Talks will begin at 2:30pm of Thursday (Nov 7) and end by 12pm on Sunday (Nov 10).
2:30pm Sebastian Casalaina-Martin (Slides)
4:00pm Karl Schwede (Slides)
9:30am Isabel Vogt
11:00am Harold Blum
1:30pm Jakub Witaszek
3:00pm Radu Laza
4:15pm Sam Grushevsky
9:30am Giulia Saccà
11:00am Iacopo Brivio
2:00pm Kevin Tucker
3:30pm Ivan Cheltsov (Slides)
6pm Poster Session (those presenting posters meet by the math lounge at 5:40pm)
6:30pm Catered Dinner
9:30am Suchitra Pande
11:00am Paolo Cascini
Harold Blum (Utah)
Title: Moduli of boundary polarized CY surface pairs
Abstract: While the theories of KSBA stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fano varieties, respectively, the case of K-trivial varieties remains less well understood. I will discuss a new approach to this problem in the case of Calabi-Yau pairs (X,D), where D is ample, in which we consider all semi-log-canonical degenerations. One challenge of this approach is that the set of such degenerations is unbounded. Nevertheless, in the case of surface pairs, we construct a projective moduli space on which the Hodge line bundle is ample. This is based on joint work with Yuchen Liu that builds on previous work with Ascher, Bejleri, DeVleming, Inchiostro, Liu, and Wang.
Iacopo Brivio (Harvard)
Title: Extension of pluricanonical forms in positive characteristic
Abstract: a famous theorem by Siu states that, if X--->D is a smooth projective family of complex manifolds over the disk, then any pluricanonical form on the central fiber X_0 can be extended to the total space. Earlier, Nakayama showed that such an extension theorem is a consequence of the MMP and Abundance conjectures. In the first part of this talk, I will discuss some counterexamples to Siu's result in positive characteristic. In the second part, I will show how one can recover Siu's theorem for deformations of good minimal models with arithmetically well behaved Iitaka fibration, as well as an "up-to-p-power" version of Nakayama's result.
Sebastian Casalaina-Martin (Colorado)
Title: Moduli spaces of cubic hypersurfaces (Slides)
Abstract: In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus will be on explaining why two natural models of the moduli space of cubic surfaces are not isomorphic, or even K-eqiuvalent. I will also discuss a related moduli space, the moduli space of cubic surfaces with a marked line.
Paolo Cascini (Imperial)
Title: Threefolds of globally F-regular type.
Abstract: As a special case of a conjecture by Schwede and Smith, we show that a smooth complex projective threefold with nef anti-canonical divisor and of globally F-regular type is weak Fano. Joint work with T. Kawakami and S. Takagi.
Ivan Cheltsov (Edinburgh)
Title: K-moduli of four qubits (Slides)
Abstract: In this talk, I will describe K-moduli of smooth Fano 3-folds in the deformation family 4-1 (in the Mori-Mukai list). Smooth members of this family are divisors of degree (1,1,1,1) in the product of 4 copies of projective line. In particular, I will present all K-polystable singular limits of these Fano 3-folds and describe the associated connected component of the K-moduli space of Fano 3-folds of anticanonical degree 24. This is a joint work Maksym Fedorchuk (Boston), Kento Fujita (Osaka) and Anne-Sophie Kaloghiros (London).
Sam Grushevsky (Stony Brook)
Title: Compact subvarieties of A_g
Abstract: We determine the maximal dimension of a compact subvariety of the moduli space of complex abelian varieties, and the maximal dimension of a compact subvariety through a generic point. Based on joint work with Mondello, Salvati Manni, and Tsimerman.
Radu Laza (Stony Brook)
Title: Isotrivial Lagrangian fibrations of compact hyper-kähler manifolds
Abstract: I will discuss some basic results on the structure of isotrivial Lagrangian fibrations for hyper-Kaehler manifolds. First, we establish a basic dichotomy: Type A (Kummer type) and Type B (Deligne-Mostow type). Then, in both cases, we prove that the smooth fibers of the fibration are isogeneous to powers of elliptic curves. We classify the Type A case and show that only the K3^n and Kum_n can occur. Finally, I will discuss some partial results towards the classification of hyper-Kaehler manifolds of Kummer type, that is birational to a quotient of an abelian variety.
