讲座人简介：

伊山修，东京大学教授，长期致力于代数表示论的研究，迄今为止已发表学术论文近百篇。论文发表期刊包含《Invent. Math.》,《Adv. Math.》,《Trans. Amer. Math. Soc.》,《Compos. Math.》,《Proc. Lond. Math. Soc.》等知名国际学术期刊。伊山修教授现任《Proc. Lond. Math. Soc.》,《Math. Z.》编委。2018年受邀在国际数学家大会上做报告。

讲座简介：

The notion of tilting complexes is basic to study the structure of derived categories of rings. The class of silting complexes complements the class of tilting objects from a point of view of mutation. For a finite dimensional algebra A over a field, 2-term silting complexes of A give rise to a simplicial complex (called the g-simplicial complex ) and a nonsingular fan in the real Grothendieck group of A (called the g-fan). For example, the g-fan of a preprojective algebra is the Coxeter fan, and the g-fan of a Jacobian algebra of a certain quiver with potential is the g-fan of the corresponding cluster algebra.

The g-fan of A is a useful combinatorial invariant which has a lot of information about representation theory of A, and therefore satisfies many nice properties. For example, the h-vector and the Dehn-Sommerville equation of the g-simplicial complex has a tilting theoretic interpretation. Also the g-fan of A is complete if and only if A has only finitely many 2-term silting complexes up to isomorphism. One of the basic problems is to classify complete g-fans. We give an answer for rank 2 case by showing that complete g-fans of rank 2 are precisely complete sign-coherent fans of rank 2. As a by-product, for each positive integer N, we give a finite dimensional algebra A of rank 2 such that the Hasse quiver of the poset of 2-term silting complexes of A has precisely N connected components. This talk is based on a series of joint works with T. Aoki, A. Higashitani, R. Kase and Y. Mizuno.