Coxeter and Dynkin Diagrams

20 October, 2022

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or ‘ADE’ Dynkin diagrams also classify finite subgroups of SU(2) and quivers with finitely many indecomposable representations.

I’m talking about Coxeter and Dynkin diagrams now in my This Week’s Finds seminars. I started in Lecture 4, which you can see on video already, and the next lecture is today.

You can read my lecture notes:

Coxeter and Dynkin diagrams.

Just as a reminder: my talks are on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building at the University of Edinburgh. The first was on September 22nd, and the last on December 1st.

To attend on Zoom, go here:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

You can see videos of my talks here.

Also, you can discuss them on the Category Theory Community Server if you go here.


This Week’s Finds – Lecture 3

8 October, 2022

 

Young diagrams are combinatorial structures that show up in a myriad of applications. Here we explain how to classify irreducible representations of classical groups, and especially the “full linear monoid” consisting of all n × n complex matrices, using Young diagrams.

For the full story on representations of the classical groups, read my paper “Young diagrams and classical groups” here:

http://math.ucr.edu/home/baez/twf/

To attend my talks on Zoom go here.


The Kuramoto–Sivashinsky Equation (Part 8)

5 October, 2022

Our paper on the Kuramoto–Sivashinsky equation is out!

• John Baez, Steve Huntsman and Cheyne Weis, The Kuramoto–Sivashinsky equation, Notices Amer. Math. Soc. 69 (2022), 1581–1583.

This equation displays chaos and an arrow of time—and we state a precise conjecture about it, which may be very hard to prove, but is easy to study numerically. Steve Huntsman has written MATLAB code to help you do this, and I can make that available if you’re interested.

So what’s the idea? The article is short so I’ll just include it here!

The Kuramoto–Sivashinsky equation

u_t = -u_{xx} - u_{xxxx} - u_x u

applies to a real-valued function u of time t \in \mathbb{R} and space x \in \mathbb{R}. This equation was introduced as a simple 1-dimensional model of instabilities in flames, but it turned out to mathematically fascinating in its own right. One reason is that the Kuramoto–Sivashinsky equation is a simple model of Galilean-invariant chaos with an arrow of time.

We say this equation is ‘Galilean invariant’ because the Galiei group, the usual group of symmetries in Newtonian mechanics, acts on the set of its solutions. When space is 1-dimensional, this group is generated by translations in t and x, reflections in x, and Galilei boosts, which are transformations to moving coordinate systems:

(t,x) \mapsto (t,x-tv)

Translations act in the obvious way. Spatial reflections act as follows: if u(t,x) is a solution, so is -u(t,-x). Galilei boosts act in a more subtle way: if u(t,x) is a solution, so is u(t,x-tv) + v.

Figure 1 — A solution u(t,x) of the Kuramoto–Sivashinsky equation. The variable x ranges over the interval [0,32\pi] with its endpoints identified. Initial data are independent identically distributed random variables, one at each grid point, uniformly distributed in [-1,1].

We say the Kuramoto–Sivashinsky equation is ‘chaotic’ because the distance between nearby solutions, defined in a suitable way, can grow exponentially, making the long-term behavior of a solution hard to predict in detail. And finally, we say this equation has an ‘arrow of time’ because time reversal

(t,x) \mapsto (-t,x)

is not a symmetry of this equation. Indeed, in Figure 1 we see that starting from random initial conditions, manifestly time-asymmetric patterns emerge. As we move forward in time, it looks as if stripes are born and merge, but never die or split. Attempting to make this precise leads to an interesting conjecture, but first we need some background.

It is common to study solutions of the Kuramoto–Sivashinsky equations that are spatially periodic, so that u(t,x) = u(t,x+L) for some L. We can then treat space as a circle, the interval [0,L] with its endpoints identified. For these spatially periodic solutions, the integral \int_0^L u(t,x) \, dx does not change with time. Applying a Galilean transform adds a constant to this integral. In what follows we restrict attention to solutions where this integral is zero. These are roughly the solutions where the stripes are at rest, on average.

