In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degenerate. As application we obtain the Bergman kernel asymptotics for adjoint semi-positive line bundles over complete K\"ahler manifolds, on the set where the curvature is positive. We also prove the asymptotics for big line bundles endowed with singular Hermitian metrics with strictly positive curvature current. In this case the full asymptotics holds outside the singular locus of the metric.

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In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes Aot= B especially when Aot= B is known to be a subset of Aot= B . In this paper we introduce a homological theory of functions that can be used to establish complexity separations, while also providing other interesting consequences. We propose to associate a topological space Aot= B to each class of functions Aot= B , such that, to separate complexity classes Aot= B , it suffices to observe a change in" the number of holes", ie homology, in Aot= B as a subclass Aot= B of Aot= B is added to Aot= B . In other words, if the homologies of Aot= B and Aot= B are different, then Aot= B . We develop the underlying theory of functions based on combinatorial and homological commutative algebra and Stanley-Reisner theory, and recover Minsky and Papert's 1969 result that parity cannot be computed by nonmaximal degree polynomial threshold functions. In the process, we derive a" maximal principle" for polynomial threshold functions that is used to extend this result further to arbitrary symmetric functions. A surprising coincidence is demonstrated, where the maximal dimension of" holes" in Aot= B upper bounds the VC dimension of Aot= B , with equality for common computational cases such as the class of polynomial threshold functions or the class of linear functionals in Aot= B , or common algebraic cases such as when the Stanley-Reisner ring of Aot= B is Cohen-Macaulay. As another interesting application of our theory, we prove a result that a priori has nothing to do with complexity separation: it characterizes when a vector subspace intersects the positive

Fundamental group is one of the most important topological invariants for general manifolds, which can be directly used as manifolds classification. In this work, we provide a series of practical and efficient algorithms to compute fundamental groups for general 3-manifolds based on CW cell decomposition. The input is a tetrahedral mesh, while the output is symbolic representation of its first fundamental group. We further simplify the fundamental group representation using computational algebraic method. We present the theoretical arguments of our algorithms, elaborate the algorithms with a number of examples, and give the analysis of their computational complexity.

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