A motivic Grothendieck–Teichmüller group

Motivic Homotopy Theory of Group Scheme Actions

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic K-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.

Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields

Motivic integration is a powerful technique to prove that certain quantities associated to algebraic varieties are birational invariants or are independent of a chosen resolution of singularities. We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general typ...

Motivic Exponential Integrals and a Motivic Thom-sebastiani Theorem

1.1. Let f and f ′ be germs of analytic functions on smooth complex analytic varieties X and X ′ and consider the function f ⊕ f ′ on X × X ′ given by f ⊕ f (x, x) = f(x) + f (x). The Thom-Sebastiani Theorem classically states that the monodromy of f ⊕ f ′ on the nearby cycles is isomorphic to the product of the monodromy of f and the monodromy of f ′ (in the original form of the Theorem [18] t...

Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields

The Motivic Fundamental Group of the Punctured Projective Line

We describe a construction of an object associated to the fundamental group of P−{0, 1,∞} in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products of Steinberg symbols in the motivic cohomology of the ground field.