All Questions

8
votes
2answers
536 views

Potential connected non-Lie subgroup

This painful question is inspired by the question "non-Lie subgroups" . Let R denote the real numbers. Let f be an discontinuous additive map from R --> R. Is it possible that the graph of f, inside ...
13
votes
7answers
7k views

Are there good programs to create mathematical pictures in svg format?

I've recently started my personal wiki to organize my notes and thoughts. I use the wiki program instiki which I believe is the same as the n-lab uses. Instiki can upload svg's. I want to be able to ...
12
votes
3answers
1k views

Hopf algebra structure on the universal enveloping algebra of a Leibniz algebra?

A Leibniz algebra L may be thought of as a noncommutative generalisation of a Lie algebra. One drops the requirement that the bracket be alternating and substitutes the Jacobi identity for the ...
25
votes
6answers
4k views

What's a reasonable category that is not locally small?

Recall that a category C is small if the class of its morphisms is a set; otherwise, it is large. One of many examples of a large category is Set, for Russell's paradox reasons. A category C is ...
20
votes
1answer
15k views

How hard is it to compute the Euler totient function?

Are there any efficient algorithms for computing the Euler totient function? (It's easy if you can factor, but factoring is hard.) Is it the case that computing this is as hard as factoring? EDIT: ...
2
votes
3answers
557 views

Variant of binomial coefficients

I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before. If so, I'd love a reference, a name, etc. First recall the following. If z is a ...
19
votes
3answers
2k views

Which rings are subrings of matrix rings?

In this question, all rings are commutative with a $1$, unless we explicitly say so, and all morphisms of rings send $1$ to $1$. Let $A$ be a Noetherian local integral domain. Let $T$ be a non-zero $...
43
votes
6answers
5k views

Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (...
2
votes
3answers
340 views

Multiplication of (0,1) matrices

is there an obvious lattice path counting interpretation for multiplying n by n (0,1) matrices ?
9
votes
3answers
1k views

Is there a stable algorithm for polynomial division (in several variables)?

Suppose you have a homogeneous ideal $I$ inside the algebra $\mathbb{C}[x_1,...,x_d]$ of complex polynomials in $d$-variables. Can one find a basis for $I$, say $\{f_1,...,f_k\}$, such that every $h \...
7
votes
3answers
466 views

internal homs and adjunctions?

This is probably an easy question. Let C be a category with (finite) products. An internal hom in C category is an object uhom(X, Z) which represents the functor: Y |-----> hom(Y x X, Z) here "...
36
votes
17answers
6k views

Canonical examples of algebraic structures

Please list some examples of common examples of algebraic structures. I was thinking answers of the following form. "When I read about a [insert structure here], I immediately think of [example]." ...
23
votes
2answers
2k views

When is a locally convex topological vector space normal or paracompact?

All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...
13
votes
7answers
3k views

Is no proof based on “tertium non datur” sufficient any more after Gödel?

There are many proofs based on a "tertium non datur"-approach (e.g. prove that there exist two irrational numbers a and b such that a^b is rational). But according to Gödel's First Incompleteness ...
42
votes
11answers
7k views

japanese/chinese for mathematicians?

I'd like to learn to read math articles in Japanese or Chinese, but I am not interested in learning these languages from usual textbooks. Exist suitable texts, specialized for the needs for reading ...
8
votes
5answers
803 views

Subgroups of a group generated by a free semigroup

Let $F$ be a free semigroup (say, $2$-generated) which is embedded in a group $G$, and suppose that $G$ (as a group) is generated by $F$. The most simple such situation would be when $G$ is a free ...
3
votes
2answers
671 views

Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ansätze do you know ...
3
votes
1answer
233 views

limits of algebraic varieties

I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject). For the kind of example I have in mind, ...
5
votes
1answer
447 views

Finiteness of Obstruction to a Local-Global Principle

Say that a projective variety V over Q satisfies the local-global principle up to finite obstruction (#) if there are only finitely many isomorphism classes of projective varieties over Q that are not ...
8
votes
1answer
582 views

Controlling Ultrapowers

Say I start with some a transitive model of a large fragment of ZFC (say enough to run Łoś' Theorem externally) and a specific set x∈M. Now let's say I'm going to pick some M-ultrafilter U on x. ...
13
votes
5answers
985 views

What kind of geometric operations “scale up” cohomology?

There's an obvious operation on the category of graded rings, given by "scaling up," multiplying the grading of every element by some fixed constant. Does anyone know of an operation on the level of ...
37
votes
8answers
4k views

Does any method of summing divergent series work on the harmonic series?

It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series,...
17
votes
4answers
1k views

What are the Benefits of Using Algebraic Spaces over Schemes?

I have heard that algebraic spaces have better formal properties than schemes. What are these benefits? Also, is there a natural way to go straight from affine schemes to algebraic spaces bypassing ...
12
votes
2answers
1k views

Is the fixed locus of a group action always a scheme?

