# All Questions

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### Ascertain properties of a new kind of rectilinear-convex set

PREABMLE TO MY QUESTION I am reading about convex sets and hulls in orthogonal/rectilinear spaces. As can be seen in this publication, for a given set of points in $\mathbb{R}^{2}$, there are many ...
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### Maximal $k$-split $k$-tori are $G(k)$-conjugate, but maximal $k$-tori are not?

Suppose $G$ is a reductive algebraic group defined over a field $k$. Is this right? If so, it means for each maximal $k$-split $k$-tori $T$ there are many maximal $k$-anisotropic ones $T'$ that ...
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### Pullback-stable model of fibrewise suspension of fibrations (in simplicial sets, or similar setting)

Given a fibration $p : Y \to X$ in simplicial sets (or any other model category), there are various ways to construct its fibrewise suspension, i.e. its suspension as an object of the slice ...
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### Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question. In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...
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### Vector bundle with a perfect pairing and ($\mathbb Z/2$, $SL_r$)-bundle

I think this is a well knowing result but I can't find any reference, Let $(E,q)$ be a vector bundle with a non degenerated quadratic form $q:E\rightarrow E^*$ with trivial determinant, suppose ...
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### Near model completeness

A theory $T$ is called near model complete if every formula is equivalent to a Boolean combination of existential formulas mod $T$. I wonder whether there is an equivalent "semantic" definition of ...
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### model theory of non-reduced schemes

In model theory one studies Boolean algebras of definable sets of complete theories. For many theories definable sets are in direct correspondence with geometric objects, for example, definable sets ...
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### Important formulas in Combinatorics

Motivation: The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...
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### Castelnuovo -Mumford regularity properties

Let $R=k[x_1,\ldots,x_n]$ be graded ring and $I,J$ be quartic square free monomial ideals in $R$. Let $I=\langle n_1,\ldots,n_s \rangle$ and $J=\langle m_1,\ldots ,m_t\rangle$ Definition: ...
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### Hashimoto Matrix (Non-backtracking operator) and the Graph Laplacian

The question is: how can we recover the graph Laplacian or its spectrum from the Hashimoto Matrix (also commonly called the Non-backtracking operator)? To make the question as self-contained as ...
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### Embedding of parallelizable closed smooth manifold

It seems that any closed parallelizable smooth $n$-manifold can be immersed in the $(n+1)$-dimensional Euclidean space (compared to the general $(2n-1)$-dimensional case). Is there any analogous ...
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### Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...
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### Computations in modular cohomology of finite groups

Let $k$ be an algebraically closed field of characteristic $p$, let $G$ be a finite group whose order is divisible by $p$, and let $H(G)$ be the commutative cohomology algebra of $G$ with coefficients ...
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### $C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
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### What is wrong with this deterministic algorithm efficiently generating large primes?

According to PolyMath (Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in ...
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### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. Motivation: I plan to use this list ...