User andreas holmstrom - MathOverflow

Answer by Andreas Holmstrom for Status of Beilinson conjectures?

2013-04-06T16:12:59Z

The survey of Nekovar tells you what was known about the Beilinson conjectures in the early 90s. Other surveys/introductions from that time include Scholl-Deninger, Soulé, Ramakrishnan (in Contemporary mathematics 83), and the volume edited by Rapoport, Schappacher and Schneider (introduction here, all articles here). Since then, not a lot has happened I think. There is a fairly recent article of Otsubo with some results for Fermat curves (published version, arXiv version). There is a nice reformulation of the conjectures in terms of Arakelov motivic cohomology, by Jakob Scholbach (see articles on his webpage). You might also want to check out some of the articles of Rob de Jeu and his coauthors. In addition, there have been various attempts at new descriptions of the Beilinson regulator, most recently by Bunke and Tamme (their work will by the way be the topic of a summer school in Freiburg in July), but this does not in itself imply any progress on the Beilinson conjectures themselves.

If you are interested in recent progress on special values in general, there are other areas where more exciting things are happening. Search for work on the Birch and Swinnerton-Dyer conjecture, the Equivariant Tamagawa number conjecture, and Weil-etale cohomology (starting point for Weil-etale: webpage of Baptiste Morin).

Handling arXiv feeds to avoid duplicates

2009-10-15T23:51:56Z

I subscribe to feeds from the arXiv Front for a number of subject areas, using Google Reader. This is great, but there is one problem: when a new preprint is listed in several subject categories, it gets listed in several feeds, which means I have to spend more time reading through the lists of new items, and due to my slightly dysfunctional memory, I often download the same preprint twice. Is there a way to get around this problem, by somehow merging the feeds, using a different arXiv site, or using some other clever trick?

(Hope this is not too off-topic, I think a good answer could be useful to a number of mathematicians. Also, I would like to tag this "arxiv" but am not allowed to add new tags.)

Answer by Andreas Holmstrom for Fubini theorem for hocolim.

2012-10-26T09:45:59Z

You might also be interested in chapter III of the paper Homotopy theory of diagrams, by Chacholski and Scherer.

Answer by Andreas Holmstrom for Background Reading for Proving Irrationality of Real Numbers

2012-10-26T09:34:44Z

You might be interested in the concept called periods. Quid already mentioned the work of Waldschmidt, but see also Wikipedia, and in particular the link at the bottom to the article of Kontsevich and Zagier.

Answer by Andreas Holmstrom for Power series expansions of $L$-series

2012-10-09T19:34:55Z

There is a related conjecture for the L-function attached to certain Galois representations, due to Colmez and stated in the beautiful survey paper Periods by Kontsevich and Zagier (see section 3.6). This survey paper refers to an Annals paper of Colmez: Périodes des variétés abéliennes à multiplication complexe (Jstor link), and to some papers of Hiroyuki Yoshida.

There is also a book by Hiroyuki Yoshida called Absolute CM-periods, where some of this might be discussed.

I have no idea whether this is related or not to the Rössler-Maillot/Kudla story mentioned above in the comments.

The conjecture as stated by Kontsevich and Zagier is for a Galois representation sending complex conjugation to minus the identity matrix. This is a rather restrictive condition, and Kontsevich-Zagier also state (without explanation) that "in general, one does not expect any interesting number-theoretic property for subleading coefficients".

Finiteness of stable homotopy groups of spheres

2012-08-26T21:33:13Z

Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or when $n$ is even and $k=2n-1$ (in which case the group is the direct sum of $\mathbb{Z}$ and a finite group). As a consequence, the stable homotopy groups $\pi_k^s$ are finite groups for $k>0$, and $\pi_0^s \cong \mathbb{Z}$.

The work of Serre was done before anyone knew about stable homotopy theory and chromatic methods, and this makes me wonder about the following questions.

Question 1: Is it possible to use methods from stable/chromatic homotopy theory to prove finiteness of stable homotopy groups of spheres directly, without having to compute any unstable homotopy groups of spheres?

Question 2: Is there any philosophical or conceptual reason for why these groups should be finite?

Answer by Andreas Holmstrom for math circles video lectures for school children?

