12
votes
2answers
572 views

Realisation of the noncommutative torus as a universal $ C^{*} $-algebra

One of the most basic examples in noncommutative geometry is the so-called noncommutative torus, denoted here by $ \mathbb{T}_{\theta} $. As far as I know, there are several equivalent constructions ...
36
votes
1answer
2k views

There's something strange about $\sqrt{d\big(j(\tau)-1728\big)}$

Given the j-function $j(\tau)$, I was looking at, $$F(\tau) = \sqrt{d\big(j(\tau)-1728\big)}$$ which appears in Ramanujan-type pi formulas. Let $C_d$ be the prime factors of the constant term of the ...
5
votes
2answers
497 views

pullback of theta divisor

Let $C$ be a smooth curve and $J$ its Jacobian. Let $p$ be a point on $C$ and $j: C \to J$ be the map $x \mapsto x-p$. Let $\theta$ be the Theta divisor on $J$, i.e. the locus $\{ x_1 + \cdots + ...
7
votes
3answers
603 views

Is the Manickam-Miklós-Singhi Conjecture solved?

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...
2
votes
1answer
217 views

Does knot Floer homology detect knot genus in rational homology spheres?

My question is the following: Does knot Floer homology detect the genus of null-homologous knot in rational homology spheres? If the answer is yes, I would like to have a reference for the ...
28
votes
6answers
2k views

Are all polynomial inequalities deducible from the trivial inequality?

I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...
33
votes
8answers
4k views

Getting nervous refereeing a paper

I am refereeing my first paper and I'm quite excited! But inexperienced and I would like to ask an advice to the Maths Community of MO. Let me tell you that I have already read Refereeing a Paper, but ...
26
votes
14answers
24k views

Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...
0
votes
1answer
210 views

Reverse optimization of a minimum cost flow network

Given an undirected graph $(V,E)$, with $W$ as the weight of each edge, and a convex cost function $F(X)$, such as $|X|^k$ ($k>1$). The cost to send $x$ unit of flow through edge $e_i$ is defined ...
63
votes
5answers
5k views

What do epimorphisms of (commutative) rings look like?

(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, ...
10
votes
3answers
507 views

Are norms intrinsically $\mathbb{R}$-valued?

Another way of phrasing this: are there any viable definitions of something which is norm-like but whose range is in a linearly ordered rig (for example) rather than $\mathbb{R}$? I have searched a ...
26
votes
11answers
22k views

Why is the gradient normal?

This is a somewhat long discussion so bear with me. There is a theorem that I have always been curious about from an intuitive standpoint. This is an issue that has been glossed over in most ...
1
vote
1answer
344 views

Brownian motion of every point in the plane

Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable? What proportion of the plane does ...
21
votes
10answers
9k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
35
votes
7answers
3k views

Why should I care about Heegaard-Floer theory?

I would like to collect a list of applications of Heegaard-Floer theory. By applications, I don't mean things like "it can detect the unknot" or "it can detect knot genus". Algorithms for these ...
15
votes
8answers
8k views

Does the exponential function have a square root?

(asked by Nathaniel Hellerstein on the Q&A board at JMM) Is there a "half-exponential" function h(x) such that h(h(x))=ex? Is it unique? Is it analytic? Related question: Is there an invertible ...
5
votes
1answer
254 views

Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$: $0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
18
votes
5answers
846 views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
35
votes
9answers
5k views

Collection of equivalent forms of Riemann Hypothesis

This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include ...
65
votes
5answers
8k views

Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?

I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...
3
votes
4answers
237 views

Bruhat order and Schubert cycles

I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ ...
5
votes
3answers
571 views

Higher dimensional Bezout via Hilbert polynomials: a reference

For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...
27
votes
6answers
3k views

Why can't proofs have infinitely many steps?

I recently saw the proof of the finite axiom of choice from the ZF axioms. The basic idea of the proof is as follows (I'll cover the case where we're choosing from three sets, but the general idea is ...
1
vote
1answer
89 views

A semi-ampleness criterion for homogeneous bundles on homogeneous spaces?

Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a ...
3
votes
1answer
122 views

Rational normal curves on Grassmanians

Consider the Grassmanian $G(k,n)$ ($k\le \frac{n}{2}$) and take its Plucker embedding. Consider now the space of all normal rational curves of degree $k$, contained in the Plucker embedding of the ...
3
votes
3answers
688 views

Mock Theta Functions

I am studying about Mock modular forms and Mock theta functions. I wonder how Zwegers connected mock theta functions with Harmonic Maass Forms? I mean, what was the philosophy/idea of Mock Theta ...
17
votes
2answers
637 views

KK-theory as a stable infinity-category and KU Mod

The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it ...
17
votes
10answers
64k views

What are the qualities of a good (math) teacher? [closed]

In forming your answer you may treat the qualifier math or maths as optional, since part of the question is whether there is anything peculiar to the subject of mathematics that demands anything ...
2
votes
0answers
103 views

A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof? Statement. ...
9
votes
4answers
987 views

“Classical” consequences of Bezout's theorem in dimensions $>2$

By Classical I mean something that could have been found before 1900 (say). A well known consequence of Bezout's theorem for plane curves is Pascal's theorem ...
15
votes
2answers
847 views

Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups. Let $p$ be a prime, and let $\mathbb{F}_p$ be the field ...
53
votes
5answers
7k views

What is sheaf cohomology intuitively?

What is sheaf cohomology intuitively? For local systems it is ordinary cohomology with twisted coefficients. But what if the sheaf in question is far from being constant? Can one still understand ...
11
votes
6answers
3k views

Introduction to Floer Theory?

Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive ...
3
votes
1answer
207 views

Sufficient conditions for a divisor to be connected on a K3 surface

Let $X$ be a K3 surface and $D$ an effective divisor such that $h^0(D)\geq2$ and $h^1(D)=0$. Is this enough to show that $D$ is connected? Any reference would also be appreciated (I looked in ...
4
votes
1answer
220 views

Cocycle condition for equivariant sheaves

Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$ : $X_n := G^n \times X$, $s_0:X_0 ...
6
votes
2answers
363 views

How many Lagrangian submanifolds?

An $n$-dimensional submanifold $L$ of a symplectic manifold $(M^{2n}, \omega)$ is called Lagrangian if $\omega|_L = 0$. I want to get some feeling about how many Lagrangian submanifolds are. For each ...
15
votes
6answers
2k views

Why chain homotopy when there is no topology in the background?

Given two morphisms between chain complexes $f_\bullet,g_\bullet\colon\,C_\bullet\longrightarrow D_\bullet$, a chain homotopy between them is a sequence of maps $\psi_n\colon\,C_n\longrightarrow ...
28
votes
8answers
3k views

What is a metric space?

According to categorical lore, objects in a category are just a way of separating morphisms. The objects themselves are considered slightly disparagingly. In particular, if I can't distinguish ...
19
votes
2answers
1k views

Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis?

This is a very open-ended question, which may or may not have a perfect answer, and for which I have a few ideas but nothing like a clear picture. However, I guess it won't hurt to ask to see if ...
4
votes
3answers
242 views

Invariants in $S^n(S^k(\mathbb{C}^w)$

Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - ...
15
votes
1answer
933 views

Singmaster's conjecture

Has any work been done on Singmaster's conjecture since Singmaster's work? The conjecture says there is a finite upper bound on how many times a number other than 1 can occur as a binomial ...
14
votes
4answers
1k views

Are any good strategies known for Erdos-Turan conjecture on additive bases of order two?

The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture. Let $ B \subseteq {\mathbb ...
10
votes
1answer
1k views

Theorem of Bryant in higher dimensions

hallo, i have the following question. i read about Bryant's theorem which sais that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically ...
3
votes
0answers
329 views

Defining the cup product in Ext using a Kunneth formula

I want to make a Kunneth product of sorts on Ext. In particular, letting $C_*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represented by maps in ...
12
votes
2answers
438 views

Classifying spaces for enriched categories

Is there a standard construction of a classifying space $BC$ for a category $C$ which is enriched which takes into account the enrichment? This is of course vague... The simplest example I can think ...
4
votes
0answers
101 views

“Pseudo-random” subsets of additive bases

We say that a subset $B \subset \mathbb{N}$ is an (asymptotic) additive basis of order $k$ if the set $kB = B + \cdots + B = \mathbb{N} \setminus C$, where $C$ is a finite set of positive integers. ...
5
votes
1answer
153 views

rigidity of eigenvalues of circular ensemble

Given a circular unitary ensemble, with the following joint density: $p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$, is the following statement true? With ...
21
votes
3answers
2k views

When is a classification problem “wild”?

I hope someone can point me to a quick definition of the following terminology. I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...
24
votes
4answers
5k views

Integrability of derivatives

Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable? I ask for pedagogical reasons. Results in ...
6
votes
1answer
434 views

Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear?

If $K$ is an imaginary quadratic field, then the $\mathbf{Z}_p$-rank of $K$ is $2$, meaning that the Galois group of the compositum of all the $\mathbf{Z}_p$-extensions of $K$ in an algebraic closure ...

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