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 ...

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 ...

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,...

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\...

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

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 ...

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....

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 ...

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 $X$ (...

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., ...

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 ...

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 ...

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

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 ...

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 $\gamma_1\in L$. ...

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 ...

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 ...

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

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)-\...

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

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 ...

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 ...

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 ...

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/...

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 ...

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/...

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 ...

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 ...

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 ...

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^{...