2012-05-16T11:20:19Z

You might want to check out the Khan Academy. They have a lot of videos on all levels of school mathematics.

Answer by Andreas Holmstrom for Number fields with same zeta function?

2011-12-30T13:47:01Z

Coming very late to the party, here is a small complement to Alex's excellent answer. There is a recent paper of Marcolli and Cornelissen (arXiv link) which among other things discusses this question. The following two points give partial answers to the question posed here:

If two number fields are Galois over $\mathbb{Q}$ and have the same zeta function, then they are isomorphic.
In general, one can say something similar if one is willing to consider all twists of the zeta function by Dirichlet characters. More precisely, assume that there is an isomorphism between the Pontryagin duals of the abelianized Galois groups of the two number fields. Assume also that whenever two characters are identified under this isomorphism, the corresponding twisted zeta functions agree. Then the two number fields are isomorphic. See Theorem 2 in the paper linked above for more details.

The introduction of the paper actually gives a rather good overview over criteria which can or cannot determine whether two number fields are isomorphic.

Is the free R-module on a sheaf of sets still a sheaf?

2011-11-14T16:30:15Z

Let $L$ be a sheaf of sets on some site $S$. Let $F$ be the presheaf obtained by composing $L$ with the free R-module functor, i.e. for any object $U$, we define $F(U)$ to be the free $R$-module on the set $L(U)$. Is $F$ a sheaf?

Answer by Andreas Holmstrom for A learning roadmap for algebraic geometry

2011-10-18T14:12:20Z

There are a few great pieces of exposition by Dieudonné that I really like. The first two together form an introduction to (or survey of) Grothendieck's EGA. The second is more of a historical survey of the long road leading up to the theory of schemes. I am sure all of these are available online, but maybe not so easy to find.

Algebraic geometry ("The Maryland Lectures", in English), MR0150140
Fondements de la géométrie algébrique moderne (in French), MR0246883
The historical development of algebraic geometry (available here or here)

Answer by Andreas Holmstrom for Lecture notes on representations of finite groups

2009-11-21T21:22:38Z

Some material from the undergrad rep theory course in Cambridge: Example sheets, A recent set of notes (by Martin), and a less recent (but very nice) set of notes (by Teleman).

A question on K_1 of an elliptic curve

2011-04-15T18:32:21Z

Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps $$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , \mathbb{R}(2) )$$ from (an Adams eigenspace of) K-theory (with rational coefficients) to Deligne cohomology of $E$. Call the first map $\iota$ and the second map $r$. Note that this map does NOT lie in the index range where the Beilinson conjectures predicts that $r$ is an isomorphism on the image of $\iota$ after tensoring with $\mathbb{R}$. Now, is anything known at all about $r$ or $r \circ \iota$, for elliptic curves in general or for some specific curve/class of curves? Unless I am mistaken, the Deligne cohomology group in question is always a one-dimensional real vector space. My main question is the following:

After tensoring everything with $\mathbb{R}$, is the the map $r \circ \iota$ zero or surjective???

I would also be interested in the following questions:

Is anything known about the two K-groups here? Finite generation? Rank? Can you write down a nonzero element?
Is the map $\iota$ injective? (This could be asked in much more generality for K-groups of regular models.)

I'd be grateful for any hints, even those based on unproven conjectures.

EDIT: Maybe one can approach this question from another point of view. I am quite sure that the following is true (have to check though). The cokernel of $r \circ \iota$ can be identified with the Gillet-Soulé arithmetic Chow group $\widehat{CH}^2(\mathcal{E}) \otimes \mathbb{R}$. Furthermore, this group is generated by arithmetic cycles of the form $(Z,g) = (0,\alpha)$, where $\alpha$ is a real harmonic $(1,1)$-form on the complex torus $E(\mathbb{C})$. So the question becomes: Do all arithmetic cycles of this form lie in the group generated by arithmetic cycles of the forms $(div(f), - \log \| f \|^2)$ and $(0, \partial u + \bar{\partial} v)$?

Zero-cycles on an arithmetic surface

2011-04-14T23:50:36Z

Could anyone give a reference for the following statement, which I believe is true.

"Let X be a regular scheme, flat over $Spec( \mathbb{Z}) $, with fiber dimension $1$. Then the Chow group $CH^2(X)$ is finite."

