The n-point configuration space of a space X (where n is a positive integer) is simply the open subset of X^n which parametrizes n-tuples of points that are pairwise distinct. The cohomology groups of such spaces (and how the homeomorphism group of X acts on these groups) are still poorly understood, despite that fact that their systematic study began already half a century ago. In this talk we focus on the case when X is a closed oriented surface. When X comes with a complex structure (so that it becomes a compact Riemann surface), the cohomology of its configuration space acquires an interesting mixed Hodge structure. We review this and discuss some of its topological repercussions. If time permits, we indicate how part of this structure relates to the space of conformal blocks.

Professor Eduard Looijenga is a Dutch mathematician who works in algebraic geometry and the theory of algebraic groups. He is a full professor of Yau Mathematical Sciences Center, Tsinghua University. He became a member of the Royal Netherlands Academy of Arts and Sciences in 1995, and in 2012 he became one of the inaugural fellows of the American Mathematical Society since his retirement from Utrecht University in 2013.