Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs

Brackets by any other name

Brackets by another name, Whitehead or Samelson products, have a history parallel to that in Kosmann-Schwarzbach's From Schouten to Mackenzie: notes on brackets. Here I sketch the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics. In contrast to the brackets of Schouten, Nijenhuis and of Gerstenhaber, which involve a relation to another graded product, in homotopy theory many of the brackets are free standing binary operations. My path takes me through many twists and turns. T…

三种风味下的弦拓扑

String topology in three flavours

本文从几何与代数两个视角研究Chas-Sullivan积和Goresky-Hingston余积，计算透镜空间上的运算并证明余积可区分三维透镜空间，同时用有理同伦论建立两种定义的等价性并讨论不变性。

On higher structure on the operadic deformation complexes ${Def}(e_n\to \mathcal{P})$

In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of operads $f\colon e_n\to\mathcal{P}$. This result generalizes the result of [T2], where the case $\mathcal{P}=\mathrm{End}_{Op}(X)$ was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of [T2] with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in [KM] and [Fr1]. We …