geometry – Page 28 – neverendingbooks

Two lecture series on absolute geometry

Published January 18, 2015 by lievenlb

Absolute geometry is the attempt to develop algebraic geometry over the elusive field with one element $\mathbb{F}_1$. The idea being that the set of all prime numbers is just too large for $\mathbf{Spec}(\mathbb{Z})$ to be a terminal object (as it is in the category of schemes).

So, one wants to view $\mathbf{Spec}(\mathbb{Z})$ as a geometric object over something ‘deeper’, the “absolute point” $\mathbf{Spec}(\mathbb{F}_1)$.

Starting with the paper by Bertrand Toen and Michel Vaquie, Under $\mathbf{Spec}(\mathbb{Z})$, topos theory entered this topic.

First there was the proposal by Jim Borger to view $\lambda$-rings as $\mathbb{F}_1$-algebras. More recently, Alain Connes and Katia Consani introduced the arithmetic site.

Now, there are lectures series on these two approaches, one by Yuri I. Manin, the other by Alain Connes.

Yuri I. Manin in Ghent

On Tuesday, February 3rd, Yuri I. Manin will give the inaugural lectures of the new $\mathbb{F}_1$-seminars at Ghent University, organised by Koen Thas.

Coffee will be served from 13.00 till 14.00 at the Department of Mathematics, Ghent University, Krijgslaan 281, Building S22 and from 14.00 till 16.30 there will be lectures in the Emmy Noether lecture room, Building S25:

14:00 – 14:25: Introduction (by K. Thas)
14:30 – 15:20: Lecture 1 (by Yu. I. Manin)
15:30 – 16:20: Lecture 2 (by Yu. I. Manin)

Recent work of Manin related to $\mathbb{F}_1$ includes:

– Local zeta factors and geometries under $\mathbf{Spec}(\mathbb{Z})$

– Numbers as functions

Alain Connes on the Arithmetic Site

Until the beginning of march, Alain Connes will lecture every thursday afternoon from 14.00 till 17.30, in Salle 5 – Marcelin Berthelot at he College de France on The Arithmetic Site (hat tip Isar Stubbe).

Here’s a two minute excerpt, from a longer interview with Connes, on the arithmetic site, together with an attempt to provide subtitles:

——————————————————

(50.36)

And,in this example, we saw the wonderful notion of a topos, developed by Grothendieck.

It was sufficient for me to open SGA4, a book written at the beginning of the 60ties or the late fifties.

It was sufficient for me to open SGA4 to see that all the things that I needed were there, say, how to construct a cohomology on this site, how to develop things, how to see that the category of sheaves of Abelian groups is an Abelian category, having sufficient injective objects, and so on … all those things were there.

This is really remarkable, because what does it mean?

It means that the average mathematician says: “topos = a generalised topological space and I will never need to use such things. Well, there is the etale cohomology and I can use it to make sense of simply connected spaces and, bon, there’s the chrystaline cohomology, which is already a bit more complicated, but I will never need it, so I can safely ignore it.”

And (s)he puts the notion of a topos in a certain category of things which are generalisations of things, developed only to be generalisations…

But in fact, reality is completely different!

In our work with Katia Consani we saw not only that there is this epicyclic topos, but in fact, this epicyclic topos lies over a site, which we call the arithmetic site, which itself is of a delirious simplicity.

It relies only on the natural numbers, viewed multiplicatively.

That is, one takes a small category consisting of just one object, having this monoid as its endomorphisms, and one considers the corresponding topos.

This appears well … infantile, but nevertheless, this object conceils many wonderful things.

And we would have never discovered those things, if we hadn’t had the general notion of what a topos is, of what a point of a topos is, in terms of flat functors, etc. etc.

(52.27)

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I will try to report here on Manin’s lectures in Ghent. If someone is able to attend Connes’ lectures in Paris, I’d love to receive updates!

Children have always loved colimits

Published January 7, 2015 by lievenlb

If Chad Orzel is able to teach quantum theory to his dog, surely it must be possible to explain schemes, stacks, toposes and motives to hipsters?

Perhaps an idea for a series of posts?

It’s early days yet. So far, I’ve only added the tag sga4hipsters (pun intended) and googled around for ‘real-life’ applications of sheaves, cohomology, and worse.

Sooner or later one ends up at David Spivak’s MIT-webpage.

David has written a book “category theory for scientists” and has several papers on applications of category theory to databases.

There’s also this hilarious abstract, reproduced below, of a talk he gave in 2007 at many cheerful facts.

If this guy ever decides to write a novel, I’ll pre-order it on the spot.

Presheaf, the cobbler.
by David Spivak

Children have always loved colimits.

