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Recall, that for two metric spaces $X$ and $Y$ the Lipschitz distance between $X$ and $Y$ is the infimum over all bi-Libschitz maps $f:X\to Y$ of $$\log(\max({\rm dil}(f),{\rm dil}(f^{-1}))).$$ Here …

There is now a great course on Commutative Algebra by Richard Borcherds on Youtube, it follows the book of David Eisenbud : Commutative Algebra with a view toward Algebraic geometry https://www.youtub …

I would like to add the following notes by Nigel Hitchin: https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Differentiable_manifolds/manifolds2014.pdf

Consider $\mathbb C^n$ with coordinates $(z_1,\dots,z_n)$, $z_j=x_j+iy_j$. Let $\omega=\sum dx_i\wedge dy_i$. Let us call by a radial diffeomorhpism $\varphi$ of $\mathbb C^n$ a diffemorphism of $\var …

Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that …

This statement about blow ups of torus is not correct. Take any aglebraic $2$-torus with a smooth curve $C$ of genus $>1$. Blow up $C^2+1$ points on $C$ and apply Theorem 1 here: https://arxiv.org/pdf …

In the book of Kra and Farkas on Riemann surfaces the following (slightly unusual) definition is given: Definition IV.3.2 (Section IV.3). Let $M$ be a Riemann surface. We will call $M$ elliptic if and …

That's true and follows from the fact a vector space with a countable dense subset can't have an uncountable number of open subsets that don't intersect pairwise. Here the vector space is the space of …

This question is quite subtle, I don't believe the answer is known in the general situation. But if you consider the case of $1$-forms on surfaces, one can completely characterise those that are harmo …

Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Is it true that the space contractible loops of this manifold can be contracted to the space of constant loops on $X$? What if $X$ is a finite dime …

By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y') …

Let $(M,g)$ be a compact manifold with a metric $g$ (not necessarily Riemannian one). Suppose that $(M,g)$ is a Lipschitz limit of a sequence of Riemannian manifolds $(M_n,g_n)$ with the sectional cur …

Consider the space $W$ of smooth time-like curves in $\mathbb{R}^{n,1}$ with fixed ends. Given $\gamma\in W$, consider the space $T_\gamma$ of all smooth normal fields along $\gamma$; one may think th …

I would like to find an appropriate reference for the following statement: Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature. Then $\pi_1(M)$ is virtually abelian. …

Recall that Kobayashi metric is defined on any complex manifold $M$. This is a pseudo-metric according to which a tangent vector $v$ at $P$ has length at most $1$ if there is holomorphic map from th …