User ilya nikokoshev - MathOverflow

Proof of Bloch-Kato conjecture of K-theory?

2010-01-01T21:15:16Z

this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture, a proof of which was announced in 2009

What exactly is the K-theory conjecture of Bloch-Kato and has it been proven?

What are dessins d'enfants?

2009-10-22T18:31:24Z

There was an observation that any algebraic curve over Q can be rationally mapped to P^1 without three points and this led Grothendieck to define a special class of these mappings, called the Children's Drawings, or, in French, Dessins d'Enfants (his quote was something like "things as simple as the drawings...").

I'm not an expert in this field, so could somebody please write more about those dessins, and what things they are related to? What's their importance? How does the cartographic group act on these?

K3 over fields other than C?

2009-10-19T19:28:54Z

How to classify K3 surfaces over an arbitrary field k?

Explanation for E_8's torsion

2009-11-01T20:06:22Z

To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is rather straightforward, but the exceptional groups are more interesting.

Any simple compact Lie group, by means of Hopf algebra theory, has the rational homology of a product $$S^a \times S^b \times \dots \times S^z$$ where the numbers are called exponents. Other than that, their cohomology could also have torsion. Now the torsion for all groups is known:

Among classical groups, only 2-torsion is possible and only for $Spin(n)$
Exceptional groups can only have 2 and 3-torsion (most do), with the exception of:
$E_8$ which has 2-, 3-, and 5- torsion.

Well, this is bound to be related to $E_8$'s Coxeter number, which is 30, but are there any hints as to why? My reference would be math-ph/0212067 but it can't relate this to Coxeter number either.

For the reference, exponents are known to be related to Coxeter number, see Kostant, The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group (google search).

Is this an open problem? Maybe yes, but maybe it's been explained, so I'm posting it as it is for now.

Most interesting mathematics mistake?

2009-10-17T14:28:43Z

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove that Euclid's parallel axiom is really ~~necessary~~ unnecessary.

But I also think there are less famous mistakes worth hearing about. So, here's a question:

What's the most interesting mathematics mistake that you know of?

This question is community wiki, meaning neither the question nor the answers receive points (which are reserved for "hard" questions). So please post as much as you like (indeed please post one answer per post so that others can upvote the ones easier), vote a lot and vote freely.

(should there be a tag 'not-math-related' or similar?)

EDIT: There is a similar question which has been closed as a duplicate to this one, but which also garnered some new answers. It can be found here:

http://mathoverflow.net/questions/77425/failures-that-lead-eventually-to-new-mathematics

What is a TMF in topology?

2009-10-11T14:54:45Z

What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?

Non-simply-connected smooth proper scheme over Z?

2010-01-07T20:48:46Z

Source

This question came up in the discussion between Kevin Buzzard and Minhyong Kim in the comments to Smooth proper scheme over Z. It was 2 weeks ago, so I took the liberty of posting it as community wiki.

Question

Is there an example of smooth proper variety $X \to \mathop{\text{Spec}}\mathbb Z$ such that $\pi_1(X) \ne 0$?

About tags

We recently had other questions of the form "Example of ... with everywhere good reduction at $\mathbb Z$" (local-global, abelian varieties). I think it would be interesting to create a tag to group these. Thoughts?

Singular K3 -- mathematical meaning?

2009-11-09T21:54:35Z

There's a very interesting text by Cumrun Vafa called Geometric Physics.

Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration:

The appearance of the Dynkin structure for the K3 singularities appears mathematically as purely “accidental”. However this accident gets explained in this duality context: One identifies the singular K3 geometries with A-D-E singularities with the points on the heterotic side with enhanced A-D-E gauge symmetry...

I look at the pictures (p.15) and I have a very simple question:

Is this vanishing K3 obtainable as a vanishing/nearby cycle functor for the cohomology of the fibration?

If it is, I will finally have an example of the abovementioned functor. If not, how to describe this K3 from a math point of view? One possible way would be to vary Kahler parameters and get a true, finite-size K3. Other descriptions?

A single paper everyone should read?

2009-10-23T18:42:05Z

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.

Do you have such an example?

Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately.

And as usual let's do one per post and vote freely, vote a lot.

What's the "Yoga of Motives"?

2009-10-23T18:52:35Z

There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new algebraico-geometric conjectures just to formulate the definition of motives.

