We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety G^r_d (C) that parametrizing linear series of degree d and dimension r on a general smooth projective curve C is indecomposable when d ≤ g(C)−1.

In this note we derive some consequences from the Mario-Vafa formula and the cut-and-join equation, these include unified simple proofs of the \lambda_g conjecture and some other Hodge integral identities. We also describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals by using the cut-and-join equation, following our proof of the Mario-Vafa formula.

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and HirzebruchRiemannRoch, and some modern development of uniformization theorems, KhlerEinstein metric and the theory of DonaldsonUhlenbeckYau on Hermitian YangMills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to bound- edness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of ε-Fano type. Furthermore, we show birational boundedness of 3-folds of ε-Fano type.

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