May 7-10, 2024 Mathematics Department The Ohio State University Columbus, OH, USA Organizers: Dave Anderson Maria Angelica Cueto Jennifer Park Stefan Patrikis (leader)

This event is funded by the Mathematics Research Institute at The Ohio State University and the National Science Foundation under RTG grant DMS 2231565.

Registration: For planning purposes, all participants (including invited speakers), are asked to register online. Applications received by March 10 will receive full consideration. Please fill out the following registration form.

About

Speakers

Schedule

Titles and abstracts

Retreat

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The RTG Conferences and retreats

The graduate and postdoctoral training supported by the RTG award is anchored on five thematic years emphasizing different aspects of our combinatorial, arithmetic, and topological approaches to study algebraic varieties. Focused topics courses and research training seminars running each year will be complemented by an RTG Workshop, followed by a Group Retreat featuring a period of intensive mathematical collaboration, and promoting community-building through a goal-oriented activity.

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Speakers

The RTG 2024 Workshop at The Ohio State University will feature two 4-lecture mini-courses by distinguished plenary speakers supported by some background talks from graduate student participants.

Daniel Litt (University of Toronto)		Alexander Petrov (Clay Mathematics Institute/IAS)

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Schedule

Details to follow.

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Titles and abstracts

Daniel Litt:

• Title: Non-abelian cohomology and applications


• Abstract: The cohomology of a complex algebraic variety X has more structure than the cohomology of a general topological space: for example, its singular cohomology with Z coefficients carries a Hodge structure, and its singular cohomology with Q_ℓ-coefficients carries a Galois action. The non-abelian cohomology of X (namely, the space of local systems on X) carries analogues of these structures; these talks will be devoted to explaining these analogues and giving applications to concrete questions in algebraic geometry.


• Prerequisites: Basics of Hodge theory, the étale fundamental group.


• Tentative plan:
  1. Pre-history: the Painlevé VI equation and its generalizations.
     Plan: discuss the Painlevé VI equation, its generalizations (the non-abelian Gauss-Manin connection), and classical and (some) recent work on the topic.
  2. The complex story: Non-abelian Hodge theory and applications.
     Plan: Introduce Corlette-Simpson-Mochizuki's non-Abelian Hodge theory. Give some basic applications, e.g. to classifying rank 2 local systems on smooth varieties and to basic questions in surface topology.
  3. ℓ-adic aspects: de Jong's conjecture, companions, and applications.
     Plan: Discuss Galois action on representations of Π₁, and applications of de Jong's conjecture and companions (e.g. integrality of cohomologically rigid local systems).
  4. Mod p aspects, and some conjectures.
     Plan: p-curvature, non-abelian Hodge theory in characteristic p. Some conjectures.

Alexander Petrov:

• Title: p-adic Riemann-Hilbert correspondence


• Abstract: When studying cohomology of algebraic varieties, it is often fruitful to consider how the structures present on the cohomology vary, as the variety varies in a family. In complex algebraic geometry, this is governed by the notion of a variation of Hodge structures. Central to the study of variations of Hodge structures is the Riemann-Hilbert correspondence: the equivalence between vector bundles with a flat connection on a smooth proper algebraic variety X over C and local systems of finite dimensional C-vector spaces on X.
  We will consider the analog of this story in p-adic geometry, mostly following the works of Scholze and Liu-Zhu, which are based on the foundational work of Fontaine on p-adic Hodge theory. For a smooth algebraic variety X over the field of p-adic numbers Q_p, we will discuss a version of the p-adic Riemann-Hilbert functor of Liu-Zhu, which attaches to every étale Q_p-local system on X a structure that is not simply a vector bundle with a flat connection, but rather a version of Deligne's notion of a t-connection. This discrepancy between p-adic and complex Riemann-Hilbert functors stems from the fact that Q_p is not algebraically closed and the study of local systems on X includes studying representations of the Galois group of Q_p — the action of the Galois group of the base field turns out to be a crucial feature in this subject. We will then talk about the notion of de Rham local systems which is a special subclass of étale Q_p-local systems to which one can meaningfully attach vector bundles with a flat connection. Local systems coming from the cohomology of a family of algebraic varieties always fall into this subclass — this is a generalization of the de Rham comparison theorem relating de Rham and étale cohomology of a variety over Q_p.


• Prerequisites: The mini-course will assume familiarity with the theory of schemes, number theory over the p-adics (such as the structure of finite extension of Q_p, and basics of p-adic analysis), and with the basics of the theory of étale cohomology — most importantly the notion of the étale fundamental group. It will also be very helpful to have seen the formulations of the comparison theorems from rational p-adic Hodge theory. The discussion will not logically depend on the classical Hodge theory, but familiarity with the statement of the Hodge decomposition and the notion of a variation of Hodge structures will be useful for the purposes of motivation and intuition. The technical framework in which the construction of p-adic Riemann-Hilbert functor takes place is rigid-analytic geometry and the theory of perfectoid spaces, but familiarity with these will not be assumed — the mini-course will not cover them thoroughly, but we will discuss examples that highlight the phenomena powering the theory.

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RTG Retreat

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Back to the main website of the RTG: "Arithmetic, Combinatorics, and Topology of Algebraic Varieties".

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