Postdoc and PhD positions – deadlines very soon

Just wanted to spread the word about some very interesting opportunities.

- PhD positions in Copenhagen. Strong research groups in Algebra and Number theory, Topology and Noncommutative geometry. Very attractive financial conditions. Deadline: June 19th. Note that Denmark is also the happiest place in the world.

- Postdoc position at University of Nottingham with Alexander Vishik. Deadline: June 5th. Some possible topics mentioned: Algebraic theory of quadratic forms, algebraic cobordism theory, linear algebraic groups and homogeneous varieties (over arbitrary fields), motives (in the sense of  Voevodsky and Chow motives), A1-homotopy theory, Milnor’s K-theory and algebraic K-theory (over arbitrary fields).

- Postdoc position in algebraic geometry in Strasbourg. Below is a quote from the email sent to the EAGER-GEN list. Deadline June 10th.  ”A post-doc position in algebraic geometry for one year (that could be extended for a second year) is available here in Strasbourg, starting from September 1, 2009. The post-doc will have no teaching duty. The following three conditions are mandatory for the candidates: (1) being less than 35 years old by December 31, 2009; (2) having obtained the Ph.D. outside France; (3) no previous long stay at the Strasbourg Math. Department. Interested people may contact Gianluca Pacienza for further details. Candidates must send their Curriculum Vitae to Gianluca Pacienza. Two letters of recommendation must be sent (again by email) separately.”

Homotopical categories and simplicial sheaves

(This is an expanded version of the 2nd part of a talk I gave last month. For the first part, see this post.)

Homotopical categories

The topic for this post is “homotopical categories”, and their role in algebraic geometry. I want to emphasize that I am very much in the process of learning about all these things, so this post is based more on interest and enthusiasm than actual knowledge. I hope to convey some of the main ideas and why they could be interesting, and come back to the details in many future posts, after having learned more. I apologize for not defining everything carefully, and for brushing the “stable” aspects of the theory, i.e. spectra and sheaves of spectra, under the carpet.

There are many different ways to speak of “homotopical categories”, and I only use this expression because I don’t know of a better thing to call them. The most well-known approach is the language of model categories, invented by Quillen and developed by many others. There are many excellent online introductions, for example Dwyer-Spalinski, Goerss-Schemmerhorn, and appendix A2 of Jacob Lurie’s book on higher topos theory, available on his webpage. Other languages are given by the many different approaches to higher categories; see the nLab page and the survey of Bergner. Still other languages include Segal categories, A-infinity categories, infinity-stacks, and homotopical categories in the precise sense of Dwyer-Hirschhorn-Kan-Smith.

Although I don’t want to go into the details of all these different homotopical/higher-categorical subtleties, I will try to list some of the basic features that “homotopical” categories typically have.

• A homotopical category should behave like a nice category of topological spaces.
• In particular, there should be a class of morphisms called weak equivalences, and:
• To any homotopical category , one should be able to associate a “homotopy category” and a functor which is universal among functors sending weak equivalences to isomorphisms. Morally, is obtained from by “formally inverting the weak equivalences”.
• A homotopical category should admit all limits and colimits, and also homotopy limits and homotopy colimits.
• A homotopical category should be enriched over some kind of spaces, i.e. for any two objects , the set should be a “space” in some sense, for example a simplicial set, a topological space, or a chain complex of abelian groups.

Simplicial objects

Before talking about algebraic geometry, we need to recall some “simplicial language”. The category is defined as follows. Objects are the finite ordered sets of the form . Morphisms are order-preserving functions , i.e. functions such that . If is any category, we define the category of simplicial -objects to be the category in which the objects are the contravariant functors from to , and the morphisms are the natural transformations of functors. There is a functor from to given by sending an object of to the corresponding constant functor, i.e. the functor sending all objects to and all morphisms to the identity morphisms of .

