Special Year 2025-2026: Arithmetic Geometry, Hodge Theory, and O-minimality

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 12, 2026 | 10:45am - 11:45am

Add to calendar 03/12/2026 10:45 03/12/2026 11:45 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: Hodge-Theoretic Anabelian Geometry Speakers: Qixiang Wang, University of Paris, Saclay More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-12 Abstract: The anabelian phenomenon may be viewed as an arithmetic analogue of Mostow rigidity: it predicts that certain varieties can be reconstructed from their arithmetic fundamental groups. A celebrated result of S. Mochizuki shows that hyperbolic curves over p-adic fields exhibit this phenomenon. Inspired by non-abelian Hodge theory, we introduce a Hodge-theoretic analogue of arithmetic fundamental groups for complex Kähler manifolds, and show that an anabelian phenomenon occurs in complex-analytic geometry. In particular, hyperbolic Riemann surfaces are uniquely determined by their Hodge-theoretic fundamental groups, yielding a complex-analytic analogue of Mochizuki’s result. If time permits, we will discuss higher-dimensional generalizations. Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract: The anabelian phenomenon may be viewed as an arithmetic analogue of Mostow rigidity: it predicts that certain varieties can be reconstructed from their arithmetic fundamental groups. A celebrated result of S. Mochizuki shows that...

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 11, 2026 | 4:00pm - 5:00pm

Add to calendar 03/11/2026 16:00 03/11/2026 17:00 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: TBA Speakers: Jacob Tsimerman, Institute for Advanced Study More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-11 Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 11, 2026 | 2:30pm - 3:30pm

Add to calendar 03/11/2026 14:30 03/11/2026 15:30 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: Some Effective Special Point Results Speakers: Gareth Jones, University of Manchester More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-10 Abstract: I'll start by discussing some work with Binyamini, Schmidt, and Thomas in which we prove a uniform Manin-Mumford result for products of CM elliptic curves. I'll show how we apply this to obtain an effective Andre-Oort result for fibre powers of the Legendre family. I'll then discuss work in progress with Schmidt, on extending this to multiplicative extensions. Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract:
I'll start by discussing some work with Binyamini, Schmidt, and Thomas in which we prove a uniform Manin-Mumford result for products of CM elliptic curves. I'll show how we apply this to obtain an effective Andre-Oort result for fibre...

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 11, 2026 | 12:00pm - 1:00pm

Add to calendar 03/11/2026 12:00 03/11/2026 13:00 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: Matrix Points on Varieties Speakers: Asvin Gothandaraman, Institute for Advanced Study More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-9 Abstract: We study a non-commutative counterpart $C_n(X)$ of the symmetric product $Sym^n X$ of a space $X$, defined as the space of nxn-matrix valued points on $X$. Our main result will be a precise formula for the cohomology of this space in great generality, but we see this as the first step in a wider non-commutative world. In particular, there is an appealing analogy with Weil restriction that we hope is illuminating and a taste of possible future directions. This is joint work with Yifeng Huang, Ruofan Jiang and Yifan Wei (who is on the job market and drove much of this current research!). Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract: We study a non-commutative counterpart $C_n(X)$ of the symmetric product $Sym^n X$ of a space $X$, defined as the space of nxn-matrix valued points on $X$. Our main result will be a precise formula for the cohomology of this space in great...

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 11, 2026 | 10:45am - 11:45am

Add to calendar 03/11/2026 10:45 03/11/2026 11:45 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: The Dynamical Schinzel-Zassenhaus Conjecture and the Transfinite Diameter of Trees Speakers: Philipp Habegger, University of Basel More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-8 Abstract: In 2019, Dimitrov proved the Schinzel-Zassenhaus Conjecture. Harry Schmidt and I extended his general strategy to cover some dynamical variants of this conjecture. One common tool in both results is Dubinin's Theorem on the transfinite diameter of hedgehogs, a star-shaped tree in the plane. I will report on joint work in progress with Harry Schmidt. We find new upper bounds for the transfinite diameter of some finite topological trees. Our trees arise from the Hubbard tree of a postcritically finite polynomial and reflect its dynamical properties. As a consequence, we prove new lower bounds for the Call-Silverman or canonical height for a class of postcritically finite polynomials. Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract: In 2019, Dimitrov proved the Schinzel-Zassenhaus Conjecture. Harry Schmidt and I extended his general strategy to cover some dynamical variants of this conjecture. One common tool in both results is Dubinin's Theorem on the transfinite...

