Conference: Lefschetz Centennial

Carlos Simpson, University of Nice: Lefschetz devissage and Hodge theory

Lefschetz proposed the study of the topology of complex algebraic varieties and gave a general method revolving around families of hyperplane sections. The topological invariants of the hyperplane sections vary in local systems over the base projective space, with singularities along the discriminant divisor, and the monodromy representations encode topological data. The fundamental group of the complement of the discriminant divisor thus plays a major role. This led to Griffiths' notion of variation of Hodge structure. Following the basic study of asymptotic properties of degenerations by Griffiths, Schmid, Clemens, Steenbrink, Cattani, Kaplan, Kashiwara, Kawai, Deligne, Saito and others, Zucker's theorem explains how to integrate the VHS coming from the Lefschetz pencil, to the Hodge-theoretic information of the original complex structure. This was generalized by Saito. These theories are inputs to the notion of nonabelian Hodge correspondence. We'll explore some recent aspects. 

Ron Donagi, UPenn

Alberto Verjovsky, UNAM: Adelic loop groups

A proalgebraic toric space over ℂ is the inverse limit of all finite branched covers over a normal toric variety, over ℂ, with branching set as the invariant divisor under the algebraic torus action. These are completions (compactifications) of the adelic abelian proalgebraic group [(ℂ*_{ℚ})]^n, , which is the profinite completion of the algebraic torus (ℂ*)ⁿ.

In the case of the Riemann sphere ℂP¹, with the standard action of ℂ*,

we obtain as proalgebraic completion the adelic projective line ℂP¹(ℚ). We define holomorphic/meromorphic functions and holomorphic vector bundles over ℂP¹(ℚ). We also introduce the adelic loop group of a Lie group G, which is the space of maps from the adele class group 𝔸/ℚ to G; we describe their properties and prove Birkhoff's factorization for these groups. We sketch the proof that the adelic Picard group of holomorphic line bundles over ℂP¹(ℚ) is isomorphic to the additive rationals  (ℚ,+), and prove the Birkhoff-Grothendieck splitting theorem for holomorphic bundles of higher rank over ℂP¹(ℚ), as sums of line bundles.

Bruno Klinger, Humboldt Universität zu Berlin: Around the Zilber-Pink conjecture

The Zilber-Pink conjecture describes the Hodge locus of subvarieties of Shimura varieties, and more generally the Hodge locus of any variation of Hodge structure. In this talk I will review recent progress towards this conjecture (based on past work with Baldi and Ullmo, work of Baldi-Urbanik,  and work in progress with Tayou).

Benjamin Bakker, University of Illinois Chicago

Tony Pantev, UPenn 

Leonardo Cavenaghi, CAMPINAS

John Morgan, Columbia University

Ken Baker, University of Miami: Morse-Novikov numbers of 3-manifolds

The Morse-Novikov number of a homotopy class of circle valued functions on a 3-manifold counts the minimum number of critical points among Morse representatives. Viewing circle valued Morse functions more coarsely as their associated handle decompositions, we recast this count as a 'handle number' and leverage the theories of generalized Heegaard splittings and sutured manifolds to advance our understanding of these counts.   We will survey key results and curious phenomena developed and observed in our work and joint works with Fabiola Manjarrez-Gutierrez.

Matt Kerr, Washington University in St. Louis

TWAS in IMSA: Jaqueline Mesquita, Universidad de Brasilia

Reception

Claire Voisin, CNRS Institut de Mathématiques de Jussieu-Paris: Universally defined cycles (Zoom)

I introduce the notion  of universally defined cycles (for smooth varieties of dimension d) and prove that any unversally defined cycle is given on  generic fibers by a polynomial in the Chern classes, which can be seen as a higher dimensional  version of the Franchetta conjecture. I will also give the motivation for the main definition and explain a conjectural extension of that result to universally defined cycles on powers of varieties of given dimension.

Ernesto Lupercio, CINVESTAV: From Quantum Toric Spaces to Motivic Rings

In this talk, I will trace a path from LVM manifolds to quantum toric spaces and their moduli, exploring their connections to sandpile groups and self-organization. This will lead to the motivation behind certain motivic rings. This is joint work with Katzarkov, Lee, and Meersseman, as well as with Verjovsky, and also Shkolnikov, and Kalinin.

Laurent Meersseman, Université d’Angers, France

Kyoung Seog Lee, POSTECH: Motivic aspects of complex analytic geometry

The motivic nature of cohomology rings of algebraic varieties is one of the key tools to study algebraic varieties. In this talk, I will discuss how to study various motivic aspects of cohomology rings of complex analytic varieties via o-minimal geometry. This talk is based on joint works with Ludmil Katzarkov, Ernesto Lupercio and Laurent Meersseman.

Frontiers Lecture: Moira Chas, Stony Brook University

Reception

Maxim Kontsevich, IHES

Dennis Sullivan, Stony Brook University

Herb Clemens, Ohio State University

Frontiers Lecture: Moira Chas, Stony Brook University: From Lefschetz to String topology

Remembering Lefschetz Presentation

Reception

Mark Andrea de Cataldo, Stony Brook: The decomposition theorem for the logarithmic Hitchin fibration

I will report on ongoing joint work with Andres Fernadez Herrero, Roberto Fringuelli and Mirko Mauri on the moduli space of semistable logarithmic principal G-Higgs bundles on a smooth curve. For any given degree d in the algebraic fundamental group of G, we exhibit a uniform description of the decomposition theorem for the corresponding Hitchin fibration of degree d logarithmic G-Higgs bundles.

Lino Grama, CAMPINAS

Nikita Nekrasov, Stony Brook

Filip Zivanovic, Simons Center for Geometry and Physics

Christian Schnell, Stony Brook

Nikita Nekrasov joint with Physics

Bruno de Oliveira, University of Miami

Rodolfo Aguilar, IMSA: Calabi-Yau vs log Calabi-Yau threefolds

We will compare the Hodge theory and the geometry of smooth projective Calabi-Yau threefolds against quasi-projective threefolds obtained by removing a smooth anti-canonical K3 surface to a smooth projective Fano threefold. Focus will be centered around Yukawa cubics, curves and Abel-Jacobi maps. Joint work with Ph. Griffiths and M. Green.

Enrique Becerra, IMSA: The stringy spectrum of orbifolds

In this talk, I will introduce the stringy spectrum of orbifolds and state its basic properties. Roughly speaking,  this is a motivic measure of orbifolds inspired in the classical Steenbrink spectrum of isolated hypersurface singularities. 

Yilong Zhang, Purdue University: Periods of elliptic-elliptic surfaces and K3 surfaces

An elliptic-elliptic surface is an elliptic surface over a genus one curve and has p_g=1. It carries a K3-type Hodge structure, and its period map dominates a 10-dimensional ball quotient (Engel-Greer-Ward 23, Greer-Zhang 24). The period image also parameterizes elliptic K3 surfaces with a marked E8 singular fiber. So, it is natural to ask if the Hodge-theoretical correspondence between an elliptic-elliptic surface and its associated K3 surface is represented by an algebraic cycle. In a joint work with Arapura and Greer, we show this is true for certain examples that arise from base change of Kummer surfaces. The construction generalizes Shioda-Inose's construction for K3 surfaces.