Some examples:

• Take ,  the category of sets. The above construction gives us the category  of simplicial sets. This category is “sort of the same as the category of topological spaces”. The precise statement is that there is a pair of adjoint functors which make  and into Quillen equivalent model categories; in particular, their homotopy categories are equivalent (as categories). For the purposes of algebraic topology, we can work with any of these categories. For example, we can define homotopy groups and various generalized homology and cohomology groups of a simplicial set. The inclusion of into corresponds to viewing a set as a discrete topological space. A weak equivalence between two simplicial sets is a morphism inducing isomorphisms on all homotopy groups.
• Take , the category of abelian groups. There is a forgetful functor from to the category , induced by the forgetful functor from to. The Dold-Kan correspondence tells us that there is an equivalence between and the category of (non-negatively graded) chain complexes of abelian groups. Under this equivalence, homotopy groups of a simplicial abelian group correspond to homology groups of a chain complex.
• Take, the category of-algebras for a commutative ring. Then there is some kind of Dold-Kan correspondence between simplicial algebras and DG-algebras. See Schwede-Shipley for precise statements.
• Take , the category of sheaves of sets on some topological space or site. Then is the category of simplicial sheaves. This category can also be viewed as the category of sheaves of simplicial sets on the site. Any category of simplicial sheaves is a “homotopical category” (I am not making this precise here). For example, one way of defining weak equivalences is to say that a morphism of simplicial sheaves is a weak equivalence iff it induces weak equivalences of simplicial sets on all stalks. 

Homotopical categories in algebraic geometry

Now to algebraic geometry. Through a few examples I want to argue that homotopical categories (in particular categories of simplicial sheaves) provide a useful and natural setting for certain aspects of algebraic geometry.

Firstly, let’s consider the general problem of viewing a cohomology theory as a representable functor. In algebraic topology, the Brown representability theorem says that any generalized cohomology group is representable, when viewed as a functor on the homotopy category of topological spaces. In other words, there is a space such that the cohomology of a space is given by , where the is taken in the homotopy category. Examples include the Eilenberg-MacLane spaces , which represent the singular cohomology groups , and the space , which represents K-theory. The existence of a long exact sequence relating the cohomology groups for various corresponds to the fact that the different Eilenberg-MacLane spaces fit together to form a so called spectrum. The Brown representability theorem is best expressed using the language of spectra, i.e. stable homotopy theory, but I want to postpone a discussion of this to a future post. An interesting aspect of Brown representability for singular cohomology is that by identifying the coefficient group with the corresponding Eilenberg-MacLane space, the two arguments of a singular cohomology group , namely the space and the coefficient group , suddenly are on equal footing. By this I mean that they both live in the same category of topological spaces, rather than in the two separate worlds of topological spaces and abelian groups, respectively.  

In classical algebraic geometry, there is no analogue of Brown representability. Most cohomology theories are of the form , where is some kind of variety, and is a sheaf of abelian groups. One may ask if there is a way to express such a cohomology group as a representable functor. In order to obtain a picture parallell to the topological picture above, a necessary requirement is to have a homotopical category in which the variety and the sheaf both live as objects, “on equal footing”. One possibility for such a category is some category of simplicial sheaves. In order to explain how this works, let us fix some category of varieties, for example the category smooth varieties over some base field . Let us also fix some Grothendieck topology on this category, for example the Zariski topology, the Nisnevich topology, the etale topology, or some flat topology. This defines a site, and we can speak of sheaves on this site, i.e. contravariant functors on , satisfying a “glueing” or “descent” condition with respect to the given topology.

Since Grothendieck, we are familiar with the idea of identifying a variety with the sheaf of sets that it represents, by the Yoneda embedding. We mentioned earlier that for any category , there is a functor . Taking to be the category of sheaves of sets, we get a functor from sheaves of sets to simplicial sheaves. In particular, any variety can be viewed as a simplicial sheaf, by composing the Yoneda embedding with the canonical functor from sheaves of sets to simplicial sheaves. 

We also want to show that a sheaf of abelian groups can be viewed as a simplicial sheaf. We can regard any abelian group as a chain complex, by placing it in degree zero, and placing the zero group in all other degrees. This gives an embedding of the category of abelian groups into the category of chain complexes, and by composing with the Dold-Kan equivalence we get a functor from abelian groups to simplicial sets. This induces a functor from sheaves of abelian groups to simplicial sheaves. More generally, any complex of sheaves of abelian groups can be viewed as a simplicial sheaf.

Now one could hope for an analogue of Brown representability, namely that the sheaf cohomology group could be expressed as , where the Hom is taken in the homotopy category of simplicial sheaves. It seems to be the case that something along these lines should be true. For example, this nLab page on cohomology seems to imply that all forms of cohomology should be of this form, at least sheaf cohomology groups of the type just described. Also, Hornbostel has proved a Brown representability theorem in the setting of motivic homotopy theory.

