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 the 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 the 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!


Hoàng Xuân Sính

20 June, 2022

During the Vietnam war, Grothendieck taught math to the Hanoi University mathematics department staff, out in the countryside. Hoàng Xuân Sính took notes and later did a PhD with him — by correspondence! She mailed him her hand-written thesis. She is the woman in this picture:



As you might guess, there’s a very interesting story behind this. I’ve looked into it, but what I found raises even more questions. Hoàng Xuân Sính’s life really deserves a good biography.

Hoàng Xuân Sính was born in 1933 in a village called Cót, one of seven children of a fabric merchant. Her mother died when she was eight years old, and she was raised by a stepmother.

She spent a lot of time sewing and designing clothes. However, this image of a magazine from 1981 shows a picture of her holding a book—and on the website where I found this, a caption in Vietnamese says “Cót village girl is passionate about math”.


She would have been 18 at the time. In this year she completed a bachelor’s degree in Hanoi, studying English and French, and then traveled to Paris for a second baccalaureate in mathematics. She stayed in France to study for the agrégation (the competitive examination for civil service) at the University of Toulouse, which she completed in 1959, before returning to Vietnam and teaching mathematics at the Hanoi National University of Education.

Grothendieck visited North Vietnam in late 1967, during the Vietnam War, and spent a month teaching mathematics to the Hanoi University mathematics department staff, including Hoàng Xuân Sính, who took the notes for the lectures. Because of the war, Grothendieck’s lectures were held away from Hanoi, first in the nearby countryside and later in Đại Từ.

After Grothendieck returned to France, he continued to teach Hoàng Xuân Sính in an exchange of letters. According to the web page of the university she founded, Thang Long University, Hoàng Xuân Sính remembers two main impressions from her contacts with Alexander Grothendieck:

1) A good teacher is a teacher who turns something difficult into something easy.

2) We should always avoid anything that is fictitious, live in accordance to our own feelings and value simple people.

She finished her thesis in 1972. Around Christmas that year, the United States dropped over 20,000 tons of bombs on North Vietnam, mainly Hanoi. So, it’s not surprising that she only defended her thesis three yearslater, when the North had almost won. But she mentions another reason. She later wrote:

I was a doctorate student during wartime. Back then, I was teaching at Hanoi Pedagogical University, there was not a mode to take leave to study for the doctorate. I taught during the day and worked on my thesis during the night under the kerosene lamp light. I wrote in French under my distant teacher’s guidance. When I got the approval from France to come over to defend, there were disagreeable talks about not letting me because they was afraid I wasn’t coming back. The most supportive person during the time was Lady Ha Thi Que—President of the Vietnamese Women Coalescent organization. Madame Que was a guerilla, without the conditions to get much education, but gave very convincing reasons to support me. She said, firstly, I was 40 years old, it is very difficult to get a job abroad at 40 years old, and without a job, how can I live? Second, my child is at home, no woman would ever leave her child… so comrades, let’s not be worried, let her go. I finished my thesis in 1972, and 3 years later with the help and struggle of the women’s organization, I was able to travel over to defend in 1975….

She went to France to defend her thesis at Paris Diderot University (also called Paris 7). Her thesis committee included not only Alexander Grothendieck but also Henri Cartan, Laurent Schwartz, Michel Zisman, and Jacques Deny.

Her thesis defense lasted two and a half hours. And soon thereafter she defended a second thesis, entitled “The embedding of a one-dimensional complex in a two-dimensional differential manifold”. I don’t know who, if anyone, directed this second thesis.

She later became the first woman mathematics professor in Vietnam — and the second came 35 years later.

In 1988, she started the first private university in Vietnam, Thang Long University in Hanoi. For a while she was not only the head, but also the janitor, bringing water to the school and sweeping floors. Later she said “When I look back at it, I thought it was the most romantic idea I’ve had.”


In 2003 she was awarded France’s Ordre des Palmes Académiques. She is still alive! I hope someone has interviewed her, or does it now. Her stories must be very interesting.

But what about her thesis?

Her thesis classified Gr-categories, which are now called ‘2-groups’ for short. A 2-group is the categorified version of a group: it’s a monoidal category where every object and morphism is invertible. (An object x is invertible if there’s an object y with x \otimes y \cong y \otimes x \cong I, where I is the unit for the tensor product.)

