Hodge theory and algebraic cycles

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The Hodge conjecture is one of the central problems of algebraic geometry. If true, it would give a strong connection between topology and algebraic geometry. The integral Hodge conjecture is a stronger statement which fails for some algebraic varieties, as shown by Atiyah and Hirzebruch. Nonetheless, the project aims to prove the integral Hodge conjecture for a large class of 3-dimensional varieties. In another direction, the integral Hodge conjecture often fails for quotients of varieties by finite groups. The project will compute the Chow ring of algebraic cycles for quotient varieties by many finite groups. This can be viewed as finding the correct replacement for the integral Hodge conjecture when it fails.Algebraic geometry is the study of shapes defined by polynomial equations. It is a central area of mathematics, at the border between arbitrary shapes (topology) and the theory of equations (algebra and computation). Users of mathematics have always had to solve systems of polynomial equations, and that remains a computationally challenging problem today. The project deals with the question of which shapes can be moved continuously to become defined by polynomial equations. There is a plausible answer, but a proof seems exceedingly hard to find. The project aims to solve the problem in the most important low-dimensional cases.