# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $W$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### “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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 - ...