Changwei Zhou | Searcharxiv

arXiv2019

Effective upper bound of analytic torsion under Arakelov metric

Given a choice of metric on the Riemann surface, the regularized determinant of Laplacian (analytic torsion) is defined via the complex power of elliptic operators: $$ \det(Δ)=\exp(-ζ'(0)) $$ In this paper we gave an asymptotic effective estimate of analytic torsion under Arakelov metric. In particular, after taking the logarithm it is asymptotically upper bounded by $g$ for $g>1$. The construction of a cohomology theory for arithmetic surfaces in Arakelov theory has long been an open problem. In particular, it is not known if $h^{1}(X,L)\ge 0$. We view this as an indirect piece of evidence that if such a cohomology theory exists, the $h^{1}$ term may be effectively estimated.

math.DG↗

arXiv2015

Bases for representation rings of Lie groups and their maximal tori

A Lie group is a group that is also a differentiable manifold, such that the group operation is continuous respect to the topological structure. To every Lie group we can associate its tangent space in the identity point as a vector space, which is its Lie algebra. Killing and Cartan completely classified simple Lie groups into seven types. Representation of a Lie group is a homomorphism from the Lie group to the automorphism group of a vector space. In general represenations of Lie group are determined by its Lie algebra and its the connected components. We may consider operations like direct sum and tensor product with respect to which the representations of G form a ring R(G). Assume the group is compact. For every Lie group we may find a maximal torus inside of it. By projecting the representation ring of the Lie group to the representation ring of its maximal torus, we may consider to express R(T) elements as sums of products of base elements with R(G) elements. In 1970s Pittie and Steinberg proved R(T) is a free module over R(G) using homological algebra methods without specifying the bases in each seven types. In this senior project I found an explicit basis for the representation ring of the maximal torus of Lie group SUn in terms of the represenation ring of the Lie group itself. I also did some computation with SO2n.

math.RT↗