IAS Mathematics: Hodge Theory and O-Minimality in Algebraic Geometry

This lecture explores Melnikov functions as higher-order approximations in Hilbert's 16th problem, particularly for polynomial Hamiltonian perturbations, building on the vanishing of first-order Abelian integrals.

This presentation discusses definable sup-norms of sections of metrized line bundles over families of arithmetic varieties, utilizing Riemann-Zariski spaces to derive new definability results for globally valued fields.

This lecture covers recent advancements in constructing completions of general period mappings, focusing on non-classical settings and generalized toroidal completions as defined by Ash–Mumford–Raport–Tai and Kato–Nakayama–Usui.

This talk introduces a Hodge-theoretic analogue of arithmetic fundamental groups for complex Kähler manifolds, demonstrating an anabelian phenomenon in complex-analytic geometry inspired by Mostow rigidity.

This lecture examines the recovery of (quasi)-period functions from period maps through differentiation, using o-minimality results to determine conditions under which this reconstruction is possible, up to an algebraic closure.

This session presents a uniform Manin-Mumford result for products of CM elliptic curves, applying it to obtain an effective Andre-Oort result for fibre powers of the Legendre family and discussing extensions to multiplicative extensions.

This talk introduces Cn(X), a non-commutative analogue of the symmetric product SymnX, presenting a precise formula for its cohomology and exploring analogies with Weil restriction in a broader non-commutative context.

Philipp Habegger presents joint work with Harry Schmidt, extending Dimitrov's proof of the Schinzel-Zassenhaus Conjecture to its dynamical variants. The presentation leverages Dubinin's Theorem on the transfinite diameter of hedgehogs and applies these concepts to Hubbard trees, yielding new lower bounds for the canonical height of postcritically finite polynomials.

This talk offers a new syntomic framework for integral canonical models of Shimura varieties, extending previous work in the abelian-type setting and contributing to the Langlands Program.

This lecture constructs a Zariski open dense locus where the Beilinson-Bloch height of Gross-Schoen and Ceresa cycles satisfies the Northcott property, establishing a generic positivity result for these heights.

This talk explores definable quotient spaces in the context of unlikely intersection problems, discussing formalisms introduced by Zilber, Bakker-Klingler-Tsimerman, and others to address the non-definability of analytic covering maps.

This lecture addresses Shafarevich's question about the holomorphic convexity of universal covers of smooth projective varieties, particularly in the linear case where the fundamental group admits an almost faithful complex linear representation.

This lecture investigates the Galois action on higher etale homotopy groups of algebraic varieties over number fields, extending known properties from etale cohomology regarding de Rham representations and Weil numbers for Frobenius eigenvalues.

This lecture demonstrates the falsity of the integral Hodge conjecture for very general abelian varieties of dimension ≥ 4, using a new combinatorial invariant of regular matroids to obstruct the algebraicity of minimal curve classes.

This talk confirms an Ax-Schanuel statement underlying Simpson's characterization of bi-algebraic subspaces in the rank one Riemann-Hilbert correspondence, thereby recovering Simpson's bi-algebraicity result.