There are things that I know about motives on some level, e.g. I know what the Grothendieck ring of varieties is or, roughy, what are the ingredients of the definition of motives.

But, how would you explain the Grothendieck's yoga of motives? What is it referring to?

Beilinson conjectures

2009-10-26T22:50:11Z

Continuing an amazingly interesting chain of answers about motivic cohomology, I thought I should learn about the Beilinson conjectures, referred there.

I have found some references, and they seem to present the conjectures from different sides, e.g. there's the statement about vanishing but then there are also connections to motivic polylogarithms.

What I miss from these articles in a general picture that would allow us to start somewhere natural. So,

how would you describe an introduction into Beilinson conjectures in motivic homotopy?

Sorry for such a loaded question — I really don't know how to make it fit MathOverflow format better. One could theoreticlly post lost of specific questions on the topic, but to ask the right questions in this case you might need to know more than I do. Also, I know there are some technical developments, e.g. the language of derived stacks, and my hope would be that somebody could make a connection to these conjectures using some clear and suitable language.

Where stands functoriality in 2009?

2009-10-19T18:13:46Z

Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s.

There's a very interesting article by Langlands called Where stands functoriality today which describes the development of the subject from Langlands' point of view as of 1990s.

But if somebody was to write an overview of the current state of Langlands functoriality, what would it say?

Answer by Ilya Nikokoshev for Number of faithful representations of a finite group

2009-11-05T21:37:10Z

I haven't heard this question before, but I would approach it in a following way: you know some basic relations between irreducible representations of a groups — e.g. their # is the # of conjugacy classes — and let's say we've written the kernels of representations and their dimensions. The generating function ^(*) that counts all representations is $((1-x^{r_1})(1-x^{r_2})\dots (1-x^{r_k}))^{-1}$ and you're asked to subtract the representations that have nontrivial kernel. This combinatorial problem is partly tractable for a given group, though I don't see a nice closed formula.

The situation simplifies for Abelian groups, and the answer you provide is in the right direction. A semisimple representation of Abelian group is a sum of characters. (I misunderstood what you asked here a bit, but bear with me.) A single character could already be faithful: for example, for $\mathbb Z_2 \times \mathbb Z_3 = \mathbb Z_6$ the irreducible character that hits all 6-roots of unity is a faithful representation. And there are $\varphi(6) = \varphi(3)\cdot\varphi(2) = 2$ of those. Another example: $\mathbb Z_2\times \mathbb Z_2$. The same formula doesn't works here as $1\cdot 1 = 1$ but the corresponding representation isn't faithful. For two-dimensional case, you positively get two characters, but there is no reason to expect that there will be a splitting into a character of exactly first summand and exactly second summand. You should, however, be able to calculate it by the method below.

So the formula $\prod\varphi(m_i)$ will correctly describe the answer only for a case of group having only one invariant factor.

Here's how to make a complete computation for an $n$-dimensional space and the case of $\mathbb Z_6$. You need to take all representations of it which are neither representations of $\mathbb Z_2$ nor of $\mathbb Z_3$. Fortunately, this is easy to write using inclusion-exclusion principle: $(1 -x)^{-5} - (1-x)^{-1} - (1-x)^{-2} + 1$ and you can simplify this or just get the first three terms:
$1+5x+20x^2 + 5x^2 - 1 - x - x^2 - 1 - 2x - 3x^2 + 1 = 0 + 2x + 21x^2 + \dots$ The 2 here stands for 2 "simple" characters of $\mathbb Z_6$, while 21, if I made no mistake, is the number of 2-dimensional representations: 19 of them are of the type "character as above $\oplus$ anything" and 2 of the type "character of $\mathbb Z_2$ $\oplus$ character of $\mathbb Z_3$ ".

Now I think using the same principle you can actually write a generating function for your sequence with an inclusion-exclusion principle for any group G. Since a representation is faithful iff it doesn't have a kernel, you can enumerate all of your normal subgroups and then find your answer as $F_G(x) - F_{G/H_1}(x) - F_{G/H_2}(x) - \dots + F_{G/H_1\cup H_2}(x) + \dots $. Here the notation $F_G(x)$ is a generating function counting all representations and $H_1, H_2, \dots$ are all maximal normal subgroups.

