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Yang Li: Lectures

June 6, 2019
TITLE: SYZ conjecture, special Lagrangian, and negative vertex

ABSTRACT: I will review some basics for the SYZ conjecture, and describe Joyce’s work on U(1) invariant special Lagrangians. Then I will mention my recent construction of the model CY metric on the negative vertex, and speculate on how to construct special Lagrangian T3 fibration on these spaces, using a strategy closely analogous to Joyce’s work.

April 11, 2019
TITLE: Taub-NUT and Ooguri-Vafa in complex dimension 3

ABSTRACT: On collapsing K3 surfaces, the Ooguri-Vafa metric is the local model for the neighbourhood of the singular SYZ fibre. It contains within itself a region modelled on the famous Taub-NUT metric. I will give a sketchy report on my recent work generalising these constructions to complex dimension 3, which is expected to be relevant for the SYZ conjecture on 3-folds.

January 10, 2019
TITLE: Taub-NUT and Ooguri-Vafa

ABSTRACT: Taub-NUT and Ooguri-Vafa metrics are S^1 invariant Calabi-Yau metrics in complex dimension 2 constructed via the Gibbons-Hawking ansatz. They feature prominantly in collapsing Calabi-Yau metrics on surfaces. After reviewing the basics we explain how a number of first principles dictate their construction. We also discuss how to identify the complex structures by explicitly constructing holomorphic functions as transcendental integrals, and how the algebraic structures arise from their functional equations.

January 11, 2019
TITLE: Taub-NUT (and Ooguri-Vafa) on Calabi-Yau 3-folds

ABSTRACT: The primary focus is on constructing a family of Taub-NUT type Calabi-Yau metrics on C^3. The tangent cone at infinity is R^4, and in particular the volume growth is not maximal. The main idea is to construct an asymptotic ansatz near infinity using the generalised Gibbons-Hawking ansatz. This ansatz is conceptually obtained by perturbing from the flat solution after incorporating topological features. If time permits we will also discuss the relevance to Ooguri-Vafa type metrics on the positive and negative vertices.

April 13, 2018
TITLE: A gluing construction of collapsing CY metrics on K3 fibred 3-folds

ABSTRACT: I will discuss the problem of describing the collapsing CY metrics on a CY 3-fold with a Lefschetz K3 fibration, from both the gluing perspective and the a priori estimate perspective. Collapsing CY metrics is a well studied subject, but most of the previous works concentrate on the behaviour away from the singular fibres, and the full description of the metric was only available in a very small number of cases, mostly relying on very favourable gluing ansatz.

From the nonlinear perspective, the essential realisation is that by restricting the type of singularities, and under some conjecture in pluripotential theory, then a small neighbourhood of the singular fibre has a local noncollapsing bound, which enables us to understand the pointed Gromov-Hausdorff limit of the singular fibre in the scale where the fibre volume is 1.

From the gluing perspective, the main geometric insight is that there should be a much finer scale near the nodal points in the fibration, where the scaled limit is a CY metric on C^3 with maximal volume growth and singular tangent cone at infinity. This model metric was previously constructed by the author in a separate work. The difficulty of the gluing lies in the coarse nature of the gluing ansatz, and the fact that the metric has many types of characteristic behaviours at different scales. We overcome this by developing a sharp linear theory, using some earlier ideas of Gabor Szeklyhidi.

April 12, 2018
TITLE: Nonlinear perspective of collapsing CY metrics on K3 fibred 3-folds

ABSTRACT: I will discuss the problem of describing the collapsing CY metrics on a CY 3-fold with a Lefschetz K3 fibration, from both the gluing perspective and the a priori estimate perspective. Collapsing CY metrics is a well studied subject, but most of the previous works concentrate on the behaviour away from the singular fibres, and the full description of the metric was only available in a very small number of cases, mostly relying on very favourable gluing ansatz.

From the nonlinear perspective, the essential realisation is that by restricting the type of singularities, and under some conjecture in pluripotential theory, then a small neighbourhood of the singular fibre has a local noncollapsing bound, which enables us to understand the pointed Gromov-Hausdorff limit of the singular fibre in the scale where the fibre volume is 1.

From the gluing perspective, the main geometric insight is that there should be a much finer scale near the nodal points in the fibration, where the scaled limit is a CY metric on C^3 with maximal volume growth and singular tangent cone at infinity. This model metric was previously constructed by the author in a separate work. The difficulty of the gluing lies in the coarse nature of the gluing ansatz, and the fact that the metric has many types of characteristic behaviours at different scales. We overcome this by developing a sharp linear theory, using some earlier ideas of Gabor Szeklyhidi.

January 12, 2018
TITLE: Mukai duality on adiabatic coassociative fibrations

ABSTRACT: We study the formal adiabatic limit of coassociative K3 fibred torsion free G_2 manifolds fibred over a contractible base, show how to put this structure on a different fibration obtained by fibrewise performing Mukai duality of K3 surfaces, and furthermore relate the gauge theories on both fibrations by a Nahm transform. This gives a mathematical interpretation to the physical speculations of Gukov, Yau and Zaslow.


June 7, 2017
TITLE: A new complete Calabi-Yau metric on C3 with maximal volume growth