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4 votes
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properly interpreting Pi_0 in the homotopy exact sequence

Define the lens space L(m,n) as the quotient of S2m+1 by the action of the cyclic group ℤn⊂S1⊂ℂ*. We can create the infinite lens space L(∞,n) by a telescoping construction ...
2 votes
1 answer
33k views

Pronunciation: Dijkstra [closed]

I know how to pronounce Dijkstra's name correctly (hear it here: http://en.wikipedia.org/wiki/Edsger_W._Dijkstra). But I'd like to know how people usually say his name. I've heard it in many ...
13 votes
5 answers
5k views

Examples and intuition for arithmetic schemes

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 ...
4 votes
1 answer
898 views

"Wick rotation" of tropical geometry

This question is related to my earlier, even more open-ended question on tropilcalization. I will give some background and ask my question at the end. On R, consider the family of commutative, ...
-3 votes
2 answers
559 views

what is logic without a proof system [closed]

Is there a proof for no proof ?
2 votes
1 answer
355 views

k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?

If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and ...
6 votes
7 answers
2k views

What can't be described by categories?

I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that ...
4 votes
3 answers
418 views

Is there a formula phi s.t. phi and not-phi have a stronger consistency?

Let Σ be an axiom system. Can there be a formula φ, s.t. Con(Σ) does not imply Con(Σ + φ) AND Con(Σ) does not imply Con(Σ + not φ) If yes, can you give me ...
5 votes
2 answers
832 views

Godel's 1st incompleteness theorem - clarification.

This should be a trivial question for people who know Gödel's 1st incompleteness theorem. I quote the statement the theorem from wikipedia: "Any effectively generated theory capable of expressing ...
2 votes
3 answers
349 views

Multiplication of (0,1) matrices

is there an obvious lattice path counting interpretation for multiplying n by n (0,1) matrices ?
5 votes
2 answers
686 views

Complexity of determining if two graphs have same cycle matroid?

Consider the following question: Input: Two graphs G1 and G2 Question: Is the cycle matroid M(G1) isomorphic to the cycle matroid M(G2) What is the complexity of this question? It is well known ...
9 votes
2 answers
840 views

How to distinguish between natural and unnatural equivalences of categories

Some equivalences of categories are constructed by explicitly giving a pair of functors that are inverses up to isomorphism. For example, the equivalence between CRing^op and affine schemes is given ...
4 votes
3 answers
768 views

Why do branches of math vary in proof styles and what category are different branches in?

Some branches of math seem to have reasoning which is more global. There is a lot of efficiency in the proofs because the reasoning transfers easily between proofs. For other branches of math, a lot ...
4 votes
3 answers
1k views

Conductors of non-abelian number fields?

Is there a definition out there of the notion of conductor of a non-abelian number field (i.e. a finite extension of Q whose Galois group is non-abelian)? If not, is there anyone you know of working ...
4 votes
3 answers
715 views

Is a holomorphic vector bundle on a projective variety locally trivial in the Zariski topology?

By the GAGA principle we know that a holomorphic vector bundle E->X is analitically isomorphic to an algebraic one, say F->X, and by definition F is locally trivial in the Zariski topology. But since ...
17 votes
4 answers
762 views

How many dimensions is it safe to get drunk in?

In Michael Lugo's blog post Variations on the drunken-bird theorem, and real-world sightings he wonders (without coming to a conclusion) what the maximum 'safe' number of dimensions to get drunk in ...
19 votes
9 answers
4k views

Learning new mathematics

While many of us have had the experience of learning mathematics informally by osmosis or more formally in classes, there are times when we have to sit down and systematically learn, without the ...
6 votes
1 answer
1k views

Specializations of Schur functions at consecutive integers

Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function. There exists a nice product formula for the principal specializations: sλ...
1 vote
2 answers
325 views

How to make commutative algebraic groups strongly dualizable?

Let's use the notation of [A=>B] for Hom(A, B). Take a 1-dimensional algebraic torus G<...
21 votes
1 answer
4k views

Equivalent forms of the Grand Riemann Hypothesis

I have long been curious about equivalent forms of the Riemann hypothesis for automorphic L-functions. In the case of the ordinary Riemann hypothesis, one gets a very good error term for the prime ...
3 votes
1 answer
270 views

L^p Idempotent multipliers in 2 dimensions

This question is suggested by that about L^p multipliers (and the answer by Michael Lacey in particular). Let E be a measurable set in the plane and XiE its characteristic function. We say E is an Lp ...
5 votes
5 answers
1k views

Quantum Computing Complexity?

After reading a recent post on Church's Thesis, I ran into Turing-Church's Strong Thesis, that may be potentially disproven by advances in Quantum Computing. Does anyone know of a good resource that ...
15 votes
3 answers
2k views

Injective proof about sizes of conjugacy classes in S_n

It's not hard to count the number of permutations in a given conjugacy class of Sn. In particular, the number of permutations in Sn whose cycle decomposition has ci i-cycles is n!/(Πi=1n ci!ici). ...
2 votes
0 answers
526 views

How much of math could be taught without using mathematical notation? [closed]

Given that mathematics is not about number, and that it is not even about the cryptic notation used to describe mathematical problems, how much of mathematics could be taught without reference to ...
2 votes
1 answer
752 views

Suprema of stochastic processes

Let X be a continuous stochastic process. I know that (t>s) P(|X(t) - X(s)|>δ) < |t-s|/δ Is it possible to say anything (e.g. estimate the decay of the tail) about Y=sup_{s \in [0,1]} |...
13 votes
3 answers
815 views

Constructing a degeneration (as a group scheme) of G_m to G_a

SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga. What is such an example?
19 votes
1 answer
1k views

Are Q-curves now known to be modular?