Hope that helps.

Update: I understand now you're interested in a closed formula for a specific case. For example, the computation by the formula above for $\mathbb Z_2\times \mathbb Z_2$ gives $(3\cdot 4)/2 -1 -1 -1 = 3$. The normal subgroups of an Abelian group are not hard to write, so it would be plausible if this could be made into a good formula.

^(*) Let me know if you're not familiar with generating functions.

Cartographic group and flat stringy connection

2009-10-23T18:05:12Z

There's a literature about dessins d'enfants (including my previous question here), and one amazing thing about them is that absolute Galois group Gal Q acts on cartographic group which, I believe, is isomorphic to letters_2 = <<A, B>> (group, freely generated by two noncommuting letters).

The funny thing about the latter group is that there is a flat connection coming from string theory defined on its group algebra, C[letters_2], which I think has the name of Knizhnik-Zamolodchikov. So, it that latter connection somehow related to Galois group?

Answer by Ilya Nikokoshev for A reading list for topological quantum field theory?

2009-10-12T17:50:45Z

Try This Week's Finds in Mathematical Physics first. It has some classic references, e.g. Atiyah's book:

The Geometry and Physics of Knots, by Michael Atiyah, Cambridge U. Press, 1990.

(update: Atiyah has many books and to my knowledge any of them is worth a look)

Why do Todd classes appear in Grothendieck-Riemann-Roch formula?

2010-01-03T22:46:49Z

Suppose for some reason one would be expecting a formula of the kind

$$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$

valid in $H^*(Y)$ where

$f:X\to Y$ is a proper morphism with $X$ and $Y$ smooth and quasiprojective,
$\mathcal F\in D^b(X)$ is a bounded complex of coherent sheaves on $X$,
$f_!: D^b(X)\to D^b(Y)$ is the derived pushforward,
$\text{ch}:D^b(-)\to H^*(-)$ denotes the Chern character,
and $t_f$ is some cohomology class that depends only on $f$ but not $\mathcal F$.

According to the Grothendieck–Hirzebruch–Riemann–Roch theorem (did I get it right?) this formula is true with $t_f$ being the relative Todd class of $f$, defined as the Todd class of relative tangent bundle $T_f$.

So, let's play at "guessing" the $t_f$ pretending we didn't know GHRR ($t_f$ are not uniquely defined, so add conditions on $t_f$ in necessary).

Question. Expecting the formula of the above kind, how to find out that $t_f = \text{td}\, T_f$ ?

You don't have to show this choice works (that is, prove GHRR), but you have to show no other choice works. Also, let's not use Hirzebruch–Riemann–Roch: I'm curious exactly how and where Todd classes will appear.

What is the exact statement of "there are 27 lines on a cubic"?

2009-10-05T13:59:55Z

I think there was a theorem, like

every cubic hypersurface in $\mathbb P^3$ has 27 lines on it.

What is the exact statement and details?

How to write math well?

2009-10-19T17:52:24Z

Let's learn about writing good mathematical texts.

For some people it could be especially interesting to answer about writing texts on Math Overflow, though I personally feel like I've already mastered a certain level in writing online answers while being hopelessly behind the curve in writing papers. So,

What is your advice in writing good mathematical texts, online or offline?

Unitary representations of SL(2, R)

2009-10-27T19:54:15Z

I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group G being SL(2, R), can be completely described and that there is a discrete and continuous part of the spectrum of L^2(G).

How are those representations described?
Do all unitary representations come from L^2(G)?
How are those related to representation of compact SO(3, R)?
What happens in the flat limit between SL(2, R) and SO(3, R)?

Also, is it possible to answer the questions above simultaneously for all Lie groups, not just SL(2, R)?

Polynomial representing prime numbers

2009-12-27T00:20:43Z

Along the lines of Polynomial representing all nonnegative integers, but likely well-known question:

is there a polynomial $f \in \mathbb Q[x_1, \dots, x_n]$ such that $f(\mathbb Z\times\mathbb Z\times\dots\times\mathbb Z) = P$ , the set of primes?

What is the base change in number theory?

2009-10-12T18:29:45Z

I'm somewhat familiar with base change in scheme theory: sometimes a property of a morphism X \to Y survives a base change f:Z\to Y, meaning that X \times_{Y} Z \to Z also has this property.

