Intersection theory and cobordism with a quadratic twist

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The aim of this project is to obtain new results in algebraic geometry via methods stemming from A1-homotopy theory. Using this theory, classical invariants with integral values can be refined into quadratic forms-valued ones, which carry new arithmetic information. The proposed research will concentrate in the following directions:(1) Refined enumerative geometry, a programme started by Levine and Kass-Wickelgren. In particular, the case of del Pezzo surfaces will be considered.(2) Use of refined invariants to provide new restrictions on fixed points of finite group actions on smooth projective varieties.(3) Properties of cohomology theories oriented by one of the groups SL, SLc, Sp, SO, O, Spin. The corresponding cobordism theories will be studied from a geometric point of view.(4) Applications of the stable Adams operations in hermitian K-theory.(5) Study of isotropic motivic categories.(6) Possible generalisations of Smirnov-Vishik "subtle Stiefel-Whitney classes" of quadratic forms to the context of other linear algebraic groups.