Representation theory of some infinite-dimensional algebras arising in continuously controlled algebra and topology

Geometry and Topology of Symmetric Point Arrangements

We investigate point arrangements $v_i\in\mathbb R^d,i\in \{1,...,n \}$ with certain prescribed symmetries. The arrangement space of $v$ is the column span of the matrix in which the $v_i$ are the rows. We characterize properties of $v$ in terms of the arrangement space, e.g. we characterize whether an arrangement possesses certain symmetries or whether it can be continuously deformed into another arrangement while preserving symmetry in the process. We show that whether a symmetric arrangement can be continuously deformed into its mirror image depends non-trivially on several factors, e.g. th…

Poset topology and homological invariants of algebras arising in algebraic combinatorics

We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. Résumé. Nous présentons une magnifique interaction entre la topologie combinatoire et l'algèbre homologique d'une classe de monoïdes qui figurent naturellement dans la combinatoire algébrique. Nous explorons plusieurs applications de cette interaction. Par exemple, …

Combinatorial Topology and the Global Dimension of Algebras Arising in Combinatorics

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operator…

由辛拓扑产生的伪阿诺索夫自等价及其在稳定性条件空间上的双曲作用

Pseudo-Anosov autoquivalances arising from Symplectic topology and their hyperbolic actions on stability conditions

本文研究树形箭图对应的N-卡拉比-丘范畴中彭纳型自等价的渐近行为，提出计算拉伸因子的方法，并证明其在稳定性条件空间上诱导作用的平移长度正下界，且实例达到该下界，实现双曲作用。

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close conn…

拓扑与幺半群表示 II：群的左正则带与 Hsiao 的有序 $G$-分拆幺半群

Topology and monoid representations II: left regular bands of groups and Hsiao's monoid of ordered $G$-partitions

本文利用拓扑方法研究 $\mathscr L$ 与 $\mathscr J$ 关系重合的 von Neumann 正则幺半群，计算不可约表示间的 $\mathrm{Ext}$，并证明 Hsiao 有序 $G$-分拆幺半群代数的 Koszul 性及其投射分解。