John Pardon
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Concurrent normals conjecture
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26 votes

Let me address the specific complaint of that review. The situation is the following. Our (bounded, open) convex set is denoted $K\subseteq\mathbb R^n$ with closure $\overline K$, and we consider ...

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Can a tangle of arcs interlock?
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23 votes

I believe there is no such locked configuration. The proof is by induction, as you suggest. Pick any arc and imagine moving it to infinity. Of course, to do this, it will have to pass through some ...

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intuition for hochschild homology
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22 votes

If you have a right $A$-module $M_A$ and a left $A$-module $_AN$, then you can form their tensor product $$M\otimes_AN:=\operatorname{coker}(M\otimes_kA\otimes_kN\xrightarrow{(m,a,n)\mapsto(ma,n)-(m,...

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What is the usual topology of $C^\infty_c(M) $
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22 votes

Topologizing $C_c^\infty(M)\subseteq C^\infty(M)$ with the subspace topology (where $C^\infty(M)$ has the Whitney topology, generated by the seminorms $\left|\sup_K\frac\partial{\partial x^\alpha}f\...

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Journals and other sources with "easy reading" papers?
20 votes

I am surprised that no one has mentioned the expository articles in the Bulletin of the AMS, which are usually excellent.

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Research directions in persistent homology
18 votes

Persistent homology has applications in symplectic geometry. For one striking example, see the recent paper Autonomous Hamiltonian flows, Hofer's geometry and persistence modules by Polterovich and ...

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Publishing conjectures
16 votes

The main contribution of the paper "The McKay conjecture and Galois automorphisms" by Gabriel Navarro is to propose various conjectures. It is published in the Annals of Mathematics. http://www.ams....

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Is there a continuous analogue of Ramanujan graphs?
16 votes

In fact, the original motivation behind Lubotzky--Phillips--Sarnak's construction of Ramanujan graphs was in analogy with modular curves $Y(N)=\mathbb H^2/\Gamma(N)$ for the principal congruence ...

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Is there a complex which computes Cech cohomology?
14 votes

While I'm sure a hocolim solution also exists (and works in general), one can avoid it, at least for compact topological spaces, by using open covers which are indexed by the points of your space $X$ (...

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Grothendieck's Galois theory without finiteness hypotheses
14 votes

In "The pro-étale topology for schemes" (http://arxiv.org/abs/1309.1198), Bhatt and Scholze introduce the pro-étale fundamental group which seems to give a good answer to your question (see, e.g., ...

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Does Physics need non-analytic smooth functions?
14 votes

In the field of mathematical general relativity, certain uniqueness results for black holes are known for real-analytic space-time but are open for smooth space-time. For example, the "No hair ...

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Number of critical points of smooth functions on $S^1$
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13 votes

Yes, in fact the number of critical points of $u$ is at least four. Expand $u:S^1\to\mathbb R$ as a Fourier series: $$u(\theta)=a_0+\sum_{i=1}^\infty a_j\cos(j\theta)+b_j\sin(j\theta)$$ Then your ...

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What is the "analytic" analogue of the valuative criterion of properness
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11 votes

No, the open disk in $\mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).

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Axiomatic characterization of virtual fundamental classes?
11 votes

Question 1 (compare virtual fundamental cycles of different perfect obstruction theories on space underlying space): There is essentially no relation between $[X]_\varphi$ and $[X]_{\varphi'}$ for ...

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Non-cyclotomic abelian extensions
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10 votes

Pick some $\gamma_1\in L\setminus\mathbb Q$ which is not a square. Pick some $\gamma\in L^\times/(L^\times)^2$ which is not fixed by $\operatorname{Gal}(L/\mathbb Q)$ and fix a lift $\gamma_1\in L$. ...

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Are dagger categories truly evil?
9 votes

What about the following definition of $\dagger$-categories (which looks non-evil, at least to my untrained eye). A $\dagger$-structure on $\mathcal C$ consists firstly of a functor $\dagger:\mathcal ...

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Lefschetz duality for non-compact relative manifolds
9 votes

Your counterexample looks correct to me. I don't see an obvious way to add hypotheses on $X$ or $A$ so that the statement called "Lefschetz duality" in the link you give becomes correct. Some ...

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Eilenberg-MacLane Spaces of "large" groups
9 votes

One place to start might be: Milnor, J. On the homology of Lie groups made discrete. Comment. Math. Helv. 58 (1983), no. 1, 72–85. http://www.ams.org/mathscinet-getitem?mr=699007

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Does the centroid depend continuously on the curve?
9 votes

Given any curve $\gamma:[0,1]\to\mathbb R^3$, any point $p$ in the convex hull of the image $\gamma([0,1])$, and any $\epsilon>0$, we can find a $\gamma':[0,1]\to\mathbb R^3$ so that $|\gamma(t)-\...

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Local structure of rational varieties
8 votes

Here is a preprint of Ilya Karzhemanov constructing counterexamples in dimensions $n\geq 4$: http://research.ipmu.jp/ipmu/sysimg/ipmu/1588.pdf

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Is Grothendieck a computer?
8 votes

I disagree with the premise of this question. Conventional computers follow a program written by a human. I think, for example, Daniel Moskovich's answer about simplicial sets is something that a ...

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Wildness of codimension 1 submanifolds of euclidean space
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7 votes

For $n\geq 5$, such a diffeomorphism exists iff the complement of $S$ has an unbounded connected component. Indeed, the existence of such a diffeomorphism implies there is an unbounded component of ...

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Heegaard genera of arithmetic 3-manifolds
7 votes

The Heegaard genus of an arithmetic hyperbolic three-manifold is bounded below by a constant times its volume. This is proved on p43 of a recent paper by Gromov and Guth (or p2601 of the published ...

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Why isn't the perfect closure separable?
7 votes

In the special case that your extension $K/F$ is of the form $K=F(\text{root of }f)$ for some irreducible polynomial $f(x)$, then there is some nice intuition as follows. If $f$ is separable, then $K/...

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Relation between "triangulated bordism", MO, and $H\mathbb{F}_2$
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6 votes

There is an implicit assumption in your question, namely that one can define a chain complex calculating the functor $X\mapsto MO_*(X)$ which is based on maps from unoriented manifolds into $X$. As ...

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Relationship between Gromov-Witten and Taubes' Gromov invariant
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6 votes

Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper: https://arxiv.org/...

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Why are quasi-isomorphisms of homotopy algebras only defined for arity 1?
6 votes

Here is a down to earth explanation, by way of analogy. A group homomorphism $\varphi:G\to H$ is an isomorphism of groups iff it is a bijection of sets. Why do we not need to say anything about the ...

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Confusion regarding Riemann-Roch for vector bundles
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6 votes

Riemann--Roch is "the same" over any field. But in any event, this isn't so helpful for you since it only shows that the Euler characteristic $h^0(X,E)-h^1(X,E)$ is the same over $\bar k$ as it is ...

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Grothendieck topologies on $\mathbb{C}$
6 votes

The subpresheaf $\mathcal S\subseteq\mathcal C^0$ is not a subsheaf. Indeed, let $U,V\subseteq\mathbb C$ be two disjoint open sets, and consider the function $f:U\cup V\to\mathbb C$ assigning $0$ to ...

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Generalization of Giroux's Theorem for Higher Dimensions?
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6 votes

Giroux proved that for every contact manifold $(M^{2n+1},\alpha)$, there exists an open book supporting the contact form. An open book consists of a codimension two submanifold $K^{2n-1}\subseteq M^{...

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