Homotopy Theory and Arithmetic Geometry: Motivic and Diophantine Aspects

9-13 July 2018
Imperial College London

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Lecture courses
Denis-Charles Cisinski Cohomological methods in intersection theory
Tomer Schlank Homotopical manifestations of rational points and algebraic cycles
Kirsten Wickelgren Arithmetic enrichments of curve counts

Guest lectures
Paul Årne Østvær TBA
Jon Pridham TBA
Vesna Stojanovska TBA

Cohomological methods in intersection theory.

Lecturer: Denis-Charles Cisinski (University of Regensburg)

Assistant: Christopher Lazda (University of Amsterdam)

Brief syllabus: Grothendieck and his school developed \(l\)-adic cohomology to prove the Weil conjectures. In particular, for each prime number \(l\), one can apply cohomological methods to \(l\)-adic cohomology in order to define invariants of schemes, such as Euler-Poincaré characteristics or \(\zeta\)-functions. This means that we interpret algebraic cycles in terms of \(l\)-adic cohomology classes. In these lectures, we shall explain how motivic sheaves, as introduced by Voevodsky, allow to understand these invariants in a more intrinsic way: in the motivic world, there is no difference between algebraic cycles and cohomology classes. We shall illustrate this by studying \(\zeta\)-functions associated to mixed motives and by studying motivic characteristic classes. A strong emphasis will be put on the dictionary between constructible functions and constructible motivic sheaves.

Homotopical manifestations of rational points and algebraic cycles.

Lecturer: Tomer Schlank (Hebrew University of Jerusalem)

Assistant: Ambrus Pál (Imperial College London)

Brief syllabus: The classical obstructions to rational points over number fields can be united and reinterpreted using étale homotopy theory. These kinds of obstructions can be applied to sections of maps of arbitrary toposes. Furthermore this more general framework can be used to understand closely related notions such as new obstructions to the existence of zero cycles, or to solutions to certain embedding problems in Galois theory. I will survey the general theory of this obstruction theory, how it can be applied in many different arithmetic problems, using either motivic or étale homotopy theory.

Arithmetic enrichments of curve counts.

Lecturer: Kirsten Wickelgren (Georgia Tech)

Assistant: Frank Neumann (University of Leicester)

Brief syllabus: We will introduce some basic notions of \(\mathbb A^1\)-homotopy theory, concentrating on extending some enumerative problems in algebraic geometry from the complex numbers to other fields. We will begin by introducing the Grothendieck-Witt group of quadratic forms over a field, and the Eisenbud-Khimshiashvili-Levine signature formula giving the local degree of a topological map over the reals. This will also introduce the notion of topological degree, and this will lead into a discussion of Morel's degree map in \(\mathbb A^1\)-homotopy theory, taking values in the Grothendieck-Witt group. We will discuss realization functors in \(\mathbb A^1\)-homotopy theory, and how the Eisenbud-Khimshiashvili-Levine signature formula is the realisation over the reals of the \(\mathbb A^1\)-degree. We will use this to give an arithmetic enrichment of the count of 1-nodal curves in appropriate families, following recent work of Marc Levine. We will discuss an Euler number valued in the Grothendieck-Witt group, and an extension of the count of the 27 lines on a cubic surface over the complex numbers to an arbitrary field from joint work with Jesse Kass.


Lecturer: Paul Årne Østvær (University of Oslo)



Lecturer: Jon Pridham (University of Edinburgh)



Lecturer: Vesna Stojanovska (University of Illinois at Urbana-Champaign)