its geometric realization and the free simplicial abelian group

• Lecture 16: The chain complex coming from the simplicial abelian group its homology, and the definition of group cohomology with coefficients in a -module.

• Lecture 17: Extensions of groups. The Jordan-Hölder theorem. How an extension of a group by an abelian group gives an action of on and a 2-cocycle

• Lecture 18: Classifying abelian extensions of groups. Direct products, semidirect products, central extensions and general abelian extensions. The groups of order 8 as abelian extensions.

• Lecture 19: Group cohomology. The chain complex for the cohomology of with coefficients in , starting from the bar construction, and leading to the 2-cocycles used in classifying abelian extensions. The classification of extensions of by in terms of

• Lecture 20: Examples of group cohomology: nilpotent groups and the fracture theorem. Higher-dimensional algebra and homotopification: the nerve of a category and the nerve of a topological space. as the nerve of the translation groupoid as the walking space with fundamental group

• Lecture 21: Homotopification and higher algebra. Internalizing concepts in categories with finite products. Pushing forward internalized structures using functors that preserve finite products. Why the ‘discrete category on a set’ functor the ‘nerve of a category’ functor and the ‘geometric realization of a simplicial set’ functor preserve products.

• Lecture 22: Monoidal categories. Strict monoidal categories as monoids in or one-object 2-categories. The periodic table of strict -categories. General ‘weak’ monoidal categories.

• Lecture 23: 2-Groups. The periodic table of weak -categories. The stabilization hypothesis. The homotopy hypothesis. Classifying 2-groups with as the group of objects and as the abelian group of automorphisms of the unit object in terms of The Eckmann–Hilton argument.

]]>First, remember the story. A subset of the plane has **diameter 1** if the distance between any two points in this set is ≤ 1. A **universal covering** is a convex subset of the plane that can cover a translated, reflected and/or rotated version of every subset of the plane with diameter 1. In 1914, the famous mathematician Henri Lebesgue sent a letter to a fellow named Pál, challenging him to find the universal covering with the least area.

Pál worked on this problem, and 6 years later he published a paper on it. He found a very nice universal covering: a regular hexagon in which one can inscribe a circle of diameter 1. This has area

0.86602540…

But he also found a universal covering with less area, by removing two triangles from this hexagon—for example, the triangles C_{1}C_{2}C_{3} and E_{1}E_{2}E_{3} here:

The resulting universal covering has area

0.84529946…

In 1936, Sprague went on to prove that more area could be removed from another corner of Pál’s original hexagon, giving a universal covering of area

0.8441377708435…

In 1992, Hansen took these reductions even further by removing two more pieces from Pál’s hexagon. Each piece is a thin sliver bounded by two straight lines and an arc. The first piece is tiny. The second is downright microscopic!

Hansen claimed the areas of these regions were 4 · 10^{-11} and 6 · 10^{-18}. This turned out to be wrong. The actual areas are 3.7507 · 10^{-11} and 8.4460 · 10^{-21}. The resulting universal covering had an area of

0.844137708416…

This tiny improvement over Sprague’s work led Klee and Wagon to write:

it does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.

However, in 2015 Philip Gibbs found a way to remove about a million times more area than Hansen’s larger region: a whopping 2.233 · 10^{-5}. This gave a universal covering with area

0.844115376859…

Karine Bagdasaryan and I helped Gibbs write up a rigorous proof of this result, and we published it here:

• John Baez, Karine Bagdasaryan and Philip Gibbs, The Lebesgue universal covering problem, *Journal of Computational Geometry* **6** (2015), 288–299.

Greg Egan played an instrumental role as well, catching various computational errors.

At the time Philip was sure he could remove even more area, at the expense of a more complicated proof. Since the proof was already quite complicated, we decided to stick with what we had.

But this week I met Philip at The philosophy and physics of Noether’s theorems, a wonderful workshop in London which deserves a full blog article of its own. It turns out that he has gone further: he claims to have found a vastly better universal covering, with area

0.8440935944…

This is an improvement of 2.178245 × 10^{-5} over our earlier work—roughly equal to our improvement over Hansen.

You can read his argument here:

• Philip Gibbs, An upper bound for Lebesgue’s universal covering problem, 22 January 2018.

I say ‘claims’ not because I doubt his result—he’s clearly a master at this kind of mathematics!—but because I haven’t checked it and it’s easy to make mistakes, for example mistakes in computing the areas of the shapes removed.

It seems we are closing in on the final result; however, Philip Gibbs believes there is still room for improvement, so I expect it will take at least a decade or two to solve this problem… unless, of course, some mathematicians start working on it full-time, which could speed things up considerably.

