Let be the set of matrices over a field F. Consider a graph G = ( , ) with as the vertex set such that two vertices A, B are adjacent if rank (A-B)=1. We study graph properties of G when F is a finite field. In particular, we show that G is a regular connected graph with diameter equal to min{m,n}; it is always Hamiltonian; it is Eulerian if and only if F has an odd number of elements.Furthermore,we determine the independence number,chromatic number and clique number of G. These results are used to characterize the graph endomorphisms of G, which extends Hua's fundamental theorem of geometry on .

I will briefly introduce two books and will talk about one of the problems---the realisation problem---discussed in the books.

报告人：陆国震教授 (北京师范大学, 美国韦恩州立大学)题目： Affine Moser-Trudinger and Adams inequalities and applications to PDEs时间：2013年1月3日(星期四)上午10：00—11：00地点：数学系致远楼10

报告人：汤善健教授(复旦大学）题目： Recent Progress on Backward Stochastic Partial Differential Equations时间：2013年1月3日(星期四)下午3：00—4：00地点：数学系致远楼10

报告人：张奇副教授(复旦大学）题目： Feynman-Kac公式的发展时间：2013年1月3日(星期四)下午4：00—5：00地点：数学系致远楼10

In these lectures we plan to discuss the history, important results and current development of the theory of motives. We plan to proceed according to the following schedule. Lecture 1( 1月4日) Overview of motive. Weil Conjecture. Grothendieck's idea. Beilinson Conjecture and Voevodsky's triangulated category of motives. Motives over fields of finite characters. Grothendieck's standard conjectures. Motivic t-structure. Absolute motives or motives over F1. Riemann Hypothesis. Lecture 2( 1月5日) Topological backgrounds. Homotopy category of topological spaces. Artiyah-Hirzebruch spectral sequences. Simplical methods. Dold-Thom theorem

报告题目：Oka's theory as a background of complex analysis and geometry报告人：Prof. Takeo Ohsawa名古屋大学（Nagoya University）报告时间：2012年12月 25，26，27 日15:30--17:00报告地点：致远楼10

报告人：Prof. Xiaochun Rong（美国Rutgers大学数学系）报告题目：Comparison Geometry On Singular Spaces报告时间：2011年12月28日（周三）上午11:00-12:00报告地点：致远楼107室欢迎广大师生参加