From a 2-group you can get two groups:

• the group G of isomorphism classes of objects, and

• the group A of automorphisms of the unit object I.

The group A is abelian, and G acts on A. But there’s one more thing! The associator can be used to get a map

a \colon G^3 \to A

The pentagon identity for the associator implies that this map obeys an equation. And this equation is familiar in the subject of group cohomology: it says a is a ‘3-cocycle’ on the group G with coefficients in A.

Even better, she showed that cohomologous 3-cocycles give equivalent 2-groups. (Equivalent as monoidal categories, that is.)

So, we can classify 2-groups using cohomology! The most exciting, least obvious part of this is the cohomology class [a] \in H^3(G,A). This is often called the ‘Sính invariant’, though I believe Hoàng is her surname, not Sính.

This connection between 2-groups and cohomology is no coincidence. It’s best understood using a bit more topology.

Any connected space with a basepoint, say X, has a fundamental group. But it also has a fundamental 2-group! This 2-group has G = \pi_1(X) and A = \pi_2(X). And if all the higher homotopy groups of X vanish, this 2-group knows everything about the
homotopy type of X, at least if X is reasonably nice, like a CW complex.

So, Hoàng Xuân Sính’s thesis sheds light on ‘homotopy 2-types’: that is, homotopy types of nice spaces with \pi_n(X) = 0 for n > 2. They are just 2-groups!

Thus, her thesis illuminated one of the simplest — yet still important — special cases of Grothendieck’s ‘homotopy hypothesis’, namely that homotopy n-types correspond to n-groupoids.

You can see Hoàng Xuân Sính’s thesis along with a handwritten
summary in English here:

Thesis of Hoàng Xuân Sính

That website also has three nice photos of Grothendieck in Vietnam. I showed a colorized version of one at the top of this article, and here is another, with Hoàng Xuân Sính at far left:



Here are the original uncolorized versions of all three:








Shannon Entropy from Category Theory

22 April, 2022

I’m giving a talk at Categorical Semantics of Entropy on Wednesday May 11th, 2022. You can watch it live on Zoom if you register, or recorded later. Here’s the idea:

Shannon entropy is a powerful concept. But what properties single out Shannon entropy as special? Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function. Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous. This is joint work with Tom Leinster and Tobias Fritz.

You can see the slides now, here. I talk a bit about all these papers:

• John Baez, Tobias Fritz and Tom Leinster, A characterization of entropy in terms of information loss, 2011.

• Tom Leinster, An operadic introduction to entropy, 2011.

• John Baez and Tobias Fritz, A Bayesian characterization of relative entropy, 2014.

• Tom Leinster, A short characterization of relative entropy, 2017.

• Nicolas Gagné and Prakash Panangaden, A categorical characterization of relative entropy on standard Borel spaces, 2017.

• Tom Leinster, Entropy and Diversity: the Axiomatic Approach, 2020.

• Arthur Parzygnat, A functorial characterization of von Neumann entropy, 2020.

• Arthur Parzygnat, Towards a functorial description of quantum relative entropy, 2021.

• Tai-Danae Bradley, Entropy as a topological operad derivation, 2021.


Categorical Semantics of Entropy

19 April, 2022

There will be a workshop on the categorical semantics of entropy at the CUNY Grad Center in Manhattan on Friday May 13th, organized by John Terilla. I was kindly invited to give an online tutorial beforehand on May 11, which I will give remotely to save carbon. Tai-Danae Bradley will also be giving a tutorial that day in person:

Tutorial: Categorical Semantics of Entropy, Wednesday 11 May 2022, 13:00–16:30 Eastern Time, Room 5209 at the CUNY Graduate Center and via Zoom. Organized by John Terilla. To attend, register here.

12:00-1:00 Eastern Daylight Time — Lunch in Room 5209.

1:00-2:30 — Shannon entropy from category theory, John Baez, University of California Riverside; Centre for Quantum Technologies (Singapore); Topos Institute.

Shannon entropy is a powerful concept. But what properties single out Shannon entropy as special? Instead of focusing on the entropy of a probability measure on a finite set, it can help to focus on the “information loss”, or change in entropy, associated with a measure-preserving function. Shannon entropy then gives the only concept of information loss that is functorial, convex-linear and continuous. This is joint work with Tom Leinster and Tobias Fritz.

