## Arithmetic Geometry and Number Theory RTG Seminar: Rational motivic path spaces

Seminar | September 25 | 3:10-5 p.m. | 891 Evans Hall

Ishai Dan-Cohen, Ben-Gurion University

Department of Mathematics

A central ingredient in Kim's work on integral points of hyperbolic curves is the “unipotent Kummer map” which goes from integral points to certain torsors for the prounipotent completion of the fundamental group, and which, roughly speaking, sends an integral point to the torsor of homotopy classes of paths connecting it to a fixed base-point. In joint work with Tomer Schlank, we introduce a space Ω of rational motivic loops, and we construct a double factorization of the unipotent Kummer map which may be summarized schematically as $\mbox {points} \to \mbox {rational motivic points} \to \Omega \mbox {-torsors} \to \pi _1\mbox {-torsors}.$ Our “connectedness theorem” says that any two motivic points are connected by a non-empty torsor. Our “concentration theorem” says that for an affine curve, Ω is actually equal to $\pi _1$.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.

yxy@berkeley.edu