Algebro-Geometric and Homotopical Methods

One pillar of our subject is the foundational work of Alexander Grothendieck, especially his introduction of K-theory in his proof of the generalized Riemann-Roch Theorem. A second is the work of Daniel Quillen who developed the foundations of algebraic K-theory and the general approach of homotopical algebra. Quillen’s higher K-groups subsume much classical as well as previously undiscovered invariants of algebraic geometry and number theory; these are necessarily very difficult to compute. Newer invariants introduced by Spencer Bloch, Andrei Suslin, and Vladimir Voevodsky have offered more challenges as well as more insight. Homotopical algebra has proved a powerful tool for the study and computation of these invariants.

In the past decade, many important inter-locking conjectures in this subject have been affirmed: the Beilinson-Lichtenbaum Conjecture, the Bloch-Kato Conjecture and the Milnor Conjecture, and the Quillen-Lichtenbaum Conjecture have all subcombed to the combined efforts of many mathematicians. The techniques used to prove these results include the newly introduced motivic homotopy theory of Fabien Morel and Vladimir Voevodsky. There are many foundational questions which remain to be answered, computations remain difficult but more accessible with new methods, and numerous outstanding conjectures sharpen the focus of this expansion. Many problems are motivated by the very strong analogy with classical homotopy, while others involve algebraic geometry and algebraic groups.

The program at the Mittag-Leffler Institute will provide a synergistic environment encompassing current research across various fields including K-theory, algebraic cycles and values of L-functions, and motivic homotopy theory.

The conference is sponsored by the Simons Foundation.