I really should know the answer to this, but I don't, so I'll ask here. A Q-curve is an elliptic curve E over Q-bar which is isogenous to all its Galois conjugates. A Q-curve is modular if it's ...
2 votes
1 answer
167 views

Triviality of the Hodge bundle for a special family of semistable curves

Let g,h be positive integers. Let E be an elliptic curve, C be a genus h curve, and D be a genus g-h-1 curve. Let c,d,e be points on (resp.) C,D, and E. Let f:CC --> E-e be the family whose fiber ...
2 votes
2 answers
626 views

What functions are not represented by their power series?

Some functions are not represented by their power series even when they are continuous and have all the necessary derivatives. What's the best characterization of these functions? Explanations at any ...
11 votes
2 answers
2k views

What is the geometric significance of Cartan's structure equations?

The Cartan structure equations for a connection and various associated 1-forms can be checked in a straightforward algebraic manner. But is there a geometric or global significance to the equations- ...
3 votes
1 answer
334 views

Prevalence of B-fields

I am wondering how B-fields, which are basic objects in Generalized Geometry, relate to the B-fields of Ben's question and the answers to it. In Generalized Geometry, the B-field is a (1,1)-form, and ...
2 votes
1 answer
276 views

Do subgroups respect the orbit-closure relation?

Suppose G is a Lie group (or algebraic group) acting on a manifold (or scheme) X, and H⊆G is a subgroup. Let x,y∈X be points such that x is in the closure of the orbit H⋅y (but not in H&...
3 votes
0 answers
804 views

Children's drawings and Seiberg-Witten curves

This physics (bear with me for a while) paper seems to say something about Gal \bar Q/Q: Children's Drawings From Seiberg-Witten Curves, hep-th/061108. Let's ...
6 votes
2 answers
606 views

What is new in MSC 2010?

Has anyone seen the new MSC 2010? I was browsing around and to my suprise there is another revision of MSC. Has anyone noticed any major changes in there? Do major journals already accept papers with ...
7 votes
2 answers
388 views

When does "splits" imply "cosplits"?

In the category of groups, there are lots of "exact sequences", e.g. 4 → H → 2, that neither split nor cosplit, where H is the eight-element group of quaternions, and lots of sequences like ...
6 votes
2 answers
849 views

Rosser's Algorithm - Musical Scales and Generalized Ternary Continued Fractions

Rosser's algorithm is typically invoked during discussions of equal temperament scales, and is a way to obtain good approximations for multiple irrational numbers simultaneously. Is there a nice, ...
3 votes
1 answer
408 views

How to do asymptotics for integrals?

What's a good way to find how fast the integral of a function is growing near a pole of the function? Here is what I mean on an example. Look at 1/z. If I want to find out how fast ∫0a 1/(z-&...
2 votes
2 answers
709 views

Minimal axiom system for a set of provable statements

I am not a mathematician, so forgive me if this question is trivial. The basic idea of my question is: For a given set of provable statements, can we find an axiom system with the smallest number of ...
13 votes
1 answer
655 views

Examples of algebraic stacks without coarse moduli space?

Keel-Mori's theorem says an algebraic stack with a finite diagonal over a scheme S has a coarse moduli space. What is an example of an algebraic stack without coarse moduli space?
25 votes
4 answers
7k views

Is the Fukaya category "defined"?

Sometimes people say that the Fukaya category is "not yet defined" in general. What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...
0 votes
1 answer
485 views

Understanding a lemma in "Loop Spaces and Langlands Parameters" article

First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction. This was actually forward-referring to ...
4 votes
2 answers
2k views

Pushforwards of Line Bundles and Stability

I recently finished reading this paper, and was wondering about a couple of things relating to theorem 1, which says that for any curve X there is a curve Y and f:Y->X such that pushforward is a ...
1 vote
1 answer
272 views

Character theory over integers

This question comes from my notes, heavily edited, thus slightly unusual structure. For Lie groups one can reformulate character theory as saying that C ⊗ K(...
8 votes
1 answer
794 views

Pseudorandom generators

Has there been any progress about constructing strong pseudorandom generators? I'm not an expert on this topic, basically everything I know is a definition of a pseudorandom generator, the idea that ...
5 votes
4 answers
666 views

Sections of a divisor on elliptic curve

I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete ...
2 votes
1 answer
978 views

What is the comultiplication of a matrix frobenius algebra?

One of the easiest examples I can think of for frobenius algebras is a plain ol' matrix algebra with tr : V → k as the co-unit (or equivalently, tr(a⋅b) as the frobenius form). This is ...
3 votes
4 answers
627 views

Has anyone studied the applications which map open sets to either open or closed sets?

Consider two topological spaces X,Y and a function f from X to Y. Are the following concepts already in use? How are they called? 1) f sends open subsets of X to either open or closed subsets of Y. ...
1 vote
1 answer
768 views

Edge-disjoint shortest paths

Let G be a simple connected graph. Let a, b, c, d be four distinct vertices of G. Is there a way to partition the above four vertices to two pairs, so that the two shortest paths between the ...
5 votes
1 answer
836 views

An inverse problem: Number fields attached to elliptic curves over Q

If I understand FC's remark under the post "Very strong multiplicity one for Hecke eigenforms," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/...
4 votes
1 answer
740 views

Eckmann-Hilton argument

The Eckmann-Hilton argument is used to prove that a doubly monoidal 0-category is a commutative monoid. If (x) is horizontal composition and . is vertical composition, and assuming that 1(x)a=a=a(x)1, ...

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