# All Questions

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### Ultrafilters and automorphisms of the complex field

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that ...
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### Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
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### Volumes of Sets of Constant Width in High Dimensions

Background The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...
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### Grothendieck's manuscript on topology

Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible ...
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614 views

### “Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
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### Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$ ...
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### Is there software to compute the cohomology of an affine variety?

I have some affine varieties whose cohomology (topological, with $\mathbb{C}$ coefficients) I would like to know. They are very nice, they are all of the form $\mathbb{A}^n \setminus \{ f=0 \}$ for ...
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### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following: The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
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### What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ...
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### Why do H_4 and M_4 have the same virtual Euler characteristic?

Here's a funny coincidence: The virtual (or "orbifold") Euler characteristic of $\mathcal M_g$ is known by the work of Harer and Zagier: one has $\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$. Now ...
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### Microlocal geometry - A theorem of Verdier

(1) In "Geometrie Microlocale", Verdier states the following theorem. Theorem: Let $E$ be a vector space and $F$ a constructible complex on $E$. Then for $\ell$ a linear form on $E$, we have a ...
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### A paper to the question, if the six dimensional sphere is a complex manifold

Hi, for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf Because I am not able to ...
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### Computer calculations in A_infinity categories?

Is there a good computer program for doing calculations in A-infinity categories? Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep ...
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In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers $\Bbb{N}$ is first order definable in $(\Bbb{Q}, ... 0answers 744 views ### Optimization problem arising from the study of zeta zeros Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ... 0answers 923 views ### Is S^2 x S^4 a complex manifold? As observed by Calabi a long time ago, the manifold$S^2\times S^4$admits an almost-complex structure (obtained by embedding it in$\mathbb{R}^7$and using the octonionic product), which however is ... 0answers 510 views ### Is there a Kan-Thurston theorem for fibrations ? Given a fibration$F \to X \to B$with all spaces path-connected. Is there a (discrete) group$G$with normal subgroup$H$such that $$H^\ast(BG;\mathcal{A}) = H^\ast(X;\mathcal{A})$$ ... 0answers 620 views ### Does a proof of Selberg's 3.2 inequality exist? A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that$$\left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ... 0answers 864 views ### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product$U\bar{U}$, with$U$a random unitary matrix and ... 0answers 430 views ### Smooth thickenings of non-smoothable manifolds It is known that any closed topological manifold is homotopy equivalent to an open smooth manifold. Question 1. What can be said about the smallest dimension of a smooth manifold that is homotopy ... 0answers 1k views ### Are there lots of integer homology three-spheres? The problem of counting combinatorial three-spheres with$N\$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and ...

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