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.
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 communitybuilding through a goaloriented activity.
The
YouTube channel of the RTG group has playlists containing a record of the talks delivered at each of the conferences. The playlist for the RTG24 conference on Local Systems in Algebraic Geometry can be found in
this link.
Speakers
The RTG 2024 Workshop at The Ohio State University will feature two 4lecture minicourses by distinguished plenary speakers supported by some background talks from graduate student participants.
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Schedule
We expect participants to arrive on Monday May 6 and depart on Friday May 10 in the afternoon.
All Talks (including the lightning round) will take place in room CH 240. Coffee breaks and the conference dinner will be held in MW 724.
Schedule of talks
Day 
Time 
Location 
Speaker/Event 
Title 
May 7 
9:20am9:30am 
CH 240 
organizers 
Welcome Remarks 
— 
9:30am10:30am 
— 
Christian Klevdal 
Litt background 1: The classical RiemannHilbert correspondence. 
— 
10:30am11:00am 
MW 724 
Coffee break 

— 
11:00am12:00pm 
CH 240 
Daniel Litt 
Litt Lecture 1 
— 
12:00pm1:30pm 

Lunch break 

— 
1:302:30pm 
CH 240 
Gleb Terentiuk 
Petrov background 1: Étale fundamental groups and local systems. 
— 
2:30pm3:00pm 
MW 724 
Coffee break 

— 
3:00pm4:00pm 
CH 240 
Alexander Petrov 
Petrov Lecture 1 
May 8 
9:30am10:30am 
— 
Yilong Zhang 
Litt background 2: Variations of Hodge structure and Higgs bundles. 
— 
10:30am11:00am 
MW 724 
Coffee break 

— 
11:00am12:00pm 
CH 240 
Daniel Litt 
Litt Lecture 2 
— 
12:00pm1:30pm 

Lunch break 

— 
1:302:30pm 
CH 240 
Alice Lin 
Petrov background 2: padic Hodge theory. 
— 
2:30pm3:00pm 
MW 724 
Coffee break 

— 
3:00pm4:00pm 
CH 240 
Alexander Petrov 
Petrov Lecture 2 
— 
4:15pm5:15pm 
MW 724 

Poster Session (more information) 
— 
5:30pm7:30pm 
MW 724 

Conference dinner 
May 9 
9:30am10:30am 
— 
Jake Huryn 
Litt background 3: Rigid local systems. 
— 
10:30am11:00am 
MW 724 
Coffee break 

— 
11:00am12:00pm 
CH 240 
Daniel Litt 
Litt Lecture 3 
— 
12:00pm1:30pm 

Lunch break 

— 
1:302:30pm 
CH 240 
Zeyu Liu 
Petrov background 3: Rigidanalytic geometry. 
— 
2:30pm3:00pm 
MW 724 
Coffee break 

— 
3:00pm4:00pm 
CH 240 
Alexander Petrov 
Petrov Lecture 3 
— 
4:00pm4:15pm 
— 

Break 
— 
4:15pm5:15pm 
— 
Ziquan Yang 
Litt background 4: Algebraic differential equations in characteristic p > 0. 
May 10 
9:20am10:20am 
CH 240 
Daniel Litt 
Litt Lecture 4 
— 
10:20am10:40am 
(outside CH 240) 
Coffee break 

— 
10:40am11:40am 
CH 240 
Andy Jiang 
Petrov background #4: More on variations of Hodge structures, period
maps. 
— 
11:40am12:00pm 
— 
Break 

— 
12:00pm1:00pm 
— 
Alexander Petrov 
Petrov Lecture 4 
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Titles and abstracts
Daniel Litt:
 Title: Nonabelian 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 nonabelian 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:
 Prehistory: the Painlevé VI equation and its generalizations.
Plan: discuss the Painlevé VI equation, its generalizations (the nonabelian GaussManin connection), and classical and (some) recent work on the topic.
 The complex story: Nonabelian Hodge theory and applications.
Plan: Introduce CorletteSimpsonMochizuki's nonAbelian Hodge theory. Give some basic applications, e.g. to classifying rank 2 local systems on smooth varieties and to basic questions in surface topology.
 ℓadic aspects: de Jong's conjecture, companions, and applications.
Plan: Discuss Galois action on representations of Π_{1}, and applications of de Jong's conjecture and companions (e.g. integrality of cohomologically rigid local systems).
 Mod p aspects, and some conjectures.
Plan: pcurvature, nonabelian Hodge theory in characteristic p. Some conjectures.
Alexander Petrov:
 Title: padic RiemannHilbert 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 RiemannHilbert 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 Cvector spaces on X.
We will consider the analog of this story in padic geometry, mostly following the works of Scholze and LiuZhu, which are based on the foundational work of Fontaine on padic Hodge theory. For a smooth algebraic variety X over the field of padic numbers Q_{p}, we will discuss a version of the padic RiemannHilbert functor of LiuZhu, 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 tconnection. This discrepancy between padic and complex RiemannHilbert 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 minicourse will assume familiarity with the theory of schemes, number theory over the padics (such as the structure of finite extension of Q_{p}, and basics of padic 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 padic 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 padic RiemannHilbert functor takes place is rigidanalytic geometry and the theory of perfectoid spaces, but familiarity with these will not be assumed — the minicourse will not cover them thoroughly, but we will discuss examples that highlight the phenomena powering the theory.
Each of the lectures given by our lectures will be preceded by talks given by some of the conference participants to cover background material. The content of each pretalk and the designated speaker can be found on this pdf file.
Additional backgroup material can be found in the notes of S. Patrikis topics course on Étale Coholomogy given at OSU in Fall 2022.
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Poster Session
On Wednesday May 8, 4:155:15pm, we will have a poster session in MW 724. You can find the title and abstracts of our presenters in this pdf file.
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RTG Retreat
The retreat will take place at Burr Oak Lodge and Conference Cente, located in Burr Oak State National Park , during May 1417, 2024.
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Back to the main website of the RTG: "Arithmetic, Combinatorics, and Topology of Algebraic Varieties".
The Ohio State University, Department of Mathematics, 231 W. 18th Avenue, Columbus, OH, 43210, USA.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the National Science Foundation.