Beilinson's height pairing vs Neron-Tate

2011-04-14T01:46:48Z

In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arithmetic intersection theory, and this note of Müller-Stach). The height pairing is supposed to generalize the Neron-Tate height pairing. Is there a precise statement in the literature where Beilinson's height pairing for an elliptic curve over $\mathbb{Q}$ is compared to the Neron-Tate pairing?

Describing the kernel of the exponential map as a homology group

2011-03-30T19:12:04Z

I am reading Deligne: Hodge III, and am puzzled by a certain statement in section 10. If anyone could give a reference or a hint for how to prove this, I would be grateful. Maybe it is obvious and I just don't see why.

We consider an extension $G$ of an abelian variety by a torus. Then Deligne claims that the kernel of the exponential map $Lie(G) \to G$ can be identified with $H_1(G, \mathbb{Z} )$. Why is this true?

Answer by Andreas Holmstrom for Books you would like to read (if somebody would just write them...)

2011-02-05T02:59:39Z

Introduction to algebraic cycles.

With lots of examples...

Answer by Andreas Holmstrom for Reference: Learning noncommutative geometry and C^* algebras

2011-02-05T02:55:33Z

A while ago Lieven Le Bruyn put together a list called "Top 10 noncommutative geometry books for newbies", with a short description/review of each book. Check this link.

References for syntomic cohomology

2009-10-16T15:28:23Z

Could anyone point to good readable references for learning about syntomic cohomology?

Dirichlet's regulator vs Beilinson's regulator

2010-08-06T22:35:32Z

Consider a number field $F$ with ring of integers $O_F$. The Beilinson regulator can in this particular setting be viewed as a map from $K_n(O_F)$ to a suitable real vector space. Here $n$ is any positive odd integer. For $n \geq 3$, there is also another regulator map, defined by Borel, and Burgos has proved that the Borel regulator (suitably normalized) is twice Beilinson's regulator. For $n=1$, the Borel regulator is not defined (as far as I understand, correct me if I'm wrong), but we do have the original and most basic example of a regulator, which is that of Dirichlet.

Question: Is there some form of comparison theorem between the Beilinson regulator and the Dirichlet regulator?

If a colimit of distinguished triangles exists, is it also a distinguished triangle?

2010-07-26T00:49:27Z

Consider the following situation in some triangulated category: We are given a collection of distinguished triangles $A_n \to B_n \to C_n \to A_n[1]$ indexed by the natural numbers, together with maps (in the obvious sense) from the n'th triangle to the (n+1)'st. If $A \to B \to C$ is the colimit of this system of triangles, is it also a distinguished triangle?

It would be really interesting to have a proof or a counterexample, or possible a proof depending on some additional hypotheses on the triangulated category in question.

Where are mathematics jobs advertised if not on mathjobs (e.g. in Europe and elsewhere)?

2009-10-17T22:55:40Z

My impression is that in the US, there is a canonical place for finding math jobs, namely mathjobs.org. For those of us who live and apply for jobs elsewhere, life is more complicated, and searching for advertised academic mathematics jobs for example in Europe can be a real hassle, with loads of different sites, different systems, and some jobs apparently advertised only on the web page of the hiring institution, or one some obscure mailing list.

So, where are academic math jobs advertised when they for some reason are not or cannot be on mathjobs.org? Of course I know of a few such places, but I am sure there must be many more.

All answers welcome, this would help me and probably many others.

Sequential colim vs sequential hocolim

2010-07-27T19:20:51Z

Suppose we have some homotopical setting in which we can speak of homotopy colimits. The setting I have in mind at the moment is that of a compactly generated triangulated category with a model, but we could look at any model category or some other homotopical context. Let $A_0 \to A_1 \to A_2 \to \cdots$ be a system of objects indexed by the natural numbers. My question is if there are theorems of the form "under such and such hypotheses, hocolim $(A_i)$ agrees with colim $(A_i)$".

Do homotopy colimits always commute with homotopy colimits?

2010-07-27T19:10:10Z

Do homotopy colimits commute with homotopy colimits? The setting I am thinking of is that of a triangulated category with a model, but it would be interesting to have more general answers as well. A good reference would also be appreciated.

