Doctoral Prize
 DR. TONI ANNALA, Institute for Advanced Study
Cohomology of Algebraic Varieties [PDF]

Various cohomology theories have played a crucial part in the study of algebrogeometric objects since the birth of modern algebraic geometry in 1960s. Unlike in algebraic topology, not all cohomology theories in algebraic geometry are homotopy invariant, complicating the efforts of defining a good "homotopy category" of varieties, which would support all reasonable cohomology theories. For homotopyinvariant cohomology theories, Morel and Voevodsky constructed such a category in late 90s, which they referred to as $A^1$homotopy category. Recently, there have been several attempts to define an analogous category that captures nonhomotopyinvariant theories as well. We have been pursuing one candidate for such a category in joint work with R. Iwasa and M. Hoyois.
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