# 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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### The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...
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Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$, and define the Fekete polynomials $$F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a.$$ Define $$f_N(X) = ... 0answers 1k views ### Is there software to compute the cohomology of an affine variety? I have some affine varieties whose cohomology (topological, with \mathbb{C} coefficients) I would like to know. They are very nice, they are all of the form \mathbb{A}^n \setminus \{ f=0 \} for ... 0answers 481 views ### Why do H_4 and M_4 have the same virtual Euler characteristic? Here's a funny coincidence: The virtual (or "orbifold") Euler characteristic of \mathcal M_g is known by the work of Harer and Zagier: one has \chi(\mathcal M_g) = \zeta(1-2g)/(2-2g). Now ... 0answers 381 views ### Class function counting solutions of equation in finite group: when is it a virtual character? Let w=w(x_1,\dots,x_n) be a word in a free group of rank n. Let G be a finite group. Then we may define a class function f=f_w of G by$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid ...
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 ...