Motivic stuff

Cohomology, homotopy theory, and arithmetic geometry

Archive for April, 2009

Lectures next week in Lisbon on Kervaire Invariant One

Posted by Andreas Holmstrom on April 28, 2009

If you are desperate to hear more about the Kervaire Invariant One problem, it might be a good idea to go to Portugal next week and listen to the first-hand account presented by Ravenel in this lecture series.

There are also lots of useful comments on the problem at the n-category cafe, including some comments of Mike Hopkins, and an explanation of the connections with exotic spheres.

Update: The video from Hopkins’ talk is available from the conference website (at the moment only as a huge mov file). 

Article in Nature News.

Some remarks by Landweber.

See also the previous post.

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The Kervaire Invariant One problem solved

Posted by Andreas Holmstrom on April 22, 2009

It seems like Mike Hopkins, Mike Hill and Doug Ravenel have solved the famous Kervaire Invariant One problem. Here is a quote from an email sent to the ALGTOP-L mailing list today: “Yesterday, at the conference on Geometry and Physics being held in Edinburgh in honor of Sir Michael Atiyah, Harvard Professor Mike Hopkins announced a solution to the 45 year old Kervaire Invariant One problem, one of the major outstanding problems in algebraic and geometric topology. This is joint work with Rochester professor Doug Ravenel and U VA postdoctoral Whyburn Instructor Mike Hill.” The whole email, containing some more background, can be found here.

Update: The slides from the lecture of Hopkins have been made available at the web page of Ranicki. Link (warning, 40MB file).

See also this post.

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Local systems and some cohomology theories at SBS

Posted by Andreas Holmstrom on April 22, 2009

The Secret Blogging Seminar have started a discussion about local systems and their connections to etale cohomology, crystalline cohomology and algebraic de Rham cohomology. Worth checking out if you haven’t already seen it.

Posted in Cohomology theories in algebraic geometry | Tagged: , , , | 1 Comment »

What is geometry?

Posted by Andreas Holmstrom on April 20, 2009

(This is an extended set of notes for Part 1 of a talk I gave a few days ago at the Young Researchers in Mathematics conference in Cambridge. Part 2 will be posted soon.)

What is geometry? What is a “space”? When is an object “geometric”? Everyone would agree that a manifold is a geometric object, and similarly for a CW complex, and probably also for a scheme. But what about a group – is it a geometric object? What about a noncommutative ring? These and other mathematical objects form categories – but when should a category be regarded as a geometric category?

The question “What is geometry” is of course very naive, but I believe it is still of some interest. First of all, it is interesting from a historical point of view to look at the answers given at various points in history, and how our idea of geometry has developed over time. Secondly, when developing new forms of geometry, where sometimes even the fundamental definitions and constructions are not completely in place, it could possibly be helpful to have spent some time reflecting on what geometry really is. The new forms of geometry I have in mind include Arakelov geometry, geometry over \mathbf{F}_1, derived algebraic geometry, and various forms of noncommutative geometry. Thirdly, one would like to understand for what kinds of objects one can define “cohomology”. For example, we can define various forms of cohomology for manifolds, schemes, Lie algebras, associative algebras, groups, rigid analytic spaces, \mathbf{C}^{*}-algebras, ring spectra, categories, stacks, operads, and many other things. What exactly do these objects have in common?

What follows is a list of suggestions for conceptual answers to the question “What is geometry?”. Of course the answers are complementary, each of them capturing some particular aspect of what geometry is. 

1. For a very long time, geometry was the same thing as Euclidean geometry, and to say or think something else was almost unheard of. Only in the 19th century did Western mathematicians begin to realize that there could actually be other forms of geometry.

2. The famous Erlangen program, formulated by Klein in 1872, gave a unification of the various types of geometry existing at the time, focusing on the notion of symmetry, and on properties invariant under symmetry groups. These ideas had a huge impact on the development of Lie theory and various other subjects in geometry and physics. 

3. There is something called “Cartan geometries” (developed by Élie Cartan), which appears to be a further generalization of the Erlangen program, including Riemannian geometry in the picture. I have not found a good online source, but there is a book by Sharpe

4. One important way of approaching geometry is to shift focus from the geometric object to some set of functions on the object. For example, one could replace a topological space by the ring of continuous complex-valued functions on the space, or replace an algebraic variety with the ring of polynomial functions on the variety. In many cases, this process gives an equivalence of categories. This approach is the standard way of introducing Grothendieck‘s schemes, and is also the basic idea of noncommutative geometry.