Question: Is the base change in number theory and derived algebraic geometry the same thing as above? What would be the examples?

Why no abelian varieties over Z?

2010-01-05T22:59:34Z

Motivation

I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form

the set $\{$ objects $\dots$ over field $K$ with good reduction everywhere except set $S$ $\}$ is finite/empty

One interesting thing he mentions is about abelian schemes in the most natural case $K = \mathbb Q$, $S$ empty. I think according to the definition we have a trivial example of relative dimension 0.

Question

Why is the set of non-trivial abelian schemes over $\mathop{\text{Spec}}\mathbb Z$ empty?

Reference

This is proven in Il n'y a pas de variété abélienne sur Z by Fontaine, but I'm asking because: (1) Springer requires subscription, (2) there could be new ideas after 25 years, (3) the text is French and could be hard to read (4) this knowledge is worth disseminating.

How to understand character sheaves

2009-10-04T22:00:31Z

There's a well-known series of articles by Lusztig about Character Sheaves. They have important connections to many things in (geometric) representation theory, e.g. 0904.1247

How to understand these for a person with less than excellent representation theory background?

Bertrand postulate

2009-10-26T23:13:23Z

I believe there was an old conjecture that there's always a prime number between N and 2N.

What's the history and how is this proven is the easiest/elementary/deepest ways?

Answer by Ilya Nikokoshev for Applications of algebraic geometry over a field with one element

2010-05-04T04:17:22Z

I'm not aware of such a theorem, but you should take a look into absolute zeta-functions, which hope to prove a lot about zeta-functions — perhaps up to Riemann's Hypothesis.

Since I don't know much, I better refer you to another MO question:

What is the field with one element?

Answer by Ilya Nikokoshev for Homology of koszul complex is finitely generated?

2010-05-03T19:56:02Z

At least for the definition I know, it seems to be because all the groups in the Koszul complex are finitely generated as $A$-modules, being explicitely constructed as sums of tensor products of $M$!

Answer by Ilya Nikokoshev for Homology of algebraic varieties in Okounkov's paper on enumerating algebraic curves

2010-05-03T18:22:52Z

It appears that you want the "rough, introductory answers", so feel fine to ask more detailed questions if anything is unclear/potentially wrong. Also, I only have very general knowledge about enumerations!

It looks like the standard moduli stack of marked curves. There are quite a lot of references obtainable by searching Internet, arXiv and Math Overflow, or check out that Wikipedia page.
He refers to topological $H_2(X)$ of the space considered as a complex manifold. $H$ can be defined as the divisor of $\mathcal O(1)$, or simply the hyperplane section under that projective embedding. It so happens that the number he's interested in is an intersection number of $C$ and $H$; you could compute it algebro-geometrically as $C\cdot H$ or topologically, doesn't matter.
Because it's 1 infinite row above and 1 infinite column left; and then he makes use of the equality $1+1=2$. After I'll read the whole paper I'll probably know this should equal to the degree $d$ :)
I'm less sure about how he gets the 9, but one way to renormalize infinity here would be to consider a whole infinite row or column to have zero area. By that logic, we have an infinite row, then an infinite colun without a piece, then 10 pieces, which adds up as $0 + 0 - 1 + 10$.

Answer by Ilya Nikokoshev for How to compute irreducible representation of Lie algebra in the framework of BBD

2010-05-03T07:04:56Z

There could be different ways to give meaning to the phrase "explicit construction".

In an algebro-geometric sense, an expicit construction comes from more classical Borel-Weil-Bott theorem of which BDD is an abstract generalization. There's a number of proof in the literature, e.g the one by Jacob Lurie (on his home page).

According to the BWB, you can get the (finite-dimensional) representation by taking the global sections of one of the equivariant bundles $ \mathcal O(\lambda)$.

Another way to construct the representation would be to start with some simple $\mathfrak g$-modules and combine them to get your represenation. In this way, BDD helps by establishing a correspondence between simple equivariant D-modules and Verma modules. Therefore, the resolution for a bundle $\mathcal O(\lambda)$ corresponds to a construction in the category of $\mathfrak g$-modules, the one called Bernstein-Gelfand-Gelfand resolution, giving rise to Weyl character formula

As an example, the $\mathfrak{sl}_2$ modules correspond to equivariant D-modules on a $\mathbb P^1$, which has two cells. Therefore, a BGG resolution for an $\mathfrak{sl}_2$-module has two terms. Since a Verma module for $\mathfrak{sl}_2$ with an integer weight $\lambda$ has (I think) exactly one vector of each weight $\lambda' \le \lambda$, you can picture it as a ray on a weight lattice; the picture then becomes [segment] = [ray] - [ray].