]]>• Riverside Mathematics Workshop for Excellence and Diversity, Friday 19 October – Saturday 20 October, 2018. Organized by John Baez, Carl Mautner, José González and Chen Weitao.

This is the first of an annual series of workshops to showcase and celebrate excellence in research by women and other under-represented groups for the purpose of fostering and encouraging growth in the U.C. Riverside mathematical community.

After tea at 3:30 p.m. on Friday there will be two plenary talks, lasting until 5:00. Catherine Searle will talk on “Symmetries of spaces with lower curvature bounds”, and Edray Goins will give a talk called “Clocks, parking garages, and the solvability of the quintic: a friendly introduction to monodromy”. There will then be a banquet in the Alumni Center 6:30 – 8:30 p.m.

On Saturday there will be coffee and a poster session at 8:30 a.m., and then two parallel sessions on pure and applied mathematics, with talks at 9:30, 10:30, 11:30, 1:00 and 2:00. Check out the abstracts here!

(I’m especially interested in Christina Vasilakopoulou’s talk on Frobenius and Hopf monoids in enriched categories, but she’s my postdoc so I’m biased.)

]]>animation by Marius Buliga

I’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.

More details will come later, but here’s the basic idea. If you’re a grad student interested in this subject, you should apply for the ‘school’. Not yet—we’ll let you know when.

]]>Dear all,

As part of a new growing community in Applied Category Theory, now with a dedicated journal

Compositionality, a traveling workshop series SYCO, a forthcoming Cambridge U. Press book series Reasoning with Categories, and several one-off events including at NIST, we launch an annual conference+school series named Applied Category Theory, the coming one being at Oxford, July 15-19 for the conference, and July 22-26 for the school. The dates are chosen such that CT 2019 (Edinburgh) and the ACT 2019 conference (Oxford) will be back-to-back, for those wishing to participate in both.There already was a successful invitation-only pilot, ACT 2018, last year at the Lorentz Centre in Leiden, also in the format of school+workshop.

For the conference, for those who are familiar with the successful QPL conference series, we will follow a very similar format for the ACT conference. This means that we will accept both new papers which then will be published in a proceedings volume (most likely a

Compositionalityspecial Proceedings issue), as well as shorter abstracts of papers published elsewhere. There will be a thorough selection process, as typical in computer science conferences. The idea is that all the best work in applied category theory will be presented at the conference, and that acceptance is something that means something, just like in CS conferences. This is particularly important for young people as it will help them with their careers.Expect a call for submissions soon, and start preparing your papers now!

The school in ACT 2018 was unique in that small groups of students worked closely with an experienced researcher (these were John Baez, Aleks Kissinger, Martha Lewis and Pawel Sobociński), and each group ended up producing a paper. We will continue with this format or a closely related one, with Jules Hedges and Daniel Cicala as organisers this year. As there were 80 applications last year for 16 slots, we may want to try to find a way to involve more students.

We are fortunate to have a number of private sector companies closely associated in some way or another, who will also participate, with Cambridge Quantum Computing Inc. and StateBox having already made major financial/logistic contributions.

On behalf of the ACT Steering Committee,

John Baez, Bob Coecke, David Spivak, Christina Vasilakopoulou

is a good approximation to the number of primes less than or equal to Numerical evidence suggests that is always greater than For example,

and

But in 1914, Littlewood heroically showed that in fact, changes sign infinitely many times!

This raised the question: when does first exceed ? In 1933, Littlewood’s student Skewes showed, assuming the Riemann hypothesis, that it must do so for some less than or equal to

Later, in 1955, Skewes showed *without* the Riemann hypothesis that must exceed for some smaller than

By now this bound has been improved enormously. We now know the two functions cross somewhere near but we don’t know if this is the first crossing!

All this math is quite deep. Here is something less deep, but still fun.

You can show that

and so on.

It’s a nice pattern. But this pattern doesn’t go on forever! It lasts a very, very long time… but not forever.

More precisely, the identity

holds when

but not for all At some point it stops working and never works again. In fact, it definitely fails for all

The integrals here are a variant of the Borwein integrals:

where the pattern continues until

but then fails:

I never understood this until I read Greg Egan’s explanation, based on the work of Hanspeter Schmid. It’s all about convolution, and Fourier transforms:

Suppose we have a rectangular pulse, centred on the origin, with a height of 1/2 and a half-width of 1.

Now, suppose we keep taking moving averages of this function, again and again, with the average computed in a window of half-width 1/3, then 1/5, then 1/7, 1/9, and so on.

There are a couple of features of the original pulse that will persist completely unchanged for the first few stages of this process, but then they will be abruptly lost at some point.