We can learn a surprising amount about these solutions by looking at the linearized equation

u_t = -u_{xx} - u_{xxxx}

We can solve this using a Fourier series

\displaystyle{  u(t,x) = \sum_{0 \ne n \in \mathbb{Z}} \hat{u}_n(t) \, e^{ik_n x} }

where the frequency of the nth mode is k_n = 2\pi n/L. We obtain

\hat{u}_n(t) = \exp\left((k_n^2 - k_n^4) t\right) \, \hat{u}_n(0)

Thus the nth mode grows exponentially with time if and only if k_n^2 - k_n^4 > 0, which happens when 0 < |n| < 2 \pi L. So, as we increase L, more and more modes grow exponentially. These appear to be the cause of chaos even in the nonlinear equation. Indeed, all solutions of the Kuramoto–Sivashinsky equation approach an attractive fixed point if L is small enough, but as we increase L we see increasingly complicated behavior, and a ‘transition to chaos via period doubling’, which has been analyzed in great detail. Interestingly, the nonlinear term stabilizes the exponentially growing modes: in the language of physics, it tends to transfer power from these modes to high-frequency modes, which decay exponentially.

Proving this last fact is not easy. However, in 1992, Collet, Eckmann, Epstein and Stubbe did this in the process of showing that for any initial data in a Hilbert space called \dot{L}^2, which consists of functions u \colon [0,L] \to \mathbb{R} such that

\int_0^L |u(x)|^2 \, dx < \infty

and

\int_0^L u(x) \, dx = 0 ,

the Kuramoto–Sivashinsky equation has a unique solution for t \ge 0, in a suitable sense, and that the norm of this solution eventually becomes less than some constant times L^{8/5}, These authors also showed any such solution eventually becomes infinitely differentiable, even analytic.

Shortly after this, Temam and Wang went further. They showed that all solutions of the Kuramoto–Sivashinsky equation with initial data in \dot{L}^2 approach a finite-dimensional submanifold of \dot{L}^2 as t \to +\infty. They also showed the dimension of this manifold is bounded by a constant times (\ln L)^{0.2} L^{1.64}. This manifold, called the inertial manifold, describes the ‘eventual behaviors’ of solutions.

Understanding the eventual behavior of solutions of the Kuramoto–Sivashinsky equation remains a huge challenge. What are the ‘stripes’ in these solutions? Can we define them in such a way that stripes are born and merge but never split or disappear, at least after a solution has had time to get close enough to the inertial manifold?

It is important to note that the visually evident stripes in Figure 1 are not regions where u exceeds some constant, nor regions where it is less than some constant. Instead, as x increases and (t,x) passes through a stripe, u(t,x) first becomes positive and then negative. Thus, in the middle of a stripe the derivative u_x(t,x) takes a negative value. One obvious thing to try, then, is to define a stripe to be a region where u_x(t,x) < c for some suitably chosen negative constant c. Unfortunately the boundaries of these regions, where u_x(t,x) = c, tend to be very rough. Thus, this definition gives many small evanescent ‘stripes’, so the conjecture that eventually stripes are born and merge but never die or split would be false with such a definition.

One way to solve this problem is to smooth u_x by convolving it with a normalized Gaussian function of x. A bit of experimentation suggests using a normalized Gaussian of standard deviation 2. Thus, let v equal u_x convolved with this Gaussian, and define a ‘stripe’ to be a region where v < 0. For the solution in Figure 1, these stripes are indicated in Figure 2. One stripe splits around t = 10, but after the solution nears the inertial manifold, stripes never die or split—at least in this example. We conjecture that this is true generically: that is, for all solutions in some open dense subset of the inertial manifold. Proving this seems quite challenging, but it might be a step toward rigorously relating the Kuramoto–Sivashinsky equation to a model where stochastically moving particles can appear or merge, but never disappear or split.

Figure 2 — The same solution of the Kuramoto–Sivashinsky equation, with stripes indicated.

Numerical calculations also indicate that generically, solutions eventually have stripes with an average density that approaches about 0.1 as L \to +\infty. This is close to the inverse of the wavelength of the fastest-growing mode of the linearized equation, (2^{3/2} \pi)^{-1} \approx 0.1125. But we see no clear reason to think these numbers should be exactly equal. Even proving the existence of a limiting stripe density is an open problem! Thus, the Kuramoto–Sivashinsky equation continues to pose many mathematical challenges.

