Let Me Think

My friend Joshua Meyers, formerly a math grad student at U. C. Riverside, is now trying to develop new scholarly institutions: alternatives to universities.

So, he’s started a project called Let Me Think. He’s gotten money from Jaan Tallinn and the Survival and Flourishing Fund to run a meeting where 20 people will stay in rural New York for two months and plan how to build these new institutions. The venue is a 93-acre site on top of a mountain, with its own lake.

It’ll happen this summer. In a while you can apply to join! Read more about it here:

• Scholarship Workshop 2023.

Here’s the schedule and goals:

Here are some of the reasons Joshua Meyers wants to build new scholarly institutions—institutions primarily for thinking, not for teaching:

A lot of people make these complaints, of course. What’s interesting is that he’s trying to do something new:

For more details on this, go here.

You can contact him at

contact@let-me-think.org

Applied Category Theory 2023

You can now submit a paper if you want to give a talk here:

• 6th Annual International Conference on Applied Category Theory (ACT2023), University of Maryland, July 31 — August 4, 2023

The Sixth International Conference on Applied Category Theory will take place at the University of Maryland from 31 July to 4 August 2023, preceded by the Adjoint School 2023 from 24 to 28 July. This event will be hybrid.

This conference follows previous events at Strathclyde (UK), Cambridge (UK), Cambridge (MA), Oxford (UK) and Leiden (NL). Applied category theory is important to a growing community of researchers who study computer science, logic, engineering, physics, biology, chemistry, social science, systems, linguistics and other subjects using category-theoretic tools. The background and experience of our members is as varied as the systems being studied. The goal of the Applied Category Theory conference series is to bring researchers together, strengthen the applied category theory community, disseminate the latest results, and facilitate further development of the field.

Submissions

We accept submissions in English of original research papers, talks about work accepted/ submitted/published elsewhere, and demonstrations of relevant software. Accepted original research papers will be published in a proceedings volume. The conference will include an industry showcase event and community meeting. We particularly encourage people from underrepresented groups to submit their work and the organizers are committed to non-discrimination, equity, and inclusion.

Original research papers intended for conference proceedings should present original, high-quality work in the style of a computer science conference paper (up to 12 pages, not counting the bibliography; more detailed parts of proofs may be included in an appendix for the convenience of the reviewers). Please use the EPTCS style files available at

http://style.eptcs.org

Such submissions should not be an abridged version of an existing journal article although pre-submission arXiv preprints are permitted. These submissions will be adjudicated for both a talk and publication in the conference proceedings.

Important dates

The following dates are all in 2023, and Anywhere On Earth.

• Submission Deadline: Wednesday 3 May

• Author Notification: Wednesday 7 June

• Camera-ready version due: Tuesday 27 June

• Conference begins: 31 July

Program committee

Benedikt Ahrens

Mario Álvarez Picallo

Matteo Capucci

Titouan Carette

Bryce Clarke

Carmen Constantin

Geoffrey Cruttwell

Giovanni de Felice

Bojana Femic

Marcelo Fiore

Fabio Gadducci

Zeinab Galal

Richard Garner

Neil Ghani

Tamara von Glehn

Amar Hadzihasanovic

Masahito Hasegawa

Martha Lewis

Sophie Libkind

Rory Lucyshyn-Wright

Sandra Mantovanni

Jade Master

Konstantinos Meichanetzidis

Stefan Milius

Mike Mislove

Sean Moss

David Jaz Myers

Susan Niefield

Paige Randall North

Jason Parker

Evan Patterson

Sophie Raynor

Emily Roff

Morgan Rogers

Mario Román

Maru Sarazola

Bas Spitters

Sam Staton (co-chair)

Dario Stein

Eswaran Subrahmanian

Walter Tholen

Christina Vasilakopoulou (co-chair)

Christine Vespa

Simon Willerton

Glynn Winskel

Vladimir Zamdzhiev

Fabio Zanasi

Organizing committee

James Fairbanks, University of Florida

Joe Moeller, National Institute for Standards and Technology, USA

Sam Staton, Oxford University

Priyaa Varshinee Srinivasan, National Institute for Standards and Technology, USA

Christina Vasilakopoulou, National Technical University of Athens

Steering committee

John Baez, University of California, Riverside

Bob Coecke, Cambridge Quantum

Dorette Pronk, Dalhousie University

David Spivak, Topos Institute

Adjoint School 2023

Are you interested in applying category-theoretic methods to problems outside of pure mathematics? Apply to the Adjoint School!

Apply here. And do it soon.

• January 9, 2023. Application Due.

• February – July, 2023. Learning Seminar.

• July 24 – 28, 2023. In-person Research Week at University of Maryland, College Park, USA

Participants are divided into four-person project teams. Each project is guided by a mentor and a TA. The Adjoint School has two main components: an online learning seminar that meets regularly between February and June, and an in-person research week held in the summer adjacent to the Applied Category Theory Conference.

During the learning seminar you will read, discuss, and respond to papers chosen by the project mentors. Every other week a pair of participants will present a paper which will be followed by a group discussion. After the learning seminar each pair of participants will also write a blog post, based on the paper they presented, for The n-Category Café. 

Projects and Mentors: 

• Message passing logic for categorical quantum mechanics – Mentor: Priyaa Srinivasan
• Behavioural metrics, quantitative logics and coalgebras – Mentor: Barbara König
• Concurrency in monoidal categories – Mentor: Chris Heunen
• Game comonads and finite model theory – Mentor: Dan Marsden

See more information about research projects at https://adjointschool.com/2023.html.

