Conference: Lefschetz Centennial

Dates: March 24-29, 2025
Location: Lakeside Village Pavilion, 1280 Stanford Dr, Coral Gables, FL 33146
Live Video Available via Zoom

To register, please click here.


IMSA is hosting a conference to honor Solomon Lefschetz, a pioneering mathematician whose work in algebraic topology, algebraic geometry, and differential equations has shaped modern mathematics. Lefschetz, who overcame adversity after losing both hands, made groundbreaking contributions that continue to influence the field. The event will bring together experts to discuss advancements in these areas and celebrate his enduring legacy.


Schedule

Monday, March 24, 2025

10:00am

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. 

11:30am

Ron Donagi, UPenn

2:00pm

Alberto Verjovsky, UNAM: Adelic loop groups

A proalgebraic toric space over  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 (*)n.

In the case of the Riemann sphere P1, with the standard action of *,

we obtain as proalgebraic completion the adelic projective line P1(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 P1(ℚ) is isomorphic to the additive rationals  (ℚ,+), and prove the Birkhoff-Grothendieck splitting theorem for holomorphic bundles of higher rank over P1(ℚ), as sums of line bundles.

3:15pm

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).

4:30pm

Benjamin Bakker, University of Illinois Chicago


Tuesday, March 25, 2025

10:00am

Tony Pantev, UPenn 

11:30am

Leonardo Cavenaghi, CAMPINAS

2:30pm

John Morgan, Columbia University

3:35pm

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.

4:40pm

Matt Kerr, Washington University in St. Louis

5:45pm

TWAS in IMSA: Jaqueline Mesquita, Universidad de Brasilia

6:45pm

Reception


Wednesday, March 26, 2025

10:00am

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.

11:30am

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.

2:00pm

Laurent Meersseman, Université d’Angers, France

3:30pm

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.

4:45pm

Frontiers Lecture: Moira Chas, Stony Brook University

5:40pm

Reception


Thursday, March 27, 2025

10:00am

Maxim Kontsevich, IHES

11:30am

Dennis Sullivan, Stony Brook University

2:00pm

Herb Clemens, Ohio State University

3:30pm

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

The speaker began her mathematical career by applying Lefschetz’s fixed-point theorem to the dynamics of surfaces. She will discuss how this work led to the study of different aspects of curves on surfaces. More precisely, each free homotopy class of closed oriented curves on a Riemann surface determines three numbers: its minimal self-intersection number, its geometric length (in a given hyperbolic metric), and its word length with respect to a fixed minimal generating set of the fundamental group. These numbers, as well as the Goldman Lie bracket of two such classes, can be explicitly computed or approximated using computational methods. In this talk, we will explore these numbers, their relationships, their computational aspects, and how this line of research led to the discovery of String Topology.

4:30pm

Remembering Lefschetz Presentation

6:15pm

Reception


Friday, March 28, 2025

10:00am

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.

11:30am

Lino Grama, CAMPINAS

2:00pm

Nikita Nekrasov, Stony Brook

3:30pm

Filip Zivanovic, Simons Center for Geometry and Physics

4:45pm

Christian Schnell, Stony Brook

5:45pm

Nikita Nekrasov joint with Physics


Saturday, March 29, 2025

9:30am

Bruno de Oliveira, University of Miami

10:45am

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.

12:00pm

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. 

1:10pm

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.


Participants

Benjamin Bakker University of Illinois Chicago Matt Kerr Washington University in St. Louis
Moira Chas Stony Brook Enrique Becerra CINVESTAV
Herb (Charles) Clemens The Ohio State University Harvey Friedman The Ohio State University
Mark Andrea de Cataldo Stony Brook Tony Pantev UPenn
Phillip Griffths University of Miami & IAS Alberto Verjovsky UNAM
Ernesto Lupercio CINVESTAV Ron Donagi UPenn
Bruno Klinger Humboldt Universität zu Berlin Carlos Simpson University of Nice
Claire Voisin CNRS Institut de Mathématiques de Jussieu-Paris Kyoung Seog Lee POSTECH
Rodolfo Aguilar University of Miami Leonardo Cavenaghi CAMPINAS
Filip Zivanovic Simons Center for Geometry and Physics John Morgan Columbia University
Laurent Meersseman Université d’Angers Jaqueline Mesquita University of Brasilia
Bruno de Oliveira University of Miami

 

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