References

[1] P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, A global attracting set for the Kuramoto–Sivashinsky equation Commun. Math. Phys. 152 (1993), 203–214.

[2] P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto–Sivashinsky equation, Physica D 67 (1993), 321–326.

[3] R. A. Edson, J. E. Bunder, T. W. Mattner and A. J. Roberts, Lyapunov exponents of the Kuramoto–Sivashinsky PDE, The ANZIAM Journal 61 (2019), 270–285.

[4] D. T. Papageorgiou and Y. S. Smyrlis, The route to chaos for the Kuramoto–Sivashinsky equation, Theor. Comput. Fluid Dyn. 3 (1991), 15–42.

[5] M. Rost and J. Krug, A particle model for the Kuramoto–Sivashinsky equation, Physica D 88 (1995), 1–13.

[6] R. Temam and X. M. Wang, Estimates on the lowest dimension of inertial manifolds for the Kuramoto–Sivashinsky equation in the general case, Differential and Integral Equations 7 (1994), 1095–1108.

[7] Encyclopedia of Mathematics, Kuramoto–Sivashinsky equation.


This Week’s Finds – Lecture 2

30 September, 2022

Young diagrams are combinatorial structures that show up in a myriad of applications. Here we explain how to classify irreducible representations of the symmetric groups Sₙ using Young diagrams. Then we introduce some ‘classical groups’ – famous groups whose irreducible representations can also be classified using Young diagrams.

For more details, read my paper “Young diagrams and classical groups” here:

http://math.ucr.edu/home/baez/twf/

To attend my talks on Zoom go here.

By the way: the video here has better resolution than the previous one. It starts out a bit too zoomed-in, but later it gets better.


This Week’s Finds – Lecture 1

27 September, 2022

Young diagrams are combinatorial structures that show up in a myriad of applications. In this talk I explain how Young diagrams classify conjugacy classes in the symmetric groups, introduce the representation theory of finite groups, and start to explain how Young diagrams classify irreducible representations of the symmetric groups.

For more details, read my paper “Young diagrams and classical groups” here:

http://math.ucr.edu/home/baez/twf/

To attend my talks on Zoom go here.


Young Diagrams and Classical Groups

16 September, 2022

Young diagrams can be used to classify an enormous number of things.   My first one or two This Week’s Finds seminars will be on Young diagrams and classical groups. Here are some lecture notes:

Young diagrams and classical groups.

I probably won’t cover all this material in the seminar. The most important part is the stuff up to and including the classification of irreducible representations of the “classical monoid” End(Cn). (People don’t talk about classical monoids, but they should.)

Just as a reminder: my talks will be on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building at the University of Edinburgh. The first will be on September 22nd, and the last on December 1st.

To attend on Zoom, go here:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

You can see videos of my talks here.

Also, you can discuss them on the Category Theory Community Server if you go here.


Seminar on “This Week’s Finds”

11 September, 2022

Here’s something new: I’ll be living in Edinburgh until January! I’m working with Tom Leinster at the University of Edinburgh, supported by a Leverhulme Fellowship.

One fun thing I’ll be doing is running seminars on some topics from my column This Week’s Finds. They’ll take place on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building, home of the Department of Mathematics. The first will be on September 22nd, and the last on December 1st.

We’re planning to

1) make the talks hybrid on Zoom so that people can participate online:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

2) make lecture notes available on my website.

3) record them and eventually make them publicly available on my YouTube channel.

4) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars: it’s here.

More details soon!

The theme for these seminars is representation theory, interpreted broadly. The topics are:

• Young diagrams
• Dynkin diagrams
• q-mathematics
• The three-strand braid group
• Clifford algebras and Bott periodicity
• The threefold and tenfold way
• Exceptional algebras

Seven topics are listed, but there will be 11 seminars, so it’s not a one-to-one correspondence: each topic is likely to take one or two weeks. Here are more detailed descriptions:

Young diagrams

Young diagrams are combinatorial structures that show up in a myriad of applications. Among other things, they classify conjugacy classes in the symmetric groups Sn, irreducible representations of Sn, irreducible representations of the groups SL(n) over any field of characteristic zero, and irreducible unitary representations of the groups SU(n).