Suppose G is an algebraic group with an action G×X→X on a scheme. Does the fixed locus (the set of points x∈X fixed by all of G) have a scheme structure? You can obviously define the ...
21
votes
4answers
3k views

Are proper linear subspaces of Banach spaces always meager?

Let X be a Banach space, and let Y be a proper non-meager linear subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-...
96
votes
11answers
22k views

“Philosophical” meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
1
vote
1answer
622 views

Quantifying Aggregate Vector Strength/Vector Arithmatic

Say I have 5 vectors and I measure the similarity of each one to a fixed reference vector using cosine similarity. But now what I want to do is understand the aggregate or collective strength of these ...
13
votes
5answers
3k views

Presheaves as limits of representable functors?

If i remember correctly, i read that given a presheaf P:C^op -> Set it is possible to describe it as a limit of representable presheaves. Could someone give a description of the construction together ...
11
votes
3answers
2k views

Non-Lie Subgroups

A result of Borel and Lichnerowicz states that the holonomy group of a connection on a principal $G$-bundle is a Lie subgroup of $G$ (Cartan had earlier asserted this, but apparently without proof). ...
24
votes
4answers
3k views

(∞, 1)-categorical description of equivariant homotopy theory

I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
15
votes
5answers
2k views

Non-commutative geometry from von Neumann algebras?

The Gelfand transform gives an equivalence of categories from the category of unital, commutative $C^*$-algebras with unital $*$-homomorphisms to the category of compact Hausdorff spaces with ...
22
votes
3answers
3k views

How do I compare the different notions of Fourier transform for sheaves?

There is a close but not perfect relationship between algebraic D-modules on C^n, constructible sheaves on C^n in the analytic topology, and \ell-adic sheaves on an n-dimensional vector space over a ...
27
votes
63answers
16k views

What's your favorite equation, formula, identity or inequality? [closed]

Certain formulas I really enjoy looking at like the Euler-Maclaurin formula or the Leibniz integral rule. What's your favorite equation, formula, identity or inequality?
9
votes
3answers
911 views

When does Tannakian theory work over affine schemes besides fields?

By 'work' I would like the correspondence between fiber functors (to finitely generated projective modules) and algebraic groups to be the same as in the field case. Specifically, if $A$ is an affine ...
14
votes
2answers
515 views

Do orbits and stable loci of group actions have natural scheme structures?

Suppose G is an algebraic group with an action G×X→X on a scheme. Then many of the usual constructions you make when you talk about group actions on sets can be made scheme-theoretically. ...
7
votes
4answers
2k views

Prestacks and fibered categories

It seems to be a well-known fact that there is a ``one-to-one correspondence'' between prestacks and fibered categories. Here a prestack (called a pseudo-functor in SGA1) means a contravariant lax ...
16
votes
2answers
1k views

What's an example of an “adjunction up to adjunction”?

(There are several dialects of 2-categorical language. Mine is the one where everything is weak by default. You might need to insert the word "weak" or "pseudo" (or even "strong", if your default is ...
7
votes
2answers
671 views

Construction of maps S^3 -> S^2 with arbitrary Hopf invariant?

The well-known Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$ has explicit constructions involving the geometry of $C^2$ and intersections of complex lines with the $3$-sphere. They don't seem ...
14
votes
6answers
1k views

Can the “physical argument” for the existence of a solution to Dirichlet's problem be made into an actual proof?

Caveat: I don't really know anything about PDEs, so this question might not make sense. In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace ...
48
votes
3answers
3k views

What are the higher homotopy groups of Spec Z ?

The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
5
votes
5answers
1k views

Bound on cardinality of a union

Suppose I have n finite sets A1 through An contained in some fixed set S, and I am given non-negative integers N and N1 through Nn such that each Ai has cardinality N, and each k-tuple intersection ...
6
votes
1answer
700 views

simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
13
votes
15answers
1k views

Most helpful heuristic?

What's the most useful piece of mathematical "folk wisdom" you've encountered? I'm talking here about things that aren't theorems, or even conjectures, or even shadows of conjectures -- just broad ...
14
votes
3answers
1k views

How exactly is Hochschild homology a monad homology?

Many texts which praise the generality of the bar construction associated to a monad, say that Hochschild homology is an example of this. What exactly is in this case the underlying endofunctor of ...
5
votes
4answers
507 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 ...
9
votes
4answers
2k views

Constructing Affine Kac-Moody Groups

Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the ...
18
votes
5answers
1k views

Simplicial Model of Hopf Map?

The Hopf fibration is a famous map S3 --> S2 with fiber S1, which is the generator in pi_3(S2). We can model this map in terms simplicial sets by taking the singular simplicial sets of these spaces ...
26
votes
6answers
1k views

Is there a topology on growth rates of functions?

I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions (0,infty) \to (0,\infty), where two functions f_1,f_2 ...
108
votes
24answers
20k views

Tools for collaborative paper-writing

I personally use a revision control system (git) to manage my own paper writing, back things up, and synchronize between different machines. However, I've found most programmer's revision control ...
64
votes
15answers
22k views

The importance of EGA and SGA for “students of today”

That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French. ...

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