This is joint work with Yoon-Joo Kim and Olivier Martin.
Suchitra Pande (Utah)
Title: Positivity of the Limit F-signature
Abstract: The F-signature of a local singularity is an invariant in positive characteristics that measures the asymptotic properties of the Frobenius map. It can be used to detect regularity, strong F-regularity and other finiteness properties in positive characteristics. Thus, the F-signature seems to play a role analogous to the local volume of KLT singularities over the complex numbers.
This talk concerns the behavior of the F-signature under the process of reduction to characteristic p >> 0 of a fixed complex singularity. Motivated by applications to the sizes of local fundamental groups, Carvajal-Rojas, Schwede and Tucker conjectured that the F-signatures remain uniformly bounded away from zero when we reduce a complex KLT singularity to large characteristics . We will present joint works with Yuchen Liu, and with Anna Brosowsky, Izzet Coskun and Kevin Tucker, in which we prove this conjecture in many new cases including for most three-dimensional KLT singularities. Notably, the key new ideas in the proof were developed in the K-moduli theory of Fano varieties.
Giulia Saccà (Columbia)
Title: Compactification of Lagrangian fibrations
Abstract: Abstract: Lagrangian fibered Hyper-K\"ahler manifolds are the natural generalization of elliptic K3 surfaces and have been used to study and construct examples of compact Hyper-K\"ahler manifolds and (possibly singular) symplectic varieties. In this talk I will talk about some compactification techniques for quasi-projective Lagrangian fibrations, with applications to the study of Prym, Intermediate Jacobian, Albanese, dual fibrations etc.
Karl Schwede (Utah)
Title: A unified theory of test ideals in mixed characteristic (Slides)
Abstract: Multiplier ideals and test ideals are ways to measure singularities in characteristic zero and p > 0 respectively. In characteristic zero, multiplier ideals are computed by a resolution of singularities, or regular alteration, by comparing the canonical module of the base and the resolution. In characteristic p > 0, test ideals were originally defined via Frobenius, but under moderate hypotheses, can be computed via a sufficiently large alteration again via canonical modules. In mixed characteristic (for example over the p-adic integers) we show that various mixed characteristic analogs of multiplier/test ideals can be computed via a single sufficiently large alteration, at least when one builds in a small perturbation term. This perturbation is particularly natural from the perspective of either almost mathematics or the theory of pairs from birational algebraic geometry. Besides unifying the three pictures, this has various applications as this theory of multiplier/test ideals satisfies all the desired formal properties. This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Kevin Tucker, Joe Waldron and Jakub Witaszek.
Kevin Tucker (UIC)
Title: Mixed characteristic analogues of Log Canonical and F-pure singularities
Abstract: The relationship between singularities in complex birational algebraic geometry and those defined via the Frobenius endomorphism in
characteristic p > 0 has captivated researchers in both fields. In recent years, this connection has been extended to the mixed characteristic setting. In this talk, I will present some results on perfectoid pure singularities, a mixed characteristic analogue of log canonical and F-pure singularities from characteristic zero and characteristic p > 0, respectively. Time permitting, we will also discuss mixed characteristic analogues of Du Bois singularities and F-injective singularities. We will see that these classes of singularities have a unified description and satisfy many expected properties. This is based on joint work with Bhatt, Ma, Patakfalvi, Schwede, Waldron, and Witaszek.
Isabel Vogt (Brown)
Title: Stability of pushforwards
Abstract: Given a finite cover of curves, it is natural to consider the vector bundles on the base that arise as pushforwards of vector bundles from the source. In this talk we will consider the stability properties of the vector bundles that arise in this way. This talk is based on joint works with Izzet Coskun and Eric Larson.
Jakub Witaszek (Princeton)
Title: Hodge Theory of Singularities in Positive Characteristic
Abstract: I will start by reviewing various Hodge-theoretic invariants of complex singularities. Then, I will discuss a way to interpret these invariants through methods of positive characteristic. This is based on joint work with Tatsuro Kawakami.