Automatically extract a bibitem (not BibTeX!) from MathSciNet?

2010-06-16T21:38:20Z

I am writing a small thing in which I am required to use manually formatted \bibitem entries rather than a BibTeX file. This takes a lot of time, and I probably get some of the formatting wrong. Is there a way of producing such entries automatically, for example from MathSciNet or from the BibTeX items created by MathSciNet? For arXiv preprints, there is a great tool from the Courant Research Centre (here) which can produce BibTeX as well as bibitem entries, and there should be something similar for MathSciNet.

Answer by Andreas Holmstrom for Sources for Bibtex entries

2010-04-07T13:40:12Z

The Courant Centre has a useful webpage which produces nice BibTeX entries for arXiv preprints, if you input a paper ID or an author ID.

Random fact: The K-theory archive has a loooong BibTeX file on the K-theory literature available for download.

Answer by Andreas Holmstrom for Kunneth formula for motivic cohomology

2009-11-26T01:59:05Z

Very briefly, I believe the following is true: Motivic cohomology does not satisfy a Kunneth formula on the level of cohomology groups, but it does satisfy a kind of Kunneth formula on the level of some suitable derived category of sheaves. This should hold true in general for any Bloch-Ogus cohomology theory, I think.

Answer by Andreas Holmstrom for What is a cohomology theory (seriously)?

2009-11-25T03:00:05Z

I promised to write a longer answer, but I simply don't have time this week - sorry. What I wanted to point our was that although the idea that "every flavor of cohomology ever considered is nothing but the study of connected components in the hom-spaces of some (oo,1)-topos" is one of the most amazing ideas ever (imho), it is still not clear (at least to me) exactly how this works in all cases, even for abelian sheaf cohomology. For example, most people seem to believe that the right (oo,1)-topos for cohomology theories in algebraic geometry should be given by A1 (or "motivic") homotopy theory, but there is nothing in the literature about representability of p-adic cohomology theories such as rigid cohomology. I believe this might be because there is some technical problem, but I am not sure. There are also other issues and examples which are not clear (to me!).

The other thing I wanted to do was to clarify various pieces of terminology related to cohomology in algebraic geometry, for example, "generalized cohomology" means different things in different articles, and there are many different notions of "universal cohomology". Maybe I can expand on this later.

One small remark: Motivic cohomology is usually thought of as the universal Bloch-Ogus cohomology, while the universal Weil cohomology should probably be pure motives with respect to rational equivalence ("probably", because it depends on what exactly you mean by "universal" and "Weil cohomology"). The two notions are closely related though.

(Aside: The reason I am very busy this week is that I suddenly find myself writing job applications, after essentially solving my thesis problem last week, and one of the main reasons I could solve my thesis problem was that I applied Urs' unified point of view on cohomology in a new setting.)

Answer by Andreas Holmstrom for How to do Computations Using the Decomposition Theorem for Perverse Sheaves

2009-11-25T01:46:31Z

There is a winter school on the decomposition theorem in Freiburg, Germany, 22-26 Feb 2010. Link.

Answer by Andreas Holmstrom for global fibrations of simplicial sheaves

2009-11-15T18:53:08Z

For model structures on simplicial sheaves, there is a difference between the Joyal-Jardine approach and the Brown-Gersten approach. This is well explained in Voevodsky's preprint: Homotopy theory of simplicial presheaves in completely decomposable topologies, available here. Briefly, the Brown-Gersten approach does not work for arbitrary sites, but it works for a class of sites defined in Voevodsky's paper - this class includes Noetherian finite-dimensional spaces. When the B-G approach works, the resulting model structure has better finiteness properties than the Joyal-Jardine model structure, which on the other hand can be defined for simplicial (pre)sheaves on any site.

Examples and intuition for arithmetic schemes

2009-10-20T13:47:52Z

How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good references? What kind of intuition do people have for such schemes?

Comment by Andreas Holmstrom

2012-08-29T17:56:21Z

Thanks for all the comments and thanks to Peter May for an excellent answer! @Ryan: Sure, a proof is a proof, and Serre's work is amazing, but I still think it is meaningful to ask for conceptual reasons for believing something. Part of my motivation is that there are many unproven conjectures suggesting that this or that cohomological invariant should be finite (or finitely generated) and it would be interesting to hear the reasons (if any!) that experts believe such conjectures.