2:30-3:00 — Coffee break.

3:00-4:30 — Operads and entropy, Tai-Danae Bradley, The Master’s University; Sandbox AQ.

This talk will open with a basic introduction to operads and their representations, with the main example being the operad of probabilities. I’ll then give a light sketch of how this framework leads to a small, but interesting, connection between information theory, abstract algebra, and topology, namely a correspondence between Shannon entropy and derivations of the operad of probabilities.

Symposium on Categorical Semantics of Entropy, Friday 13 May 2022, 9:30-3:15 Eastern Daylight Time, Room 5209 at the CUNY Graduate Center and via Zoom. Organized by John Terilla. To attend, register here.

9:30-10:00 Eastern Daylight Time — Coffee and pastries in Room 5209.

10:00-10:45 — Operadic composition of thermodynamic systems, Owen Lynch, Utrecht University.

The maximum entropy principle is a fascinating and productive lens with which to view both thermodynamics and statistical mechanics. In this talk, we present a categorification of the maximum entropy principle, using convex spaces and operads. Along the way, we will discuss a variety of examples of the maximum entropy principle and show how each application can be captured using our framework. This approach shines a new light on old constructions. For instance, we will show how we can derive the canonical ensemble by attaching a probabilistic system to a heat bath. Finally, our approach to this categorification has applications beyond the maximum entropy principle, and we will give an hint of how to adapt this categorification to the formalization of the composition of other systems.

11:00-11:45 — Polynomial functors and Shannon entropy, David Spivak, MIT and the Topos Institute.

The category Poly of polynomial functors in one variable is extremely rich, brimming with categorical gadgets (e.g. eight monoidal products, two closures, limits, colimits, etc.) and applications including dynamical systems, databases, open games, and cellular automata. In this talk I’ll show that objects in Poly can be understood as empirical distributions. In part using the standard derivative of polynomials, we obtain a functor to Set × Setop which encodes an invariant of a distribution as a pair of sets. This invariant is well-behaved in the sense that it is a distributive monoidal functor: it acts on both distributions and maps between them, and it preserves both the sum and the tensor product of distributions. The Shannon entropy of the original distribution is then calculated directly from the invariant, i.e. only in terms of the cardinalities of these two sets. Given the many applications of polynomial functors and of Shannon entropy, having this link between them has potential to create useful synergies, e.g. to notions of entropic causality or entropic learning in dynamical systems.

12:00-1:30 — Lunch in Room 5209

1:30-2:15 — Higher entropy, Tom Mainiero, Rutgers New High Energy Theory Center.

Is the frowzy state of your desk no longer as thrilling as it once was? Are numerical measures of information no longer able to satisfy your needs? There is a cure! In this talk we’ll learn about: the secret topological lives of multipartite measures and quantum states; how a homological probe of this geometry reveals correlated random variables; the sly decategorified involvement of Shannon, Tsallis, Réyni, and von Neumann in this larger geometric conspiracy; and the story of how Gelfand, Neumark, and Segal’s construction of von Neumann algebra representations can help us uncover this informatic ruse. So come to this talk, spice up your entropic life, and bring new meaning to your relationship with disarray.

2:30-3:15 — On characterizing classical and quantum entropy, Arthur Parzygnat, Institut des Hautes Études Scientifiques.

In 2011, Baez, Fritz, and Leinster proved that the Shannon entropy can be characterized as a functor by a few simple postulates. In 2014, Baez and Fritz extended this theorem to provide a Bayesian characterization of the classical relative entropy, also known as the Kullback–Leibler divergence. In 2017, Gagné and Panangaden extended the latter result to include standard Borel spaces. In 2020, I generalized the first result on Shannon entropy so that it includes the von Neumann (quantum) entropy. In 2021, I provided partial results indicating that the Umegaki relative entropy may also have a Bayesian characterization. My results in the quantum setting are special applications of the recent theory of quantum Bayesian inference, which is a non-commutative extension of classical Bayesian statistics based on category theory. In this talk, I will give an overview of these developments and their possible applications in quantum information theory.

Wine and cheese reception to follow, Room 5209.