A classical theorem of Lagrange says that every natural
number can be written as a sum of four squares. We can generalize
this beautiful result in two ways. The one way is horizontal
generalization (Cauthy's polygonal number theorem) which says that
every natural number can be written as a sum of k k-gonal numbers.
The other way is vertical generalization (Hilbert-Waring problem)
which asserts that every natural number can be written as a sum
of 4 squares
9 cubes
19 fourth powers
37 fifth powers
and so on.
Ps. This talk is elementary. The speaker thinks that undergraduates
or even some good high school students can understand the materials.
So it is strongly recomended for undergraduate students to attend
this talk.