Comment by Andreas Holmstrom

2011-04-21T19:21:59Z

Thanks to both of you for the clarifications! To profilesdroxford5: It is possible that I in the future might want to use your construction (2) for an example in some paper, and if so I would like to give you due credit. If you prefer me to put your real name there, please just drop me an email.

Comment by Andreas Holmstrom

2011-04-19T15:15:19Z

Thanks, that is very helpful. However, although surely the cokernel you refer to is "big", it could still in principle be a very big torsion group, right? To your comment (1), the answer is yes, but is it not equivalent? Is anything known about whether the groups $K'_0(\mathcal{E}_p)^{(1)}$ can contain a non-torsion element?

Comment by Andreas Holmstrom

2011-04-16T21:59:12Z

Hej Daniel - bra fråga, men svårt att svara inom 600 tecken!! Vore för övrigt kul att ses nån gång, våra intressen är ju inte alltför olika. Är i Uppsala ibland.

Comment by Andreas Holmstrom

2010-08-26T22:04:05Z

Hi Urs, this may or may not be useful to you, but there is a proof of properness for a certain model structure on commutative monoids in symmetric spectra. See Hornbostel (arXiv:1005.4546, Thm 3.17), and the reference he gives to Shipley: A convenient model structure... (article available on her webpage). Maybe some idea used in this proof could also be useful in your setting.

Comment by Andreas Holmstrom

2010-08-07T15:05:30Z

Hi Robin, sure, both the Beilinson regulator and the Dirichlet regulator are defined on $K_1(O_F)$, but the former is defined in terms of Gillet's general theory of Chern classes, and the latter in terms of the explicit logarithm formula found in any algebraic number theory textbook, and the question is if both definitions agree, or if they differ by some constant factor.

Comment by Andreas Holmstrom

2010-07-27T19:24:02Z

Tom, I asked two new questions based on your reply, in the hope that more people will see the questions. I would still be very interested in what exactly was wrong and what was right in your original answer though.

Comment by Andreas Holmstrom

2010-07-27T00:33:57Z

Aha, I guess I never had to think of uncountable homotopy colimits so far ;-) and thanks, I will check the reference to May.

Comment by Andreas Holmstrom

2010-07-26T21:01:00Z

Actually I was not completely sure myself what I meant. I am trying to understand this for a specific application and wasn't sure which case I need. When writing the question I was thinking of a situation where the colimit happens to exist, but I also had in mind a homotopy colimit. Now I learnt from Tom that they coincide, which was a big surprise. About your last question: As far as I understand one can define sequential hocolim in any triangulated category, without assuming existence of a model, as a cone of the shift map on the direct sum of all terms, or something like that.

Comment by Andreas Holmstrom

2010-07-26T20:44:19Z

I am not sure what compactly generated actually means for a model category which is not stable (and in the latter case I interpret compactly generated as saying that the homotopy category is compactly generated as a triangulated category). Do you know of a more general definition?

Comment by Andreas Holmstrom

2010-07-26T18:36:12Z

That it an excellent answer. Do you have a reference for the statement that sequential colim equals hocolim in a compactly generated stable model category? Or is it easy to prove? Regarding triangulated categories not of the above form, I am quite sure they never occur in nature, but there are some artificial examples of Muro/Schwede/Strickland.

Comment by Andreas Holmstrom

2010-07-26T09:06:49Z

Maybe an argument in this spirit could work, but one question would be the following: Does homotopy colimits commute with homotopy colimits in general?

Comment by Andreas Holmstrom

2010-07-26T07:08:04Z

Yes, I'm interested in the motivic stable homotopy category over some base scheme. Most statements that hold in the topological setting should hold there as well.

Comment by Andreas Holmstrom

2010-06-16T23:15:47Z

Hi Matthew, the bbl file was very useful. Thanks a lot, and hope all is well with you. (and thanks to Willie as well!)

Comment by Andreas Holmstrom

2010-03-03T06:56:05Z

Have you tried asking Waldspurger himself for an electronic copy? His email address is here: people.math.jussieu.fr/~waldspur