Lecture Series by V. Voevodsky

Abstract: In these lectures I will tell about my work on two related but separate subjects. The first one is mathematical population genetics. I will describe a simple model which is useful for the study of the relationship between the history of a population and its genetic properties. While the positive results obtained in the framework of this model may have little use because of the model's simplicity the negative results are likely to remain valid for more complex real world populations.

The second subject can be described as a categorical study of probability theory where "categorical" is understood in the sense of category theory. Originally, I developed this approach to probability to get a better understanding of the constructions which I had to deal with in population genetics. Later it evolved into something which seems to be also interesting from a purely mathematical point of view. On the elementary level it gives a category which is useful for the work with probabilistic constructions involving complicated combinations of stochastic processes of different types. On a more advanced level, applying in this context the old idea of a functor as a generalized object one gets a better view of the relationship between probability and the theory of (pre-)ordered topological vector spaces. This leads to the third topic mentioned in the title. But I am only beginning to understand this connection.

Some Information:

Vladimir Voevodsky
2002 Fields Medalist

Vladimir Voevodsky is a Professor at the Institute for Advanced Study in Princeton. He was awarded 2002 Fields Medal in Beijing for developing a new cohomology theory for algebraic varieties. The last lead to the proof of Milnor's and recently of Bloch Kato conjectures.

Voevodsky, whose field within mathematics is algebraic geometry, is known for his work in the homology theory of schemes, algebraic K-theory, and interrelations between algebraic geometry and algebraic topology. He has been concerned with a synthesis of algebraic geometry and homotopy theory, two major branches of modern mathematics. His interest has moved recently to Math Biology.