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（可能无穷维和可能整体无穷维）温柔代数的所有挠类进行了分类。我们的分类是采用简单的曲线和叠层紧凑的实二维曲面，这样的工作是潜在的有用的其他领域，如拓扑福谷类别温柔代数是至关重要的。与阿达奇，我们研究复合物的布劳尔图代数。我们采用拓扑技术类似的Khovanov-Seidel的结果来构建这些代数的pretilting复形。我们还计算了它们的自同态环，并研究了它们是倾斜的可能性。与Iyama和Marczinzik，我们研究了precluster倾斜理论的推广和极小Auslander-Gorenstein。特别地，我们的结果统一了Marczinzik关于特殊双正合根对称代数的研究，并与Miemietz一起研究了平坦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局部化联系起来，并发现域上的余代数的局部化理论可以适当地推广到平面2-范畴中的余代数对象;这个理论也适用于其他领域的兴趣设置，例如monoidal范畴的模范畴的研究。我们调查这些复合物的斜线同源性，特别是，我们证明了一个猜想的威尔顿的延伸。

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.