John C. Baez

I know no branch of math that makes me allergic to examples. For example in category theory it's crucial to have lots of examples to get a deep understanding of the concepts.

I prefer 〇 because it's a framed picture of nothing.

From Geylang, which is where you find all the prostitutes and also many small Buddhist temples.

So no, Singapore is not a "theme park where everything is forbidden but the streets are squeaky clean". It's a lot more complex and interesting.

From downtown during the Hungry Ghost Month ceremonies:

In general the Western image of Singapore (scarily oppressive) is quite different from my experience of it when I lived there for 2 years.

From Little India:

The Singapore Botanic Gardens is huge, and fun to wander around in, and my sister (who was a horticulturalist) would have spent all day there if I didn't finally demand that we leave. "Leave" - pardon the pun.

Sample:

Now, if you never wanted to do algebraic topology for varieties, there might be other reasons you'd want to consider "sheaves on a category" as special examples of presheaves on that category, and thus need the concept of Grothendieck topology. But that was the historical reason.

It all depends on where you're coming from. I have no idea what an "ionad" is or what makes them "bounded". But a category with a Grothendieck topology arises naturally when we want to generalize open covers of varieties to "etale covers" - allowing us to do algebraic topology for varieties.

Neat! I didn't remember that painting, so here it is for anyone curious:

en.wikipedia.org/wiki/File:La...

Self-portrait of Manet holding an easel and a paintbrush: https://en.wikipedia.org/wiki/Self-Portrait_with_Palette_(Manet)

When Manet came out with this painting in 1882, some critics mocked him for his poor understanding of perspective. Some said he was going senile.

It was, in fact, his last major painting. But he was a genius, and he was going... whoosh... over their heads, just like he went over mine.

(8/n, n=8)

The upper left corner of Manet's painting Un bar aux Folies Bergère shows the green shoes and pinkish legs of someone standing on a trapeze, with massed crowds of revelers below.

Before someone explained this painting to me, I didn't see it - I wasn't really looking at it. I didn't even see the green shoes of the trapeze artist!

I can often grasp music quite quickly. But paintings often fail to move me until someone explains them.

(7/n)

YouTube video by ZCZ Films Manet's A Bar at Folies Bergere

The woman in the painting was actually a real person, known as Suzon, who worked at the Folies-Bergère in the early 1880s. For his painting, Manet posed her in his studio.

(6/n)

www.youtube.com/watch?v=Ye5k...

A young barmaid behind the bar at the famous French nightclub Folies-Bergère is staring forward in a bored, introspective way. Behind her is a mirror which reflects the lively scene. At the top left you see the green shoes of a trapeze artists. More importantly, at right you see the back of the barmaid, and see that she is face to a man who leans close and seems to be requesting something. This is Manet's famous painting "A Bar at the Folies-Bergère": https://en.wikipedia.org/wiki/A_Bar_at_the_Folies-Berg%C3%A8re

Many of the barmaids at the Folies Bergère also served as prostitutes. Standing behind the oranges, the champagne and a bottle of Bass ale, the woman appears just as much a commodity as these other things. But she doesn't accept this: she is coldly aloof from her objectification.

(5/n)

A young barmaid behind the bar at the famous French nightclub Folies-Bergère is staring forward in a bored, introspective way. Behind her is a mirror which reflects the lively scene. At the top left you see the green shoes of a trapeze artists. More importantly, at right you see the back of the barmaid, and see that she is face to a man who leans close and seems to be requesting something. This is Manet's famous painting "A Bar at the Folies-Bergère": https://en.wikipedia.org/wiki/A_Bar_at_the_Folies-Berg%C3%A8re

The trickery allowed Manet to make a deep point. You think the barmaid is looking at you. The other customer thinks he's looking at him. That is her job. She is literally SPLIT.

In fact she's not looking at anyone. She is completely detached - perhaps bored, perhaps introspective.