There are many other phenomena in algebraic geometry which also seem to indicate that categories of simplicial sheaves might be more natural to study than the smaller categories of schemes and varieties we typically consider. Some examples (longer explanations of these will have to wait until future posts):

• It seems to be the case that almost any geometric object generalizing the concept of a variety can be thought of as a simplicial sheaf. Examples: Simplicial varieties, stacks, algebraic spaces.
• Deligne’s groundbreaking work on Hodge theory in the 70s (see Hodge II and Hodge III) uses in a crucial way that the singular cohomology of a complex variety can be defined on the larger category of simplicial varieties. Simplicial varieties are special cases of simplicial sheaves, and I believe it should be true that functors on simplicial varieties can be extended to simplicial sheaves.
• Simplicial varieties/schemes also pop up naturally in other settings. For example, Huber and Kings need K-theory of simplicial schemes for their work on the motivic polylogarithm.
• As already indicated, simplicial sheaves appears to be the most natural domain of definition for many different kinds of cohomology theories.
• Morel and Voevodsky’s  A1-homotopy theory (also known as motivic homotopy theory) is based on categories of simplicial sheaves for the Nisnevich topology.
• Brown showed that Quillen’s algebraic K-theory can be thought of as “generalized sheaf cohomology”, where the coefficients is no longer a sheaf of abelian groups, but a simplicial sheaf.
• The work of Thomason relating algebraic K-theory and etale cohomology uses the language of simplicial sheaves.
• Simplicial sheaves provide a natural language for “resolutions”. For example, it gives a unified picture of the two methods for computing sheaf cohomology: Cech cohomology and injective resolutions.
• Simplicial sheaves seems to be the most natural language for descent theory.
• Toen’s work on higher stacks can be formulated in terms of simplicial sheaves. 
• Homotopy categories of simplicial sheaves can be thought of a generalization of the more classical derived categories of sheaves. The homotopical point of view seems to clarify some unpleasant aspects of the classical theory of triangulated categories. 

See also the nLab entry on motivation for sheaves, cohomology, and higher stacks.

Questions

I hope to come back to many of these examples in detail. For now, I  just want to list a few questions which I find intriguing. 

• To define a category of simplicial sheaves, we must choose a Grothendieck topology. How does this choice affect the properties of the category we obtain? Morel and Voevodsky work with the Nisnevich topology, Huber and Kings work with the Zariski topology, and Toen (at least sometimes) works with some flat topology. For some purposes, it seems to be the case that we don’t need a topology at all, instead we can just work with simplicial presheaves. What is the role of the Grothendieck topology? 
• Most of the above examples are developed for varieties over a base field of characteristic zero. Based on the above, it seems reasonable to believe that simplicial sheaves are useful in this case, but what if the base scheme is field of characteristic p, a local ring, a Dedekind domain, or something even more general? Is it the case that simplicial sheaves is the most natural language for understanding cohomology theories for arithmetic schemes, such as schemes which are flat and of finite type over ? Are simplicial sheaves important in number theory/Arakelov theory/geometry over the field with one element? What are the obstacles to “doing homotopy theory over an arithmetic base”?

Obviously I hope that there will be interesting answers to these questions, but I am still completely in the dark as to what these answers might be.

Wolfram Alpha launched next week?

A friend showed me this today, and I just find it so cool: Check out the demo video for Wolfram Alpha, a new “computational knowledge engine”. It is apparently being launched very soon – the webpage says “May” and some other sources say May 18th. It can do all sorts of things, like compute integrals, tell you stuff about the human genome, or tell you precisely where the International Space Station is at any given time (by solving the relevant differential equation). Although most of the answers you get would have been available through Google as well, Wolfram Alpha works in a different way, exhibiting some rudimentary form of “intelligence”. Am not sure just how much maths it can actually do, but I’ll try it out as soon as possible.

Lectures next week in Lisbon on Kervaire Invariant One

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.

The Kervaire Invariant One problem solved

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.

Local systems and some cohomology theories at SBS

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.

What is geometry?

(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 , 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, -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 (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  and hence to . 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, -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.

Doing mathematics online

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?

Videos from the Grothendieck conference

A few months ago I posted some notes from the talks at the Grothendieck conference (Maltsiniotis, Bloch, Katz, Toen). Now the IHES has generously made all the talks available online here. The list of all talks can be found at the conference webpage.

The Chow ring

Matt DeLand recently posted a very nice introduction to the Chow ring and Chern classes on Rigorous Trivialities. We will need this material when talking more about Weil cohomology and pure motives, and I will simply refer to his post, and to this introduction by Gillet, for all things related to Chow groups/rings.