# All Questions

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### Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?

A very specific case of Reed's Conjecture Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic ...
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### Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
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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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### How many algebraic closures can a field have?

Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
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### What is the current understanding regarding complex structures on the 6-sphere?

In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
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### Example of a compact set that isn't the spectrum of an operator

This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity: Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
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### What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link). Let me first ...
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### What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...
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### Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that {$L_1,\dots,L_m$} ...
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### What are the potential applications of perfectoid spaces to homotopy theory?

This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ...
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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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### A three-line proof of global class field theory?

There is an idea (I think originally due to Tate) that class field theory is fundamentally a consequence of Pontrjagin duality and Hilbert Theorem 90. I'm curious whether this can phrased using modern ...
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It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(... 0answers 2k views ### Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required? This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of ... 0answers 1k views ### What is the three-dimensional hyperbolic volume of a four-manifold? Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ... 0answers 1k views ### Functor that maps to both$KO^n$and$KO^{-n}$(my question is also meaningful for complex K-theory, but since Kn(X) is always isomorphic to K-n(X), it's less interesting) I start by recalling the analytic definition of KO-theory: The following ... 0answers 810 views ### Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus? Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ... 0answers 2k views ### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product$U\bar{U}$, with$U$a random unitary matrix and ... 0answers 922 views ### Computer calculations in A_infinity categories? Is there a good computer program for doing calculations in A-infinity categories? Explicit calculations in A-infinity categories are an important, useful, yet very tedious task. One has to keep track ... 0answers 1k views ### Mikhalkin's tropical schemes versus Durov's tropical schemes In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ... 0answers 1k views ### Two-convexity ⇒ Lefschetz? Assume that$\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$. Is it true that any component of ... 1answer 2k views ### Computing Self-Intersections with Complex Analysis It is possible to find the winding number of a path$C \subset \mathbb{C}$using complex analysis: $$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of$f(z) = 0$inside a close ... 0answers 718 views ### Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function? The adèles$\mathbb A$arise naturally when considering the Berkovich space$\mathcal M(\mathbb Z)$of the integers. Namely, they are the stalk$\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$... 0answers 449 views ### Chern character of a Representation Let$G$be a finite group. Under the identification of the representation ring$R_{\mathbb{C}}(G)$with the equivariant K-theory$KU^0_G(\ast)$of the point, followed by Atiyah-Segal completion-... 0answers 963 views ### Converse of the Archimedean property of the sphere In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ... 0answers 593 views ### Are there infinite versions of sporadic groups? The classification of finite simple groups states roughly that every non-abelian finite simple group is either alternating, a group of Lie type, or a sporadic group. For each of the groups of Lie ... 0answers 1k views ### Peano Arithmetic and the Field of Rationals 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}, +,...
(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 ...