曲面の組合せ論によるブラウアーグラフ代数の導来圏の研究

利用Demonet，我们对(可能是无穷维的，也可能是全局维无限的)柔化代数的所有扭转类进行了分类。我们的分类是在紧致定向的实二维曲面上使用简单的曲线和薄片；这样的工作在其他领域也有潜在的用处，例如拓扑Fukaya范畴，其中柔和代数是中心重要的。我们使用类似于Khovanov-Seidel结果的拓扑技巧来构造这些代数的预倾斜复形。利用Iyama和Marczinzik，我们研究了簇前倾斜理论和极小Auslander-Gorenstein理论的推广。特别地，我们的结果统一了Marczinzik以前在特殊双列Gendo-对称代数研究方面的工作。我们利用Mihemetz研究了平面2-范畴的2-表示的短正合列的概念。我们将这些概念与交换范畴的回忆联系起来，发现了域上余代数的局部化理论在FIAT-2-范畴中的余代数对象上的适当推广；该理论也适用于其他领域的兴趣设置，如研究么半群范畴的模范畴。我们研究了这些复形的斜同调；特别地，我们证明了Wildon猜想的一个推广。

With Demonet, we classify all torsion classes of (possibly infinite dimensional and possibly global dimension infinite) gentle algebras. Our classification is to employ simple curves and laminations on compact oriented real-two-dimensional surfaces; such work is potentially useful in other areas such as topological Fukaya categories where gentle algebras are of central importance.With Adachi, we study complexes of Brauer graph algebras. We employ topological techniques similar to results of Khovanov-Seidel to construct pretilting complexes of these algebras. We also calculate their endomorphism rings, and investigate the possiblity of them being tilting.With Iyama and Marczinzik, we study generalisation of precluster tilting theory and minimal Auslander-Gorenstein. In particular, our results unify a previous work with Marczinzik on the study of special biserial gendo-symmetric algebras.With Miemietz, we study the notion of short exact sequences for 2-representations of fiat 2-categories. We relate these notions with recollement of abelian categories, and found appropriate generalisation of localisation theory for coalgebras over a field to coalgebras objects arising in fiat 2-categories; this theory is applicable to setting of interests in other fields such as the study of module-category of a monoidal category.With Wong, we study p-complexes of permutation modules in the setting of modular representations over the symmetric groups. We investigate the slash homologies of these complexes; in particular, we prove an extension of a conjecture of Wildon.