Talk
What is… Hilbert’s 17th problem?
Talk by Segev Gonen Cohen
Date: 12.03.24 Time: 16.30 - 17.30 Room:
''Suppose that we are given a polynomial which can be expressed as a sum of squares of polynomials with real coefficients, for example f(X,Y,Z) = (X-4Y)^2 + (17Z^3 - 4XYZ)^2. Then it is clear that no matter what (real) inputs we put in, we will get something positive. Minkowski, in his PhD defence, asked about the converse - suppose we have a polynomial that takes only positive values, must this be because it is a sum of squares? Hilbert, who was sat in the audience, realised that the answer is no; but modified the question slightly, and famously included it as the 17th in a list of open problems presented at the 1900 ICM in Paris. The main part of this talk will be devoted to the solution to this problem, relying on some algebraic foundations of Artin, and a nice model-theoretic proof due to Robinson. Time permitting we will look at modern day generalisations of this question, partial solutions, and (depending on audience interest) their applications to the Connes Embedding Problem, Kazhdan’s Property (T), norm computability, and more. No knowledge of any of these topics will be assumed.''