Dynkin diagrams

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably Coxeter groups, lattices having such groups as symmetries, and simple Lie algebras. The simply laced Dynkin diagrams also classify the Platonic solids and quivers with finitely many indecomposable representations. This tour of Coxeter and Dynkin diagrams will focus on the connections between these structures.

q-mathematics

A surprisingly large portion of mathematics generalizes to something called q-mathematics, involving a parameter q. For example, there is a subject called q-calculus that reduces to ordinary calculus at q = 1. There are important applications of q-mathematics to the theory of quantum groups and also to algebraic geometry over Fq, the finite field with q elements. These seminars will give an overview of q-mathematics and its
applications.

The three-strand braid group

The three-strand braid group has striking connections to the trefoil knot, rational tangles, the modular group PSL(2, Z), and modular forms. This group is also the simplest of the Artin–Brieskorn groups, a class of groups which map surjectively to the Coxeter groups. The three-strand braid group will be used as the starting point for a tour of these topics.

Clifford algebras and Bott periodicity

The Clifford algebra Cln is the associative real algebra freely generated by n anticommuting elements that square to -1. Iwill explain their role in geometry and superstring theory, and the origin of Bott periodicity in topology in facts about Clifford algebras.

The threefold and tenfold way

Irreducible real group representations come in three kinds, a fact arising from the three associative normed real division algebras: the real numbers, complex numbers and quaternions. Dyson called this the threefold way. When we generalize to superalgebras this becomes part of a larger classification, the tenfold way. We will examine these topics and their applications to representation theory, geometry and physics.

Exceptional algebras

Besides the three associative normed division algebras over the real numbers, there is a fourth one that is nonassociative: the octonions. They arise naturally from the fact that Spin(8) has three irreducible 8-dimensional representations. We will explain the octonions and sketch how the exceptional Lie algebras and the exceptional Jordan algebra can be constructed using octonions.


Joint Mathematics Meetings 2023

24 August, 2022

This is the biggest annual meeting of mathematicians:

Joint Mathematical Meetings 2023, Wednesday January 4 – Saturday January 7, 2023, John B. Hynes Veterans Memorial Convention Center, Boston Marriott Hotel, and Boston Sheraton Hotel, Boston, Massachusetts.

As part of this huge meeting, the American Mathematical Society is having a special session on Applied Category Theory on Thursday January 5th.

I hear there will be talks by Eugenia Cheng and Olivia Caramello!

You can submit an abstract to give a talk. The deadline is Tuesday, September 13, 2022.

It should be lots of fun. There will also be tons of talks on other subjects.

However, there’s a registration fee which is pretty big unless you’re a student or, even better, a ‘nonmathematician guest’. (I assume you’re not allowed to give a talk if you’re a nonmathematician.)

The special session is called SS 96 and it comes in two parts: one from 8 am to noon, and the other from 1 pm to 5 pm. It’s being run by these participants of this summer’s Mathematical Research Community on applied category theory:

• Charlotte Aten, University of Denver
• Pablo S. Ocal, University of California, Los Angeles
• Layla H. M. Sorkatti, Southern Illinois University
• Abigail Hickok, University of California, Los Angeles

This Mathematical Research Community was run by Daniel Cicala, Simon Cho, Nina Otter, Valeria de Paiva and me, and I think we’re all coming to the special session. At least I am!


Symposium on Compositional Structures 9

9 July, 2022

The Symposium on Compositional Structures is a nice informal conference series that happens more than once a year. You can now submit talks for this one.

Ninth Symposium on Compositional Structures (SYCO 9), Como, Italy, 8-9 September 2022. Deadline to submit a talk: Monday 1 August 2022.

Apparently you can attend online but to give a talk you have to go there. Here are some details:

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. Previous SYCO events have been held in Birmingham, Strathclyde, Oxford, Chapman, Leicester and Tallinn.

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. Submissions is easy, with no formatting or page restrictions. The meeting does not have proceedings, so work can be submitted even if it has been submitted or published elsewhere. You could submit work-in-progress, or a recently completed paper, or even a PhD or Masters thesis.

