# All Questions

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### Why do polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

Conjecture. Let $P(x),Q(x) \in \mathbb{R}[x]$ be two monic polynomials with non-negative coefficients. If $R(x)=P(x)Q(x)$ is $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ ...
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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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### A = B (but not quite); 3-d arrays with multiple recurrences

Many years ago, I discovered the remarkable array (apparently originally discovered by Ramanujan) 1 1 3 2 10 15 6 40 105 105 24 196 700 1260 945 ...
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### 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 ...
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### Can a regular icosahedron contain a rational point on each face?

The title says it all: Is there a (regular) icosahedron containing a rational point on each of its faces? For other Platonic solids, the affirmative answer is easy. Indeed, regular tetrahedra, cubes, ...
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### Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
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### Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
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### 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 ...
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### Functor that maps to both $KO^n$ and $KO^{-n}$

(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting) I start by recalling the analytic definition of KO-theory: The following ...
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Assume that $\Omega$ is an open simply connected set in $\mathbb R^n$ (two-convexity) if 3 faces of a 3-simplex belong to $\Omega$ then whole simplex in $\Omega$. Is it true that any component of ...