Mar Gonzalez
Date: May 16, 2000
Special topics: Differential Geometry and Functional Analysis
Commitee: Alice Chang (chair), Edward Nelson and Hale Trotter.
Length: 2 hours and a half, more or less.
The exam started at 9am. When I arrived, Trotter was not there yet, so I
had to wait. Then they sent me outside to discuss what they wanted to
ask. Finally, we began. They decided the order of the topics: Algebra,
Complex, Real, Functional and Geometry.
Trotter asked all the questions in algebra. He started with: What can
you say about finite groups? ('a bit' vague question...). Then he
suggested me to talk about Sylow theorems, and some applications (groups
of order 15, p-groups...), simple groups...
We changed to finite fields, and I had to list some properties. Trotter
gave me some specific examples.
I had to talk about the structure theorem for abelian groups, and a bit
about field extensions. The last thing was to describe the ideals of the
ring of square matrices.
The algebra part was over. Nelson and Chang then asked me about complex:
Fundamental theorem of algebra, Liouville's formula, Maximum modulus
principle...
Nelson wanted a version of Phragmen-Lindelof (an analytic function on
the strip -1