Conference: Hodge theory, Birational Geometry & Atoms

Tony Pantev, University of Pennsylvania

Leonardo Cavenaghi, ICMS: Recent progresses in the theory of atoms

In this talk we report on some new results related to the theory of atoms obtained jointly with L. Katzarkov and M. Kontsevich, which will appear in the ``9.5 lectures in the theory of atoms'' we are working together.

Ivan Cheltsov, University of Edinburgh: G-birationally rigid cubic threefolds

A Fano variety X equipped with an action of a group G is called G-birationally rigid if X is a G-Mori fibred space (over a point) and X is not G-birational to any other G-Mori fibre space. In this talk I will classify all pairs (X,G) consisting of a (possibly singular) cubic threefold X and a subgroup G of its automorphism group such that X is G-birationally rigid.

Robert Śmiech, University of Edinburgh: G-birational rigidity of del Pezzo threefolds of degree 4

For a Fano variety equipped with an action of a (finite) group G, one can consider a G-equivariant Mori Program along with related notions, like G-solidity or G-birational rigidity. One of the most interesting questions that can be posed in such a setting is as follows: if H is a subgroup of G and X is a H-birationally rigid H-Fano, is X G-birationally rigid? In my talk I will present the results of my research into this question for del Pezzo threefolds of degree 4.

Refreshments

Yuri Tschinkel, NYU Courant & the Simons Foundation

Thorgal Hinault, CalTech: Unfolding of equivariant F-bundles. Comparison of Gross-Siebert and Keel-Yu mirror constructions

Constantin Teleman, UC Berkeley 

Alexander Perry, University of Michigan

Daniel Pomerleano, UMass Boston

Enrique Becerra, CINVESTAV: The Twisted Chiral de Rham Complex and the Elliptic Genus of Isolated Hypersurface Singularities

This talk presents a construction of the elliptic genus for isolated hypersurface singularities using the twisted chiral de Rham complex. By orbifoldizing the BRST cohomology through a specific automorphism, we define a chiral elliptic genus suitable for Landau-Ginzburg models. We demonstrate that the semiclassical limit of this invariant recovers the Steenbrink spectral polynomial, providing a generalization of the Borisov-Libgober construction for the non-weighted homogeneous case. Finally, we briefly discuss how the modularity of the elliptic genus provides insights into the distribution of spectral numbers.

Jenia Tevelev, UMass Amherst: Two-Ray Games of Fano Manifolds

Conjectural atomic semi-orthogonal decompositions of Fano manifolds predict that a Fano manifold with two extremal contractions should admit two compatible semi-orthogonal decompositions related by mutation. One of the contractions is often easier to understand than the other, and this can be used to produce a nontrivial semi-orthogonal decomposition for the second contraction by addition and subtraction. I will discuss several such categorical "two-ray games," including examples arising from moduli spaces of vector bundles on curves and moduli spaces of orthosymplectic complexes on K3 surfaces.

Paul Hacking, UMass Amherst: Moduli of Calabi--Yau 3-folds and mirror symmetry

Recent work of Bakker-Filipazzi-Mauri-Tsimerman constructs compactifications of moduli spaces of polarized Calabi--Yau manifolds generalizing the Baily-Borel compactification for K3 surfaces, proving conjectures of Green-Griffiths-Laza-Robles. We conjecture that the BFMT compactification for Calabi Yau 3-folds coincides with the compactification proposed by Morrison in 1993 based on mirror symmetry in a neighborhood of a large complex structure limit point. In particular, the boundary strata near the limit can be understood in terms of the birational geometry of the mirror.  We will present a correspondence between degenerations and contractions of the mirror supporting our conjecture.

Shayoun Bai, MIT: Quantum connection and arithmetics

Quantum connections, which are flat connections constructed from Gromov-Witten theory, have led to tremendous developments in enumerative geometry and beyond. Traditionally, the coefficient field is either Q or C. In these two lectures, I will outline some recent progress on studying the quantum connection as a flat connection over fields of positive characteristics and p-adic fields, which are closely tied with arithmetic objects involving p-curvature, Frobenius structure, and Fontaine-Laffaille modules. This is based on joint works with Lee, Pomerleano, and Seidel.

Andrew Harder, Lehigh University

Paolo Cascini, Imperial College London: Birational boundedness of stable families

Ron Donagi, University of Pennsylvania

Brendan Hassett, Brown University

Giovanni Neto, ICMS Sofia

Pedro Muniz, ICMS Sofia

Coll FAPESPI - IMSA

Jacqueline Mesquite, Universidade de Brasilia

Followed by Refreshments

Vladimiro Benedetti, Université Côte d'Azur: Quantum cohomology and irrationality of Gushel-Mukai fourfolds

Jérémy Guéré, University of Grenoble

Zhijia Zhang, NYU Courant

Paul Horja, IMSA: Dubrovin connection residues and categorical resolutions