非常简单的表示：克利福德主题的变化

Variations on the theme of Zariski's Cancellation Problem

This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss several topics inspired by this exploration, including the problem of classifying a class of affine algebraic groups that are naturally singled out in studying the conjugacy problem for algebraic subgroups of the Cremona groups.

Variations on a theme by Higman

We propose elementary and explicit presentations of groups that have no amenable quotients and yet are SQ-universal. Examples include groups with a finite classifying space, no Kazhdan subgroups and no Haagerup quotients.

Complete reducibility: Variations on a theme of Serre

In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--Röhrle, the usual notion of $G$-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of $G$. We show that other variations of this notion, such as relative complete reducibility and $σ$-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of…

The uncertainty principle: variations on a theme

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho--Stark uncertainty principle, and Meshulam's non-abelian uncertainty principle, have little to do with the structure of the Fourier transform itself. Rather, all of these results follow from very weak properties of the Fourier transform (shared by numerous linear operators), namely that it is bounded as an operator $L^1 \to L^\infty$, and that it is unitary. Using a single, simple proof template, and only these (or weaker) properties, we obtain some new …

Baer定理主题的新变体

New variations on the theme of Baer's theorem

本文证明，对于广义可解群，若下中心级数第s项与上中心级数第t项的交的商群具有有限秩r，则第s+t项下中心群具有有限且受(r,s,t)界定的秩；类似地，将有限秩条件替换为有界大小的Chernikov群也得到相应结果，推广了经典Baer定理。

克莱恩与唐金主题变奏曲

Variations on a theme of Cline and Donkin

设$N$是群$G$的正规子群。对于每个$g\in G$，只要$Q$等价于$g$对$Q$的扭曲$Q^g$，$N$模块$Q$就是$G$稳定的.如果$N$对$Q$的作用扩展为$G$对$Q$的作用，则$Q$显然是$G$稳定的，但反过来不一定成立。约化代数群的模表示理论中的一个著名猜想$G$断言，$G$的Frobenius核的（显然是$G$ -稳定的）投射不可分解模（PIM）$Q$具有$G$ -模结构。对于一般的模块$Q$，知道$Q$的有限直和$Q^{\oplus n}$具有兼容的$G$ -模块结构有时也是有用的。在本文中，这种性质被称为数值稳定性。在最近的工作（arXiv：0909.5207v2）中，作者建立了PIM特殊情况下的数值稳定性。在本文中，我们提供了一个更一般的背景下的结果，工作组计划和一个合适的版本的$G$稳定性，称为强$G$稳定性的背景下。在我们的研究结果中，这里是一个同调障碍的存在下，一个$G$模块结构，强$G$稳定的模块，张量积的方法来杀死的障碍。