The first feature is that F(0) = 1/2. In the original pulse, the point (0,1/2) lies on a plateau, a perfectly constant segment with a half-width of 1. The process of repeatedly taking the moving average will nibble away at this plateau, shrinking its half-width by the half-width of the averaging window. So, once the sum of the windows’ half-widths exceeds 1, at 1/3+1/5+1/7+…+1/15, F(0) will suddenly fall below 1/2, but up until that step it will remain untouched.

In the animation below, the plateau where F(x)=1/2 is marked in red.

The second feature is that F(–1)=F(1)=1/4. In the original pulse, we have a step at –1 and 1, but if we define F here as the average of the left-hand and right-hand limits we get 1/4, and once we apply the first moving average we simply have 1/4 as the function’s value.

In this case, F(–1)=F(1)=1/4 will continue to hold so long as the points (–1,1/4) and (1,1/4) are surrounded by regions where the function has a suitable symmetry: it is equal to an odd function, offset and translated from the origin to these centres. So long as that’s true for a region wider than the averaging window being applied, the average at the centre will be unchanged.

The initial half-width of each of these symmetrical slopes is 2 (stretching from the opposite end of the plateau and an equal distance away along the x-axis), and as with the plateau, this is nibbled away each time we take another moving average. And in this case, the feature persists until 1/3+1/5+1/7+…+1/113, which is when the sum first exceeds 2.

In the animation, the yellow arrows mark the extent of the symmetrical slopes.

OK, none of this is difficult to understand, but why should we care?

Because this is how Hanspeter Schmid explained the infamous Borwein integrals:

∫sin(t)/t dt = π/2

∫sin(t/3)/(t/3) × sin(t)/t dt = π/2

∫sin(t/5)/(t/5) × sin(t/3)/(t/3) × sin(t)/t dt = π/2…

∫sin(t/13)/(t/13) × … × sin(t/3)/(t/3) × sin(t)/t dt = π/2

But then the pattern is broken:

∫sin(t/15)/(t/15) × … × sin(t/3)/(t/3) × sin(t)/t dt < π/2

Here these integrals are from t=0 to t=∞. And Schmid came up with an even more persistent pattern of his own:

∫2 cos(t) sin(t)/t dt = π/2

∫2 cos(t) sin(t/3)/(t/3) × sin(t)/t dt = π/2

∫2 cos(t) sin(t/5)/(t/5) × sin(t/3)/(t/3) × sin(t)/t dt = π/2

…

∫2 cos(t) sin(t/111)/(t/111) × … × sin(t/3)/(t/3) × sin(t)/t dt = π/2But:

∫2 cos(t) sin(t/113)/(t/113) × … × sin(t/3)/(t/3) × sin(t)/t dt < π/2

The first set of integrals, due to Borwein, correspond to taking the Fourier transforms of our sequence of ever-smoother pulses and then evaluating F(0). The Fourier transform of the sinc function:

sinc(w t) = sin(w t)/(w t)

is proportional to a rectangular pulse of half-width w, and the Fourier transform of a product of sinc functions is the convolution of their transforms, which in the case of a rectangular pulse just amounts to taking a moving average.

Schmid’s integrals come from adding a clever twist: the extra factor of 2 cos(t) shifts the integral from the zero-frequency Fourier component to the sum of its components at angular frequencies –1 and 1, and hence the result depends on F(–1)+F(1)=1/2, which as we have seen persists for much longer than F(0)=1/2.

• Hanspeter Schmid, Two curious integrals and a graphic proof,

Elem. Math.69(2014) 11–17.

I asked Greg if we could generalize these results to give even longer sequences of identities that eventually fail, and he showed me how: you can just take the Borwein integrals and replace the numbers 1, 1/3, 1/5, 1/7, … by some sequence of positive numbers

The integral

will then equal as long as but not when it exceeds 1. You can see a full explanation on Wikipedia:

• Wikipedia, Borwein integral: general formula.

As an example, I chose the integral

which equals if and only if

Thus, the identity holds if

However,

so the identity holds if

or

or

On the other hand, the identity fails if

so it fails if

However,

so the identity fails if

or

or

With a little work one could sharpen these estimates considerably, though it would take more work to find the *exact* value of at which

first fails.

]]>• Tai-Danae Bradley, *What is Applied Category Theory?*

Abstract.This is a collection of introductory, expository notes on applied category theory, inspired by the 2018 Applied Category Theory Workshop, and in these notes we take a leisurely stroll through two themes (functorial semantics and compositionality), two constructions (monoidal categories and decorated cospans) and two examples (chemical reaction networks and natural language processing) within the field.

Check it out!

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