Jungkai Alfred Chen (National Taiwan University). Introduction to classification of threefolds of general type. Friday, Oct 4, 1:30-2:30pm. Huxley 140.
Abstract: In higher dimensional algebraic geometry, the following three types of varieties are considered to be the building blocks: Fano varities, Calabi-Yau varieties, and varities of general type. In the study of varieties of general type, one usually work on “good models” inside birtationally equivalent classes. Minimal models and canonical models are natural choices of good models.
In the first part of the talk, we will try to introduce some aspects of geography problem for threefolds of general type, which aim to study the distribution of birational invariants of threefolds of general type. In the second part of the talk, we will explore more geometric properties of those threefolds on or near the boundary. Some explicit examples will be described and we will compare various different models explicitly.
Mark Andrea de Cataldo (Stony Brook University). The P=W Conjecture in Non Abelian Hodge Theory. Friday, Oct 11, 1:30-2:30pm. Huxley 140.
Abstract: The complex singular cohomology groups of a projective manifold can be described in at least three ways via the de Rham Theorem and the Hodge Decomposition. By taking into account integral cohomology, we obtain three different descriptions of the singular cohomology groups with coefficients in the non-zero complex numbers GL1. Now, replace GL1 with a complex algebraic reductive group G, e.g. GLn. The Non Abelian Hodge Theory of Corlette, Simpson et al. establishes a natural homeomorphism between three distinct complex algebraic varieties parametrizing three different kinds of structures on the projective manifold associated with the reductive group: representations of the fundamental group into G, flat algebraic G-connections, G-Higgs bundles. The case G=GL1, which is Abelian, recaptures the de Rham and Hodge Decomposition. The three complex algebraic varieties of Non Abelian Hodge Theory have naturally isomorphic cohomology groups. However, by taking into account their distinct structures of algebraic varieties, the cohomology groups carry additional distinct structures. The P=W Conjecture seeks to relate two of these structures, at least in the case of compact Riemann surfaces. This talk is devoted to introducing the audience to this circle of ideas and related developments.
Kento Fujita (Osaka University). On the coupled Ding stability for log Fano pairs. Friday, Oct 18, 1:30-2:30pm. Huxley 140.
Abstract: In the last decade, there are significant progresses about the Calabi problem for Fano varieties. I will review a part of the history, and then I will explain its generalization to a coupled setting. This is a joint work with Yoshinori Hashimoto.
Daniel Platt (Imperial). Numerical approximations of harmonic 1-forms on real loci of Calabi-Yau manifolds. Friday, Oct 25, 1:30-2:30pm. Huxley 140.
Abstract: For applications, it is desirable to have Calabi-Yau manifolds of complex dimension three which contain a real three-dimensional submanifold with a nowhere vanishing harmonic 1-form. The harmonic equation depends on the Calabi-Yau metric, which is not known explicitly. Currently, one conjectural example of such a manifold based on the SYZ conjecture exists. In the talk, I will explain a second conjectural example motivated by numerical approximations using neural networks. The numerical approximation works by first computing an approximate Calabi-Yau metric, and then computing a harmonic form for this approximate metric. This is based on an approach by Donaldson for computing approximate Calabi-Yau metrics that I will briefly review. I will also compare the results with a proven non-example, i.e. one where a harmonic 1-form exists, but it is guaranteed to have zeros. Such examples are interesting for M-theory in Physics. This is based on arXiv:2405.19402, which is joint work with Mike Douglas, Yidi Qi, and Rodrigo Barbosa. Time permitting I will comment on using an approximate harmonic 1-form in a numerically verified proof to rigorously show existence of such a 1-form. This is work in progress with Mike Douglas, Javier Gómez-Serrano, Fabian Lehmann, Yidi Qi, and Freid Tong.
Yang Li (Cambridge). Special Lagrangian pair of pants. Friday, Nov 1, 1:30-2:30pm. Huxley 140.
Abstract: The familiar pair of pants is a special Lagrangian in (C^*)^2 (after hyperk\”ahler rotation). We will discuss the generalisation of pair of pants to all higher dimensions, using a combination of PDE and geometric measure theory techniques. This provides the local model for the more general picture of special Lagrangians in SYZ fibrations fibred roughly over a tropical hypersurface.
Hannah Tillmann-Morris (Leipzig). Birational mirrors to pairs of Laurent polynomials. Friday, Nov 8, 1:30-2:30pm. Huxley 140.
