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Song Sun: Lectures

January 7, 2020
TITLE: Singular Hermitian-Yang-Mills connections and reflexive sheaves

ABSTRACT: In higher dimensional gauge theory it is important to understand essential singularities of finite energy Yang-Mills instantons. I will discuss joint work with Xuemiao Chen (University of Maryland), which studies the model case of Hermitian-Yang-Mills connections, and the results say that the singularities have unique tangent cones which can be characterized in terms of certain local algebro-geometric invariants of reflexive sheaves.

April 9, 2019
TITLE: Small complex structure limit of Calabi-Yau hypersurfaces

ABSTRACT: This will be a synopsis of the talks I gave on the previous two meetings. The main purpose is to explain a relatively complete result for understanding the metric collapsing of a generic splitting of a family of smooth Calabi-Yau hypersurfaces into the union of two Fano hypersurfaces.

January 8, 2019 and January 10, 2019
TITLE: Degeneration of Calabi-Yau metrics under complex structure degenerations

ABSTRACT: We will describe some progress towards understanding complex structure limit of Calabi-Yau metrics via glueing constructions. The main ingredient involves constructing Calabi-Yau metrics with torus symmetry (with fixed loci), by studying the dimension reduced equation (i.e. Gibbons-Hawking ansatz and its non-linear generalization due to Matessi) and Dirac type solutions to the linearized equation. Based on arXiv:1807.09367 (with Hein, Viaclovsky, Zhang) and more recent results and work in progress (with Zhang).

September 13, 2018
TITLE: Collapsing of hyperkahler metrics on K3 surfaces

ABSTRACT: Hyperkahler metrics on K3 surfaces are prototypical examples of compact Ricci-flat metrics with special holonomy. I will explain some known results in this field and describe a new gluing construction, joint with Hans-Joachim Hein, Jeff Viaclovsky and Ruobing Zhang, of a family of hyperkahler metrics on K3 surfaces with multiscale collapsing phenomenon.

September 13, 2017
TITLE: Singularities of Hermitian-Yang-Mills connections

ABSTRACT: In this talk we discuss how to understand the tangent cones of Hermitian-Yang-Mills connections in terms of the local algebraic data on the underlying reflexive sheaf. (Joint work with Xuemiao Chen)

January 10, 2017 and January 12, 2017
TITLE: Isolated conical singularities of Calabi-Yau varieties

ABSTRACT: I will talk about joint work with Hans-Joachim Hein on the conical behavior of Ricci-flat metrics on Calabi-Yau varieties with certain types of isolated singularities. Interesting examples include hypersurfaces with nodal singularities. In the more technical part I will try to sketch the main steps involved in the proof.
January 10:

January 12:

September 9, 2016
TITLE: Degeneration of Calabi-Yau metrics

ABSTRACT: The complex structure moduli space of Calabi-Yau manifolds can be compactified using the Gromov-Haudorff topology, and a central question is to understand the structure of these Gromov-Hausdorff limits. We will focus on the “non-collapsing” case, and explain the connection with algebraic geometry (joint work with S. Donaldson), and the recent example of a compact Calabi-Yau manifold with isolated conical singularities (joint work with H. Hein). A well-known application of the latter is the existence of special Lagrangian spheres on the smoothing of nodal Calabi-Yau varieties.