We propose to continue the study of algebraic cycles in the cohomology rings of finite groups, as initiated in the candidate's PhD thesis. (A cohomology class is called algebraic if it is Poincare dual to the fundamental class of an algebraic manifold mapping into the space under consideration, here the classifying space of a finite group).
We have been able in our previous work to do a lot of explicit computations with these. We have been led to formulate a conjecture, which predicts the behaviour of the rings we study. Roughly speaking, it says that the algebraic cycles are completely determined by the abelian subgroups of the group considered and their conjugacies.
We propose to examine this conjecture on new examples and hopefully, prove it. This involves many branches of mathematics: topology, group theory, and algebraic geometry (which is a new thing, the cohomology of groups not being traditionally associated directly with algebraic geometry). We have already developed a number of original techniques to tackle the problem, including the use of complex cobordism and the Steenrod algebra.
It would be invaluable help for the researcher to benefit from the experience of the mathematical team at the Universitat Autonoma de Barcelona. People there have worked on similar problems recently, and Carles Broto, our scientist in charge, knows our field particularly well.
Call for proposal
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