Organizers:

• Ana Luiza Tenorio 
• Angeline Aguinaldo
• Elena Di Lavore 
• Nathan Haydon

ACT 2023

Here’s a bit of information about the conference Applied Category Theory 2023, and the associated Adjoint School—a school on applied category theory.  Many students have gotten into applied category theory by attending this school!

They will both take place at the University of Maryland in College Park, Maryland. Here are the dates:

• Adjoint School: July 24–28, 2023.
• Applied Category Theory conference: July 31–August 4, 2023.

Applications for the Adjoint School are due January 9, 2023. You can apply for this school here, and see information about the various mentors you might work with here.

This Week’s Finds – Lecture 10

 

This was the last of this year’s lectures on This Week’s Finds. You can see all ten lectures here. I will continue in September 2023.

This time I spoke about quaternions in physics and Dyson’s ‘three-fold way’: the way the real numbers, complex numbers and quaternions interact. For details, try my paper Division algebras and quantum theory.

One cute fact is how an electron is like a quaternion! More precisely: how quaternions show up in the spin-1/2 representation of SU(2) on ℂ².

Let me say a little about that here.

We can think of the group SU(2) as the group of unit quaternions: namely, 𝑞 with |𝑞| = 1. We can think of the space of spinors, ℂ², as the space of quaternions, ℍ. Then acting on a spinor by an element of SU(2) becomes multiplying a quaternion on the left by a unit quaternion!

But what does it mean to multiply a spinor by 𝑖 in this story? It’s multiplying a quaternion on the right by the quaternion 𝑖. Note: this commutes with left multiplications by all unit quaternions.

But there are some subtleties here. For example: multiplying a quaternion on the right by 𝑗 also commutes with left multiplication by unit quaternions. But 𝑗 anticommutes with 𝑖:

𝑖𝑗 = −𝑗𝑖

So there must be an ‘antilinear’ operator on spinors which commutes with the action of SU(2): that is, an operator that anticommutes with multiplication by 𝑖. Moreover this operator squares to -1.

In physics this operator is usually called ‘time reversal’. It reverses angular momentum.

You should have noticed something else, too. Our choice of right multiplication by 𝑖 to make the quaternions into a complex vector space was arbitrary: any unit imaginary quaternion would do! There was also arbitrariness in our choice of 𝑗 to be the time reversal operator.

So there’s a whole 2-sphere of different complex structures on the space of spinors, all preserved by the action of SU(2). And after we pick one, there’s a circle of different possible time reversal operators!

So far, all I’m saying is that quaternions help clarify some facts about the spin-1/2 particle that would otherwise seem a bit mysterious or weird.

For example, I was always struck by the arbitrariness of the choice of time reversal operator. Physicists usually just pick one! But now I know it corresponds to a choice of a second square root of -1 in the quaternions, one that anticommutes with our first choice: the one we call 𝑖.

At the very least, it’s entertaining. And it might even suggest some new things we could try: like ‘gauging’ time reversal symmetry (changing its definition in a way that depends on where we are), or even gauging the complex structure on spinors.

This Week’s Finds – Lecture 9

 

In this talk I explained the quaternions and octonions, and showed how to multiply them using the dot product and cross product of vectors.

For more details, including a proof that octonion multiplication obeys |ab|=|a||b|, go here:

• Octonions and the Standard Model (Part 2).

This was one of a series of lectures based on my column This Week’s Finds.

 

 

This Week’s Finds – Lecture 8

 

In this talk I explained the E₈ root lattice and how it gives rise to the ‘octooctonionic projective plane’, a 128-dimensional manifold on which the compact Lie group called E₈ acts as symmetries. I also discussed how some special root lattices give various notions of ‘integer’ for the real numbers, complex numbers, quaternions and octonions.

For more, read my paper Coxeter and Dynkin diagrams.

This was one of a series of lectures based on my column This Week’s Finds.

Mathematics for Humanity: a Plan

I’m working with an organization that may eventually fund proposals to fund workshops for research groups working on “mathematics for humanity”. This would include math related to climate change, health, democracy, economics, AI, etc.

I can’t give details unless and until it solidifies.

However, it would help me to know a bunch of possible good proposals. Can you help me imagine some?

A good proposal needs:

• a clearly well-defined subject where mathematics is already helping humanity but could help more, together with

• a specific group of people who already have a track record of doing good work on this subject, and

• some evidence that having a workshop, maybe as long as 3 months, bringing together this group and other people, would help them do good things.

I’m saying this because I don’t want vague ideas like “oh it would be cool if a bunch of category theorists could figure out how to make social media better”.

I asked for suggestions on Mathstodon and got these so far:

• figuring out how to better communicate risks and other statistical information,

• developing ways to measure and combat gerrymandering,

• improving machine learning to get more reliable, safe and clearly understandable systems,

• studying tipping points and ‘tipping elements’ in the Earth’s climate system,

• creating higher-quality open-access climate simulation software,

• using operations research to disrupt human trafficking.

Each topic already has people already working on it, so these are good examples. Can you think of more, and point me to groups of people working on these things?

---

The International Centre for Mathematical Sciences is indeed going ahead with this plan! Read more about it here:

• Mathematics for Humanity.

This Week’s Finds – Lecture 7

 

Coxeter and Dynkin diagrams classify some of the most beautiful objects in mathematics. In this talk I went through all the connected Dynkin diagrams and say how they correspond to compact simple Lie group— which happen to be act as symmetries of geometrical structures built using the real numbers, complex numbers, quaternions and octonions!

For more, read my paper Coxeter and Dynkin diagrams.

This was one of a series of lectures based on my column This Week’s Finds.

Seminar on “This Week’s Finds”

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 S_n, irreducible representations of S_n, 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 F_q, 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 Cl_n 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.