David Roberts scratch 2017

This page is intended to be a dumping ground for thoughts. No guarantee is made for correctness. On the other hand, I would be happy to discuss stuff from here if you have ideas. See David Roberts for my contact details. See also: scratch 2014, scratch 2015, scratch 2016

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2017

September

26

In Section 3.1 of

• William Stein, Modular Forms: A Computational Approach pdf

it is given (bottom of page 37) the push-pull moduli space interpretation of Hecke operators on modular forms over the complex numbers. $Y(N)$ is the space of isogenies of degree $N$ (or rather, of pairs consisting of an elliptic curve together with an order-$N$ subgroup). Also, it describes the action of Hecke operators on $H_1(X_0(N),\mathbb{Z})$, which includes pullback of homology class.

See also section A.4.2, which gives a more general modular subgroup but also push-pull interpretation, with the information about which way to do it.

13

The paper

• Steve Hofmann, Marius Mitrea, Michael Taylor, Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains, The Journal of Geometric Analysis, 17 (2007) issue 4, 593–647, doi:10.1007/BF02937431, (pdf)

gives the result (Theorem 4.1) that locally strongly Lipschitz domains in $\mathbb{R}^n$ are preserved under $C^1$ diffeomorphisms. Also (Theorem 4.3), for any $C^1$ atlas on a topological manifold, any locally strongly Lipschitz domain relative to that atlas (?what does this mean?) is locally strongly Lipschitz relative to any other atlas compatible with the first.

Ideas for MSC codes

• 58D15 Manifolds of mappings (under Global analysis, analysis on manifolds)
• 46T10 Manifolds of mappings (under Functional analysis/ Nonlinear functional analysis)
• 58C15 Implicit function theorems; global Newton methods (under Global analysis, analysis on manifolds)

10

Regarding the question below, perhaps only under bi-Lipschitz maps? And also worth considering is the non-invertible case (cf images/inverse images of convex sets under appropriately metric maps)

9

A map $f\colon A \to B$ between metric spaces $(A,d)$ and $(B,\delta)$ is $\alpha$-Hölder bi-continuous (see Athanase Papadopoulos, Weixu Su, Thurston’s metric on Teichmüller space and isomorphisms between Fuchsian groups 2012. hal-00740320, https://arxiv.org/abs/1210.2676) if

$$\frac{d(a,a')^{1/\alpha}}{C} \leq \delta(f(a),f(a')) \leq C d(a,a')^\alpha$$

For a metric space $(M,d)$ with closed $F$ in $M$, the Frerick condition (see Theorem 3.16 of Frerick, Extension operators for spaces of infinite differentiable Whitney jets, J. reine angew. Math. 602 (2007), 123-154) for $F$ is:

$\forall$ compact $K \subset F$,

$\exists \epsilon_0,\ \rho \gt 0,\ r \geq 1$, (so only depending on $K$) such that:

$\forall z\in \partial(K) \cap \partial(F)$ and $\forall 0\lt\epsilon\lt\epsilon_0$,

$\exists x \in F$ such that: $B(x,\rho\epsilon^r) \subset F \cap B(z,\epsilon)$.

Q: is the Frerick condition preserved, under $\alpha$-Hölder bi-continuous maps?