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研究生研讨班 2022-2023
发布人:许全华  发布时间:2022-09-20   浏览次数:683

研究生研讨班是由数学研究院研究生自行组织的学术交流活动,面向数学专业的所有研究生和高年级本科生开放。研讨班旨在为各位研究生同学提供一个拓展学术视野、锻炼学术交流能力和增进相互了解的平台。每次研讨班由一位主讲同学就个人的科研方向或已完成的科研成果做一场报告,报告形式及内容应使得不同研究方向的师生均易于理解。


研讨班每月组织一至两次,通常时间在周一下午14:30,地点在明德楼B201-1。
如果您有兴趣参加我们的研讨班或做学术报告,或有问题需要咨询,欢迎同研讨班组织者联系。


组织者:姚永胜(yaoyongsheng000 AT 126.com),杨福林(fulinyoung AT hit.edu.cn)


报告安排


2022年10月10日, 14:30-15:30

报告人: 田永强 (中南大学)

题目: Dirac Operators in Riemannian Geometry

摘要

As is well known, the Dirac operator plays a crucial role in Alain Connes’ noncommutative geometry. In this talk, we will revisit the construction of Dirac operators on Riemannian spin manifolds. Some basic knowledge of classical geometry is required.
Working on a spectral triple (A,H,D), i.e. the noncommutative generalization of a Riemannian spin manifold, places you into the operator framework. So, in order to get some non-trivial results on it, both summability and regularity concerning the abstract Dirac operator D (self-adjoint, possibly unbounded) are usually assumed to be good enough. However, life is not smooth, especially when your algebra A is not ‘smooth’ either. Suppose now we have a nice algebra acting on Hilbert space H, then how to construct a proper Dirac operator D to guarantee the summability and regularity? There is no routine method in the noncommutative setting. This motivates us to look for some inspirations from the starting point: Riemannian geometry! And we will provide a few examples.


2022年9月22日, 14:30-15:30

报告人: 白婧 (哈尔滨工业大学)

题目: Primitivity for random quantum channels

摘要

We have considered the primitive index for random quantum channels, which means the minimal natural number n such that the Choi state/matrix of the n-fold composition of channels is full rank. In the previous work, several upper bounds for the index have been obtained, which can be used to construct the so-called (quantum) Wielandt inequality. We note that the optimal upper bound is still an open problem. In our work, we have shown a generic lower bound for the index when the channels are randomly chosen. Our main method is the graphical Weingarten calculus, introduced by Collins and Nechita. Moreover, our result is closely related to the injectivity of the representation of matrix product states. Perez-Garcia et al. claimed that a similar lower bound for the injectivity could be numerically verified, and our result provides a rigorous proof for their argument.