Abstract: Mirror symmetry gives a correspondence between Fano varieties and certain Laurent polynomials, translating the classification of Fano varieties up to deformation into a combinatorial problem. To understand the classification using this correspondence, we ask the question: ‘Which pairs of Laurent polynomials (f,g) are mirrors to pairs of Fano varieties that are related by a blow-up X_g –> X_f?’ The combinatorial criteria on (f,g) we are looking for will generalise the relationship between the fans of two toric varieties related by a toric blow-up.
I will present a new method of constructing Fano mirrors to Laurent polynomials, which uses mirror constructions from the Gross—Siebert program. This will include the construction of a birational morphism between the mirrors to a pair of Laurent polynomials in two variables that satisfies certain combinatorial conditions.
Maria Yakerson (Sorbonne). An alternative to spherical Witt vectors. Friday, Nov 15, 1:30-2:30pm. Huxley 140.
Abstract: Witt vectors of a ring form a “bridge” between characteristic p and mixed characteristic: for example, Witt vectors of a finite field Fp is the ring of p-adic integers Zp. Spherical Witt vectors of a ring is a lift of classical Witt vectors to the world of higher algebra, much like sphere spectrum is a lift of the ring of integers.
In this talk we will discuss a straightforward construction of spherical Witt vectors of a ring, in the case when the ring is a perfect Fp-algebra. Time permitting, we will further investigate the category of modules over spherical Witt vectors, and explain a universal property of spherical Witt vectors as an E1-ring. This is joint work with Thomas Nikolaus.
Mirko Mauri (École Polytechnique). Hodge-to-singular correspondence for reduced curves. Friday, Nov 22, 1:30-2:30pm. Huxley 140.
Abstract: We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes. This is based on a joint work with Luca Migliorini and Roberto Pagaria.
Yin Li (Uppsala). Hypertoric convolution algebras as Fukaya categories. Friday, Nov 29, 1:30-2:30pm. Huxley 140.
Abstract: Given a simple polarized hyperplane arrangement in the d-dimensional Euclidean space, Braden-Licata-Proudfoot-Webster defined a convolution algebra, which we call the hypertoric convolution algebra, whose center gives the cohomology of the hypertoric variety associated to the hyperplane arrangement. It is conjectured by Lauda-Licata-Manion that the universal flat graded deformation of the hypertoric convolution algebra, regarded as a formal A infinity algebra, is quasi-isomorphic to (the endomorphism algebra of) the partially Fukaya category of the complement of the complexification of the hyperplane arrangement in the d-dimensional complex vector space, equipped with a stop determined by the polarization. We present a proof of this conjecture and explain its relation with Lekili-Segal’s conjecture on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions. This is based on joint works with Sukjoo Lee, Siyang Liu and Cheuk-Yu Mak.
Alexander Polishchuk (University of Oregon). Feigin-Odesskii Poisson brackets and del Pezzo surfaces. Friday, Dec 6, 1:30-2:30pm. Huxley 140.
Abstract: The talk is based on joints works with Zheng Hua, Nikita Markarian and Eric Rains. Feigin-Odesskii Poisson brackets are remarkable Poisson brackets on projective spaces associated with simple vector bundles on elliptic curves. It turns out that when elliptic curves vary in the anticanonical linear system on a surface and the corresponding vector bundles are restrictions of a fixed exceptional bundle forming an exceptional pair with the structure sheaf, one gets a linear subspace of compatible Poisson brackets on a projective space. I’ll discuss some results around this construction, including the fact that for every relatively prime (r,d), r>0, on every del Pezzo surface of degree 5 there exists an exceptional pair (O,V), with V of rank r and degree d.
Martin Kalck (University of Graz). Derived categories of singular varieties. Friday, Dec 13, 1:30-2:30pm. Huxley 140.
Abstract: The derived category of coherent sheaves is an invariant that captures the cohomological information of a variety. For smooth varieties, a lot is known about the structure and symmetries of these categories.
In particular, for certain varieties, (parts of) these categories can be described via exceptional sequences and, more generally, semi-orthogonal decompositions.
We report on recent progress on finding similar descriptions for derived categories of certain singular projective varieties.
We are particularly interested in semi-orthogonal decompositions containing components equivalent to derived categories of finite dimensional associative (but typically non-commutative) algebras.