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 10, 2026 | 4:00pm - 5:00pm

Add to calendar 03/10/2026 16:00 03/10/2026 17:00 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: A Syntomic Perspective on Integral Canonical Models Speakers: Alex Youcis, University of Toronto More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-7 Abstract: Since Langlands's earliest paper on his now famous program, canonical integral models of Shimura varieties have occupied a central role in modern number theory. Steady progress has been made in the intervening 50 years toward the correct formulation of the notion of 'canonical', the study of the remarkable properties of such models, and ultimately of their construction. In this talk, I will discuss work with Madapusi (continuing previous work of Imai--Kato--Youcis in the abelian-type setting) on giving a new, and what we believe to be more robust, notion of 'canonical' using recent advances in $p$-adic Hodge theory due to Bhatt--Scholze, Drinfeld, Bhatt--Lurie, and others. In particular, we give constructions of models of pre-abelian type Shimura varieties for primes $p>2$, extending and more conceptually framing previous constructions of Kisin in the abelian-type setting. We moreover recapture the models for arbitrary Shimura varieties for $p\gg 0$ recently obtained by Bakker--Shankar--Tsimerman from this perspective. Finally, we discuss how this new notion of canonicity developed here provides the correct foundations to prove many strong results about such models, including Néron-type mapping properties and algebraization questions of the following kind: which $p$-divisible groups over $\overline{\mathbb{F}}_p$ arise as $A[p^\infty]$ for an abelian variety $A$? Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract: Since Langlands's earliest paper on his now famous program, canonical integral models of Shimura varieties have occupied a central role in modern number theory. Steady progress has been made in the intervening 50 years toward the correct...

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 10, 2026 | 2:30pm - 3:30pm

Add to calendar 03/10/2026 14:30 03/10/2026 15:30 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: Heights of Gross-Schoen and Ceresa Cycles Speakers: Ziyang Gao, Institute for Advanced Study More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-6 Abstract: In this talk, we construct a Zariski open dense locus in $M_g$ on which the Beilinson-Bloch height of the Gross-Schoen and Ceresa cycles is a Weil height, i.e. it has a lower bound and satisfies the Northcott property. This implies a generic positivity result of the heights of these cycles. This is joint work with Shouwu Zhang. Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract: In this talk, we construct a Zariski open dense locus in $M_g$ on which the Beilinson-Bloch height of the Gross-Schoen and Ceresa cycles is a Weil height, i.e. it has a lower bound and satisfies the Northcott property. This implies a...

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 10, 2026 | 12:00pm - 1:00pm

Add to calendar 03/10/2026 12:00 03/10/2026 13:00 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: TBA Speakers: Thomas Scanlon, University of California, Berkeley More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-5 Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 10, 2026 | 10:45am - 11:45am

Add to calendar 03/10/2026 10:45 03/10/2026 11:45 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: Constructing Holomorphic Functions on Universal Coverings of Complex Algebraic Varieties Speakers: Yohan Brunebarbe, University Bordeaux More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-4 Abstract: Which complex analytic spaces can arise as the universal cover of a complex algebraic variety? Motivated by this question, Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex — that is, whether there exists a proper holomorphic map from the universal cover of X to a Stein space. This was established in the linear case — when the fundamental group of X admits an almost faithful complex linear representation — by Eyssidieux–Katzarkov–Pantev–Ramachandran, using tools from non-abelian Hodge theory. In this talk, I will discuss a generalization of Shafarevich’s question to the case of non-compact algebraic varieties. This is joint work with Ben Bakker and Jacob Tsimerman. Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract: Which complex analytic spaces can arise as the universal cover of a complex algebraic variety? Motivated by this question, Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex —...

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Workshop on Recent Developments in Hodge Theory and O-minimality

March 09, 2026 | 4:00pm - 5:00pm

Add to calendar 03/09/2026 16:00 03/09/2026 17:00 Workshop on Recent Developments in Hodge Theory and O-minimality use-title Topic: Galois Action on Higher etale Homotopy Groups Speakers: Alexander Petrov, MIT More: https://www.ias.edu/math/events/workshop-recent-developments-hodge-theory-and-o-minimality-3 Abstract: Given an algebraic variety over a number field F, one can attach to it its etale cohomology groups, etale fundamental group, and higher etale homotopy groups, all equipped with an action of the absolute Galois group of F. The Galois action on etale cohomology is known to satisfy several special properties: for example, it is de Rham at places above p (where we consider etale cohomology with Q_p-coefficients) and eigenvalues of Frobenius elements are Weil numbers. Analogous facts, appropriately formulated, also hold for the Galois action on the fundamental group. On the contrary, Galois action on higher etale homotopy groups turns out to fail some of these properties. In this talk, I will discuss the observation that (dual of) higher etale homotopy groups of varieties over numbers fields often contain subrepresentations that are not de Rham at p. In our examples this stems from the difference between the cohomology of an arithmetic group and its pro-finite completion, and I will also discuss this (well-known) phenomenon, which turns out to behave very differently depending on whether the ambient reductive group of the arithmetic group is of Hodge type. This talk is based on joint works with Lue Pan and George Pappas. Simonyi Hall 101 a7a99c3d46944b65a08073518d638c23

Abstract: Given an algebraic variety over a number field F, one can attach to it its etale cohomology groups, etale fundamental group, and higher etale homotopy groups, all equipped with an action of the absolute Galois group of F. The Galois action...