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 . Friday, Nov 1, 1:30-2:30pm. Huxley 140.
Abstract:
Hannah Tillmann-Morris (Leipzig). Friday, Nov 8, 1:30-2:30pm. Huxley 140.
Abstract:
Maria Yakerson (Sorbonne). Friday, Nov 15, 1:30-2:30pm. Huxley 140.
Abstract:
Mirko Mauri (École Polytechnique). Friday, Nov 22, 1:30-2:30pm. Huxley 140.
Abstract:
Yin Li (Uppsala). Friday, Nov 29, 1:30-2:30pm. Huxley 140.
Abstract:
Alexander Polishchuk (University of Oregon). Friday, Dec 6, 1:30-2:30pm. Huxley 140.
Abstract:
Martin Kalck (University of Graz). Friday, Dec 13, 1:30-2:30pm. Huxley 140.
Abstract: