Title:    Two Local-global Theorems and a Powerful Formula
Author:                    Zhi-Wei Sun
          Department of Mathmatics, Nanjing University
          Nanjing 210093, People's Republic of China
          zwsun@nju.edu.cn    http://pweb.nju.edu.cn/zwsun

Abstract

    A conjecture of Paul Erd\"os states that a system of $k$
residue classes covers all the integers if it covers those integers
from $1$ to $2^k$. Motivated by this conjecture we obtain two local-global
results
one of which is concerned with sums of periodic arithmetical maps to 
an arbitrary abelian group. We also present a powerful formula
for polynomials over a ring, which implies some deep results on zero-sums
(e.g. Olson's theorem on the Davenport constant of an abelian $p$-group).  
This talk is essentially self-contained, and we will show how some deep
results 
can be deduced via very simple (but definitely nontrivial) ideas.