(4/n)

A diagram showing, from above, a woman behind a rectangular table and in front of a mirror. Lines emanating from the "viewpoint" show what we see in Manet's painting A Bar at the Folies-Bergère. We see the woman, but not the man directly in front of her, because the viewpoint is far to the right. In the reflection, we see both the woman and the man. This is a computer-generated diagram by Malcolm Park, created with the assistance of Darren McKimm: https://www.getty.edu/art/exhibitions/manet_bar/bar_diagram.html

This diagram shows how the perspective works in Manet's famous painting Un bar aux Folies Bergère. We are viewing the woman at an angle, and while the man is outside our field of view, his reflection can be seen.

Astounding! But it's not just a technical feat.

(3/n)

A photographic reconstruction of Manet's famous painting A Bar at the Folies-Bergère. A barmaid stands behind a counter with a rectangular mirror behind here. You can see her reflection to her right, and in the reflection you also see that she is facing a man, who is not visible directly. This astoundingly clever reconstruction was created by Malcolm Park, here: https://www.getty.edu/art/exhibitions/manet_bar/looking_glass.html

Here is Malcolm Park's reconstruction of the scene in Manet's painting. How is it possible?

In fact the woman is viewed not head-on, but from an angle! While the man cannot be seen directly, his reflection is visible!

In my next post, I'll show you a diagram that explains how this works.

(2/n)

A young barmaid behind the bar at the famous French nightclub Folies-Bergère is staring forward in a bored, introspective way. Behind her is a mirror which reflects the lively scene. At the top left you see the green shoes of a trapeze artists. More importantly, at right you see the back of the barmaid, and see that she is face to a man who leans close and seems to be requesting something. This is Manet's famous painting "A Bar at the Folies-Bergère": https://en.wikipedia.org/wiki/A_Bar_at_the_Folies-Berg%C3%A8re

The perspective looks all wrong. You're staring straight at this barmaid, but her reflection in the mirror is way off to right. Even worse, her reflection is facing a guy who doesn't appear in the main view!

But in 2000, a researcher showed this perspective is actually possible!

(1/n)

Moods are like muscles. If you don't use them, they atrophy!

Schur Functors and Categorified Plethysm It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a "pl...

I've written a couple of papers on how Core(FinSet) being the coproduct of all the symmetric groups helps us understand representations of symmetric groups, and how those are related to symmetric functions.

Unfortunately the first has an unnecessarily scary title.

arxiv.org/abs/2106.00190

I'm waiting for beer-flavored chocolate cake.

The cool thing about the subjunctive is that you can skip the "if":

"Oh, if only I were able to use the subjunctive mood correctly!"

shrinks to

"Oh, were I only able to use the subjunctive mood correctly!"

But nowadays this comes across as "flexing". 😢

Indeed, kids aren't scared away from sports by athletes....

Google Maps Find local businesses, view maps and get driving directions in Google Maps.

It's near one of the main eastern entrances of the Navajo Nation, coming from the south east.

www.google.com/maps/place/C...

Interesting! I believe in these other branches but it doesn't affect my action at all. The only thing I can do is be virtuous, wise, etc. - thus increasing the amplitude of branches where things go well.

My favorite is the trading post in Cameron, just because they've got a nice hotel and restaurant. There's something cool about sleeping in a trading post and having blue corn pancakes for breakfast.

There may actually be a serious mistake in this section - I actually forget! 😢 But the basic idea is that any group is a quandle via conjugation and if you differentiate conjugation in a Lie group you get the Lie bracket on the Lie algebra (a known fact).

The canonical rack of something?

Lie 2-Algebras We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. ...

Some of Chapter 3 here might be good to read, to get more of an idea about quandles and racks. Ignore the other stuff!

arxiv.org/abs/math/040...

My student Alissa Crans did her thesis on quandles, racks and shelves - among other things. Great stuff!

Network Theory (Part 1) As a mathematician who has gotten interested in the problems facing our planet, I’ve been trying to cook up some new projects to work on. Over the decades I’ve spent a lot of time study…

What I really like is how @nattyover.bsky.social explained some of my original dream of using category theory for "green mathematics", and how this dream lives on in some applications to epidemiology and biology, but remains unfulfilled.

(4/n, n=4)

johncarlosbaez.wordpress.com/2011/03/04/n...