Non-Abelian Hodge Theory and Applications

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Katzarkov 9700605 This project supports work on three problems on non-abelian Hodge theory and its applications. The first problem is concerned with work on the Shafarevich conjecture. The principal investigator will work on non-abelian analogs of the theorems of the (p,p) cycles, of the fixed pard theorem, and of the nonabelian Clemens Schmidt sequence. The second problem concerns a study of algebraic cycles on algebraic surfaces. The principal investigator will use the action of a certain infinite dimensional Heizenberg algebra on the kernel of the Albanese map to try to prove the Bloch conjecture. The third problem concerns work on families of K3 surfaces. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.