Beginning with his Harvard Ph.D. thesis, Voevodsky has had the goal of creating a homotopy theory for algebraic varieties amenable to calculations as in algebraic topology.

In one of his most influential papers, A Weil (1949), proved the "Riemann hypothesis for curves over functions fields", an analogue in positive characteristic algebraic geometry of the classical Riemann hypothesis. In contemplating the generalization of this theorem to higher-dimensional varieties (subsequently proved by P Deligne (1974) following foundational work of A Grothendieck), Weil recognized the importance of constructing a cohomology theory with good properties. One of these properties is functoriality with respect to morphisms of varieties. J-P Serre showed with simple examples that no such functorial theory exists for abstract algebraic varieties which reflects the usual (singular) integral cohomology of spaces. Nevertheless, Grothendieck together with M Artin (1972, 1973) developed étale cohomology which succeeds in providing a suitable Weil cohomology theory provided one considers cohomology with finite coefficients (relatively prime to residue characteristics). This theory, presented in J Milne's book 'Étale cohomology' (1980), relies on a new formulation of topology and sophisticated developments in sheaf theory. In the present paper, the authors offer a very different solution to the problem of providing an algebraic formulation of singular cohomology with finite coefficients. Indeed, their construction is the algebraic analogue of the topological construction of singular cohomology, thereby being much more conceptual. Their algebraic singular cohomology with (constant) finite coefficients equals étale cohomology for varieties over an algebraically closed field. The proof of this remarkable fact involves new topologies, new techniques, and new computations reminiscent of the earlier work of Artin and Grothendieck.

To this outside observer, one of the most significant strands in the recent history of algebraic geometry has been the search for good cohomology theories of schemes. Each new cohomology theory has led to significant advances, the most famous being étale cohomology and the proof of the Weil conjectures. In a beautiful tour de force, Voevodsky has constructed all reasonable cohomology theories on schemes simultaneously by constructing a stable homotopy category of schemes. This is a triangulated category analogous to the stable homotopy category of spaces studied in algebraic topology; in particular, the Brown representability theorem holds, so that every cohomology theory on schemes is an object of the Voevodsky category. This work is, of course, the foundation of Voevodsky's proof of the Milnor conjecture. The paper at hand is an almost elementary introduction to these ideas, mostly presenting the formal structure without getting into any proofs that require deep algebraic geometry. It is a beautiful paper, and the reviewer recommends it in the strongest terms. The exposition makes Voevodsky's ideas seem obvious; after the fact, of course. One of the most powerful advantages of the Voevodsky category is that one can construct cohomology theories by constructing their representing objects, rather than by describing the groups themselves. The author constructs singular homology (following the ideas of A Suslin and Voevodsky (1996), algebraic K-theory, and algebraic cobordism in this way. Throughout the paper, there are very clear indications of where Voevodsky thinks the theory needs further work, and the paper concludes with a discussion of possible future directions.

Vladimir Voevodsky made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties. ... For about forty years mathematicians worked hard to develop good cohomology theories for algebraic varieties; the best understood of these was the algebraic version of K-theory. A major advance came when Voevodsky, building on a little-understood idea proposed by Andrei Suslin, created a theory of "motivic cohomology". In analogy with the topological setting, there is a relationship between motivic cohomology and algebraic K-theory. In addition, Voevodsky provided a framework for describing many new cohomology theories for algebraic varieties. One consequence of Voevodsky's work, and one of his most celebrated achievements, is the solution of the Milnor Conjecture, which for three decades was the main outstanding problem in algebraic K-theory. This result has striking consequences in several areas, including Galois cohomology, quadratic forms, and the cohomology of complex algebraic varieties.

Voevodsky's achievements are remarkable. First, he has developed a general homotopy theory for algebraic varieties. Second, as part of this general theory, he has formulated what appears to be the "correct" motivic cohomology theory and verified many of its remarkable properties. Third, as an application of this general approach, he has proved a long-standing conjecture of John Milnor relating the Milnor K-theory of a field to its étale cohomology (and to quadratic forms over the field).

We start with geometry, the category of topological spaces. We invent something about this geometrical world using our basically visual intuition. The notion of pieces comes exclusively from visual intuition. We somehow abstract it and re-write it in terms of category theory which provides this connecting language. And then we apply in a new situation, in this case in the situation of algebraic equations which is purely algebraic. So what we get is some fantastic way to translate geometric intuition into results about algebraic objects. And that is from my point of view the main fun of doing mathematics.

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.