Algebraic groups, quadratic forms and related topics

In the early 19th century a young French mathematician E. Galois laid the foundations of abstract algebra by using the symmetries of a polynomial equation to describe the properties of its roots. One of his discoveries was a new type of structure, formed by these symmetries. This structure, now called a “group”, is central to much of modern mathematics. The groups that arise in the context of classical Galois theory are finite groups. Galois died in a duel at the age of 20; his work was not understood or recognized during his lifetime. It took much of the rest of the 19th century for his ideas to be rediscovered, absorbed and applied in other contexts. In the context of differential equations, these ideas were advanced by E. Picard, who, following a suggestion of S. Lie, assigned a Galois group to an ordinary differential equation. This group is no longer finite. It naturally acts on the n-dimensional complex vector space V of holomorphic solutions to the equation. In modern language, the Galois groups that arose in Picard’s theory are algebraic subgroup of GL(V ). This construction was developed into differential Galois theory by J. F. Ritt and E. R. Kolchin in the 1930s and 40s. Their work was a precursor to the modern theory of algebraic groups, founded by A. Borel, C. Chevalley, J.-P. Serre, T. A. Springer, and J. Tits starting in the 1950s. From the modern point of view algebraic groups are algebraic varieties, with group operations given by algebraic morphisms. Linear algebraic groups can be embedded in GLn for some n, but such an embedding is no longer a part of their intrinsic structure. Borel, Chevalley, Serre, Springer and Tits used algebraic geometry to establish basic structural results in the theory of algebraic groups, such as conjugacy of maximal tori and Borel subgroups, and the classification of simple linear algebraic groups over an algebraically closed field. Considerations in number theory, among others, require the study of algebraic groups over fields that are not necessarily algebraically closed. This more general setting was the primary focus for much of the work discussed in the workshop. In the 1960s J. Tate and J.-P. Serre developed a theory of Galois cohomology. Serre published his influential lecture notes on this topic in 1964; they have been revised and reprinted several times since then. Galois cohomology can be viewed as an important special case of étale cohomology, In the 1970s the work of H. Bass, J. Tate and Milnor, established connections among Milnor K-theory, Galois cohomology, and graded Witt rings of quadratic forms. In particular, Milnor asked whether (in modern language) Milnor K-theory modulo 2, is isomorphic to Galois cohomology with F2 coefficients. A more general question, with 2 replaced by an odd prime, was posed in subsequent work of Bloch and Kato and became known as the Bloch-Kato conjecture. Since the 1980s there has been rapid progress in the theory of algebraic groups due to the introduction of powerful new methods from algebraic geometry and algebraic topology. This new phase began with the Merkurjev-Suslin theorem which settled a long-standing conjecture in the theory of central simple algebras, using a combination of techniques from algebraic geometry and K-theory. The Merkurjev-Suslin theorem was a starting point of the theory of motivic cohomology constructed by V. Voevodsky. Voevodsky developed a homotopy theory in algebraic geometry similar to that in algebraic topology. He defined a (stable) motivic homotopy category and used it to define new cohomology theories such as motivic cohomology, K-theory and algebraic cobordism. Voevodsky’s use of these techniques resulted in the solution of the Milnor conjecture for which he was awarded a Fields Medal in 2002. For a discussion of the history of the Milnor conjecture