5. Closely related to the previous item is the idea of defining geometry as the study of locally ringed spaces. In their really nice introduction to algebraic geometry, Demazure and Gabriel define a geometric space to be a locally ringed space

6. One could “define” a geometric category as a category admitting an interesting functor to \mathbf{Hot} (the homotopy category of topological spaces). Some examples to motivate this approach: For categories of (well-behaved) topological spaces with some extra structures, e.g. smooth manifolds, there is a forgetful functor to \mathbf{Top} and hence to \mathbf{Hot}. For the category of groups, and more generally the category of small categories, we have the classifying space functor. For the category of (non-negatively graded) chain complexes of abelian groups, we have the Dold-Kan correspondence, which gives a functor to simplicial abelian groups and hence to Hot (more about this example in Part 2 of the talk).

7. Another way of “defining” a geometric category could be: A category admitting some notion of cohomology. The problem with this definition is of course that it is hard to define what exactly we mean by cohomology, but it should be a functor to some abelian category, producing long exact sequences and spectral sequences in ways similar to what we observe in topology and algebraic geometry.

8. There are various approaches to “homotopical categories”, and we could define geometry as the study of these categories. The most well-known approach is probably Quillen’s notion of model categories. There are many other approaches and languages as well, for example various notions of infinity-categories, homotopical categories in the sense of Dwyer-Hirschhorn-Kan-Smith, higher stacks, Segal categories, simplicial sheaves, simplicial categories, A^{\infty}-categories, and more. I will say more about this in Part 2 of the talk.

9. In the 60s, Lawvere developed the concept of a “theory”. As a special case, there is something called a “geometric theory”, which could maybe serve as a way to define what geometry is. For more about this, see the online book by Barr and Wells, in particular sections 4.5 and 8.3.

10. Some people would argue that everything is geometry.

Some remarks: There are probably many other approaches to answering the question we started with. It seems to me that a good definition of geometry should (1) allow for noncommutative structures, and (2) agree with the principle that everything algebraic is also geometric. 

In Part 2 of the talk (to be posted soon!) I will say something about point no 8, homotopical categories, and try to show that they can be useful in algebraic geometry.

Posted in Talks | Tagged: , , | 29 Comments »

Doing mathematics online

Posted by Andreas Holmstrom on April 12, 2009

As many others have already noted, the Tricki project is now on the verge of going live, thanks to the efforts of Tim Gowers, Alex Frolkin, and Olof Sisask. This seems to be a sign among many that the Internet can and will have a profound impact on how mathematical research is done, and it is intriguing to speculate about how communication technology will change the way we do mathematical research in the coming decades (and centuries).

The Internet has of course already changed a lot of things, many of which we already take for granted. We enjoy the advantages of preprint servers such as the arXiv. Lots of basic mathematical knowledge is instantly available at Wikipedia, and I was recently very happy to discover the nLab. Some people are writing a book about stacks in an online collaborative project. Others create resource pages on specific subjects, like motivic homotopy theory. Needless to say, there are lots of math blogs, and lots of online books and lecture notes. Tim Gowers once proposed a site with alternative maths reviews. There are various useful databases, like the Sloane’s Online Encyclopedia of Integer Sequences and John Cremona’s tables of elliptic curves, both available through SAGE. Although the choice of subject matter prevented me from taking part, I very much liked the idea of the polymath experiment.

So what will the future bring? All mathematical definitions implanted in a chip in your brain? Quantum computers emulating the brain of Grothendieck? Computers actually inventing new mathematics? One can only guess about these and other developments, but in the shorter run it seems to me that a key question will be to find a way to make all of the mathematics literature available online, for free. Having been blogging for a few months now, it is very impractical to not being able to link to articles, just because the author has given away the copyright to a huge profit-hungry company, or because the article only exists in paper form. The same problem must face anyone trying to implement any form of open online collaborating. Some very clever people developed Spotify for music lovers, making almost all music available for free while keeping the music industry happy. Who will do the same for maths lovers, and for other scientists?

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