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

• logical methods in computer science, including classical and quantum programming, type theory, concurrency, natural language processing and machine learning;
• graphical calculi, including string diagrams, Petri nets and reaction networks;
• languages and frameworks, including process algebras, proof nets, type theory and game semantics;
• abstract algebra and pure category theory, including monoidal category theory, higher category theory, operads, polygraphs, and relationships to homotopy theory;
• quantum algebra, including quantum computation and representation theory;
• tools and techniques, including rewriting, formal proofs and proof assistants, and game theory;
• industrial applications, including case studies and real-world problem descriptions.

Important dates

All deadlines are 23:59 Anywhere on Earth.

Submission deadline: Monday 1 August
Author notification: Monday 8 August 2022
Symposium dates: Thursday 8 and Friday 9 September 2022

Submission instructions

Submissions are by EasyChair, via the SYCO 9 submission page:

https://easychair.org/my/conference?conf=syco9

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

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. Deferred submissions can be re-submitted to any future SYCO meeting, where they will not need peer review, and where they will be prioritised for inclusion in the programme. Meetings will be held sufficiently frequently to avoid a backlog of deferred papers.

If you have a submission which was deferred from a previous SYCO meeting, it will not automatically be considered for SYCO 9; you still need to submit it again through EasyChair. When submitting, append the words “DEFERRED FROM SYCO X” to the title of your paper, replacing “X” with the appropriate meeting number. There is no need to attach any documents.

Programme committee

The PC chair is John van de Wetering, Radboud University. The Programme Committee will be announced soon.

Steering committee

Ross Duncan, University of Strathclyde
Chris Heunen, University of Edinburgh
Dominic Horsman, University of Oxford
Aleks Kissinger, University of Oxford
Samuel Mimram, École Polytechnique
Simona Paoli, University of Aberdeen
Mehrnoosh Sadrzadeh, University College London
Pawel Sobocinski, Tallinn University of Technology
Jamie Vicary, University of Cambridge


Compositional Modeling with Decorated Cospans

27 June, 2022

It’s finally here: software that uses category theory to let you build models of dynamical systems! We’re going to train epidemiologists to use this to model the spread of disease. My first talk on this will be on Wednesday June 29th. You’re invited!

Compositional modeling with decorated cospans, Graph Transformation Theory and Practice (GReTA) seminar, 19:00 UTC, Wednesday 29 June 2022.

You can attend live on Zoom if you click here. You can also watch it live on YouTube, or later recorded, here:

Abstract. Decorated cospans are a general framework for composing open networks and mapping them to dynamical systems. We explain this framework and illustrate it with the example of stock and flow diagrams. These diagrams are widely used in epidemiology to model the dynamics of populations. Although tools already exist for building these diagrams and simulating the systems they describe, we have created a new software package called StockFlow which uses decorated cospans to overcome some limitations of existing software. Our approach cleanly separates the syntax of stock and flow diagrams from the semantics they can be assigned. We have implemented a semantics where stock and flow diagrams are mapped to ordinary differential equations, although others are possible. We illustrate this with code in StockFlow that implements a simplified version of a COVID-19 model used in Canada. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.

My talk is at a seminar on graph rewriting, so I’ll explain how the math applies to graphs before turning to ‘stock-flow diagrams’, like this one here:

Stock-flow diagrams are used to create models in epidemiology. There’s a functor mapping them to dynamical systems.

But the key idea in our work is ‘compositional modeling’. This lets different teams build different models and then later assemble them into a larger model. The most popular existing software for stock-flow diagrams does not allow this. Category theory to the rescue!

This work would be impossible without the right team! Brendan Fong developed decorated cospans and then started the Topos Institute. My coauthors Evan Patterson and Sophie Libkind work there, and they know how to program using category theory.

Evan started a seminar on epidemiological modeling – and my old grad school pal Nate Osgood showed up, along with his grad student Xiaoyan Li! Nate is a computer scientist who now runs the main COVID model for the government of Canada.

So, all together we have serious expertise in category theory, computer science, and epidemiology. Any two parts alone would not be enough for this project.

And I’m not even listing all the people whose work was essential. For example, Kenny Courser and Christina Vasilakopoulou helped modernize the theory of decorated cospans in a way we need here. James Fairbanks, Evan and others designed AlgebraicJulia, the software environment that our package StockFlow relies on. And so on!

Moral: to apply category theory to real-world problems, you need a team.

And we’re just getting started!