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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 ...
5k views

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 ...
3k views

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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Consider the Fibonacci polynomials $$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j}$$ and the Lucas polynomials $$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ... 0answers 588 views 3-colorings of the unit distance graph of \Bbb R^3 Let \Gamma be the unit distance graph of \Bbb R^3: points (x,y) form an edge if |x,y|=1. Let (A,B,C,D) be a unit side rhombus in the plane, with a transcendental diagonal, e.g. A = ... 0answers 761 views Grothendieck's Period Conjecture and the missing p-adic Hodge Theories Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over \mathbb Q to \mathbb Q-vectors spaces. They come with the ... 0answers 511 views Why do H_4 and M_4 have the same virtual Euler characteristic? Here's a funny coincidence: The virtual (or "orbifold") Euler characteristic of \mathcal M_g is known by the work of Harer and Zagier: one has \chi(\mathcal M_g) = \zeta(1-2g)/(2-2g). Now ... 0answers 788 views Why are there so few quaternionic representations of simple groups? Having spent many hours looking through the Atlas of Finite Simple Groups while in Grad school, I recall being rather intrigued by the fact that among the sporadic groups, only one (McLaughlin as I ... 0answers 808 views Derivative of Class number of real quadratic fields Let \Delta be a fundamental quadratic discriminant, set N = |\Delta|, and define the Fekete polynomials$$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$Define$$ f_N(X) = ...
Suppose you are given two graphs $G$ and $H$ and are told that one of the following two situations occurs. Either they are isomorphic, or one of the graphs contains a Hamilton cycle and the other ...