What is a topos?

2009-10-04T22:20:36Z

According to Higher Topos Theory math/0608040 topos is

a category C which behaves like the category of sets, or (more generally) the category of sheaves of sets on a topological space.

Could one elaborate on that?

How to get product on cohomology using the K(G, n)?

2009-10-27T21:37:59Z

This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map K(G,n) x K(G,n) --> K(G,n) that endows cohomology with an additive structure.

Question: what's the most geometric way to show the existence of maps K(G,n) x K(G,m) --> K(G,n+m) that endow cohomology with multiplicative structure?

Comment by Ilya Nikokoshev

2010-05-05T19:31:38Z

Not really: you can't get all divisors that way. That's a fascinating story you can search for in courses and textbooks on something like "algebraic geometry" or "geometry of surfaces".

Comment by Ilya Nikokoshev

2010-05-04T04:13:27Z

That should be $X\cap H$ above.

Comment by Ilya Nikokoshev

2010-05-04T04:13:04Z

Every time you have $X$ in projective space, you have a divisor $X\cup H$ on $X$ where $H$ is a hyperplane... Take a look at any book on projective varieties.

Comment by Ilya Nikokoshev

2010-05-03T20:01:43Z

Thanks! I misread the question as being about finitely presented modules. As Robin Chapman correctly says, the question is stated needs some additional assumptions!

Comment by Ilya Nikokoshev

2010-05-03T19:58:59Z

I don't think you'll get anything this way that you couldn't get by Verma modules... although that, again, depends on what types of constructions you're looking for. Perhaps you should really split the question into different ones...

Comment by Ilya Nikokoshev

2010-05-03T18:03:31Z

There's no reason the paper couldn't be linked, let's just do that!

Comment by Ilya Nikokoshev

2010-04-25T06:01:10Z

I don't think it's fair to characterize Shannon's information theory as completely new. In fact, the idea of a qualitative matric of information was quite well-known under the name of enthropy in physics (statistical mechanics was developed in 19th-first half of 20th century). Shannon introduced that concept into math and formalized it; but he didn't pretend to invent the idea, and even used the same name!

Comment by Ilya Nikokoshev

2010-03-02T10:04:37Z

+1. Quite an amazing paper!

Comment by Ilya Nikokoshev

2010-03-02T09:52:06Z

That looks like an interesting theorem! Could you provide a link for those new to it, please?

Comment by Ilya Nikokoshev

2010-02-10T18:12:25Z

Actually, I first write this as a statement, but then decided to change to question, since I didn't check it carefully. Will think a bit more and post!

Comment by Ilya Nikokoshev

2010-02-04T23:34:33Z

It's true, but the question was stated in geometric terms, so the answer should also include identification of letters with cycles. You can't have this for free: if your map is that simple, the expressions for cycles must be a bit more involved then mine :)

Comment by Ilya Nikokoshev

2010-02-04T19:34:10Z

@norondion, please don't take the closure personally. If the linked answer is not sufficient for some reason (e.g. it only deals with curves), you should just provide more background in the question body.

Comment by Ilya Nikokoshev

2010-02-04T19:30:25Z

I vote to close as duplicate. I also changed the title of the question linked to by David to make it reflect the question/answer more accurately.

Comment by Ilya Nikokoshev

2010-02-04T19:27:18Z

@Ben, I retitled your question to reflect more general second question; it includes the first one as well. Feel free to revert!

Comment by Ilya Nikokoshev

2010-02-04T13:05:45Z

They discuss the "plus/minus problem" as well: We prove polylogarithmic (C(log n)d) upper and lower bounds for the number of hypercubes needed to tile a hypercuboid in either of the above senses. In more than two dimensions this is a huge improvement on anything previously known. It was not even known whether an n × 1 × 1 cuboid could be tiled with fewer than n cubes in the plus/minus sense. (page 3)