Whether it be sorting their blocks according to color, gluing a pair of googly eyes and a pipe-cleaner onto a piece of yellow construction paper, or simply eating a peanut butter sandwich, colimits play a huge role in their lives.

But what happens when their category doesn’t have enough colimits?

In today’s ”ownership” society, what usually happens is that the parents upgrade their child’s category to a Presheaf category. Then the child can cobble together crazy constructions to his heart’s content.

Sometimes, a kid comes up to you with an FM radio she built out of tinkertoys, and says
”look what I made! I call it ’182 transisters, 11 diodes, 6 plastic walls, 3 knobs,…’”

They seem to go on about the damn thing forever.

Luckily, Grothendieck put a stop to this madness.

He used to say to them, ever so gently, ”I’m sorry, kid. I’m really proud of you for making this ’182 transistors’ thing, but I’m afraid it already has a name. It’s called a radio.

And thus Grothendieck apologies were born.

Two years later, Grothendieck topologies were born of the same concept.

In this talk, I will teach you to build a radio (that really works!) using only a category of presheaves, and then I will tell you about the patent-police, known as Grothendieck topologies.

God willing, I will get through SGA 4 and Lurie’s book on Higher Topos Theory.”

Further reading:

David Spivak’s book (old version, but freely available) Category theory for scientists.

The published version, available from Amazon.

Can one explain schemes to hipsters?

Published January 5, 2015 by lievenlb

Nature (the journal) asked David Mumford and John Tate (of Fields and Abel fame) to write an obituary for Alexander Grothendieck.

Probably, it was their first experience ever to get a paper… rejected!

What was their plan?

How did they carry it out?

What went wrong?

And, can we learn from this?

the plan

Mumford and Tate set themselves an ambitious goal. Although Nature would have been happiest with a purely biographical note, they seized the opportunity to explain three ‘simple’ things to a wider audience: (1) schemes, (2) category theory, and, (3) cohomology…

“Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck’s work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology.”

1. Schemes

Here, the basic stumbling block, as Mumford acknowledged afterwards, is of course that most people don’t know what a commutative ring is. If you’ve never encountered a scheme before in broad daylight, I’m not certain this paragraph tells you how to recognise one:

“… In simplest terms, he proposed attaching to any commutative ring (any set of things for which addition, subtraction and a commutative multiplication are defined, like the set of integers, or the set of polynomials in variables x,y,z with complex number coefficients) a geometric object, called the Spec of the ring (short for spectrum) or an affine scheme, and patching or gluing together these objects to form the scheme. …”

2. Categories

Here they do a pretty good job, I think. They want to explain Grothendieck’s ‘functor of points’ and the analogy they used with several measuring experiments is neat:

“… Grothendieck used the web of associated maps — called morphisms — from a variable scheme to a fixed one to describe schemes as functors and noted that many functors that were not obviously schemes at all arose in algebraic geometry.

This is similar in science to having many experiments measuring some object from which the unknown real thing is pieced together or even finding something unexpected from its influence on known things….”

3. Cohomology

Here, Mumford “hoped that the inclusion of the unit 3-sphere in $\mathbb{C}^2- \{ (0,0) \}$ would be fairly clear to most scientists and so could be used to explain the Mike Artin’s breakthrough that $H^3_{et}(\mathbb{A}^2 – \{ (0,0) \}) \not= 0$.”

I’d love to know the fractional odds an experienced bookmaker would set in case someone (not me!) wants to bet on them successfully getting this message across.

“… Using complex coordinates (z,w), a plane has four real dimensions and taking out a point, what’s left is topologically a three dimensional sphere. Following the inspired suggestions of Grothendieck, Artin was able to show how with algebra alone that a suitably defined third cohomology group of this space has one generator, that is the sphere lives algebraically too. Together they developed what is called étale cohomology at a famous IHES seminar. …”

the aftermath

The good news is that Nature will still publish the Tate-Mumford obit, is some form or another, next week, on januari 15th. According to Mumford they managed to sneak in three examples of commutative rings in passing: polynomials, dual numbers and finite fields.

what went wrong?

The usual?

We mathematicians are obsessed with getting definitions right. We truly believe that no-one can begin to understand the implications of an idea if we don’t teach them the nitty gritty details of our treasured definitions first.

It appears that we are alone on this.

Did physicists smack us in the face with the full standard-model Lagrangian, demanding us to digest the minute details of it first, before they could tell us they had discovered the Higgs boson?

No, most scientists know how to get a message across. You need 3 things:

– a catchy name (the ‘God Particle’)

– good graphics (machines at CERN, collision pictures)

– a killer analogy (the most popular, in relation to the Higgs particle, seems to be “like Maggie Tatcher walking into a room”…)