# All Questions

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### Why is there no Cayley's Theorem for rings?

Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...
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### Fibered knot with periodic homological monodromy

It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act ...
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### quotient by finite group actions that are smooth

Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero. Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$. We assume that ...
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### K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?
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### self-dual representations

Let $V$ be a finite-dimensional irreducible representation (complex or $\ell$-adic) of a group $G$ (compact Lie group or algebraic group etc.). Does there always exist a linear character $\rho$ of ...
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### Natural number properties as uninterpreted functions in first order logic

Can we express the following property of natural numbers as FOL. The property given below is only indicative, I am more interested in knowing how the concepts such as "infinitely many X exists for so ...
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### A differentiable approximation to the minimum function

Suppose we have a function $f : \Re^N \rightarrow \Re$ which, given a vector, returns the value of its smallest element. How can I approximate $f$ with a differentiable function(s)?
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### Monomial orderings in noncommutative setting

An ordering of monomials in the free associative algebra $k\langle x_1,\ldots,x_n\rangle$ is called a monomial ordering (EDIT: it seems that an equally common term used in this context is "term ...
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### Homology with Coefficients

We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in ...
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### Singular chains as an HZ-module spectrum

For $R$ any ring and $H R$ its Eilenberg-MacLane spectrum -- a ring spectrum -- there is an equivalence between the $\infty$-categories of $H R$-module spectra and that of unbounded chain complexes of ...
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### Geometric quantization: why are the prequantum operators self-adjoint?

I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical ...
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### How has modern algebraic geometry affected other areas of math?

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...
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### Advice for pure-math Phd students [closed]

Pursuing a Phd in pure math can be a daunting task. A number of students who begin a Phd in pure math don't complete it, and there are high-quality dissertations and those which are not so high ...
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### Nilpotent operator of the Weyl algebra

For a research project I'm currently working on, I came across the following problem: Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where ...
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### Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus? Is $\lambda x$ the same as $\tau_x$?

Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus ? Whether $\lambda {x}$ is same as $\tau _{x}$ ? Are the rules of Meta-Mathematics (Criteria of Substitution, ...
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### Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $SL(2,C)$-character variety

Does the following statement: "Let $G$ be a finitely generated group and let $X(G)$ be the $SL(2,\mathbb{C})$-character variety of $G$. Suppose $X(G)$ contains an irreducible component ...
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### Picard groups of quartic K3 surfaces

Does anyone know where I can find examples of quartic K3 surfaces for which the Picard group is known? I'm really interested in examples where there are explicit constructions of the divisors ...
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### What is the optimal size in the finite axiom of symmetry?

Freiling's axiom of symmetry states that if you assign to each real number $x$ a countable set $A_x\subset\mathbb{R}$, then there should be two reals $x,y$ for which $x\notin A_y$ and $y\notin A_x$. ...
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### Identifying the stacks in Devinatz-Hopkins-Smith

I read the Devinatz-Hopkins-Smith proof of the nilpotence conjectures last year, and while I followed along sentence to sentence I don't think I understood much of the motivating reasons for why what ...
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### Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A\$\dfrac{\partial}{\partial ...