时间：11月9日（周四）下午2点，2017年

地点：数学楼401报告厅

报告人：张福振教授（美国诺瓦东南大学数学系）

题目1： 中美高等教育漫谈 （个人体会和经历）

摘要：根据在中国和美国两国作为学生和教育工作者的体会和经历，用大量的实例对比中美高等教育。

题目2:   Tensors and Combinatorial Properties of Tensors

摘要: We begin with the definition of a tensor (in algebra) and then focus on the tensors by which we mean multi-dimensional arrays (or hypermatrices) of real numbers.  A square matrix is doubly stochastic if its entries are all nonnegative and each row and column sum is 1. A celebrated result known as  Birkhoff's theorem about doubly stochastic matrices states that an n x n matrix is doubly stochastic if and only if it is a convex combination of some n x n permutation matrices (a.k.a Birkhoff polytope). The Birkhoff polytope of n x n stochastic matrices in R^{n^2} is of dimension (n-1)^2 with n^2 facets and n! vertices.

We consider the generalization of the Birkhoff's theorem in higher dimensions. An n x n x n stochastic tensor is a nonnegative array (hypermatrix) in which every sum over one index is 1. A permutation tensor can be identified with a Latin square (vice versa). We study the polytope of all these tensors, the convex set of all tensors with some positive diagonals, and the polytope generated by the permutation tensors. We present lower and upper bounds for the number of vertices of the polytopes, and discuss further questions on the topic.  

Determinant and permanent are basic and important functions of n x n matrices. We attempt to define these for tensors. More generally, we will consider defining the generalized matrix functions for tensors.

报告人简介：张福振教授在中国获学士和硕士学位，在美国获博士学位。他是矩阵分析及其应用领域领头的专家之一。他对文化和教育问题也有敏锐的观察和独到的见解。