2022 Morningside Center of Mathematics Geometry Summer School
Aug 15 - Aug 20, 2022. Hosted by Morningside Center of Mathematics, CAS via Zoom. Please register so that we can get an estimate of the scale and send out further notices.
Aug 15 - Aug 20, 2022. Hosted by Morningside Center of Mathematics, CAS via Zoom. Please register so that we can get an estimate of the scale and send out further notices.
Lefschetz fibrations in symplectic topology and applications
Convexity and symplectic dynamics
Monge-Ampère type equations and Hodge classes
Singularities of special Lagrangian submanifolds
Higgs bundles and harmonic maps
Soft methods in higher codimension area-minimizing surfaces
Introduction to extending calibration pairs
Given a smooth affine algebraic variety over the complex numbers, a holomorphic Morse function on it defines a so-called Lefschetz fibration. It turns out that such a notion can be vastly generalized, and it is quite useful in the study of symplectic manifolds. The four lectures of this minicourse will cover the following topics: 1. Lefschetz pencils, vanishing cycles, Picard-Lefschetz formula, and monodromy transformation; 2. Floer cohomology groups associated with symplectic Lefschetz fibrations; 3. Exceptional collections, Serre functor, and Hochschild invariants; 4. A recent generalization of Thurston’s construction of pseudo-Anosov mapping classes to higher dimensional symplectic manifolds using symplectic Lefschetz fibration techniques.
Consider classical phase space \(\mathbb{R}^{2n}\), with \(n\) position coordinates and \(n\) momentum coordinates. Hamilton's formulation of classical mechanics tells us that any smooth function \(H:\mathbb{R}^{2n} \to \mathbb{R}\) determines a dynamical system on \(\mathbb{R}^{2n}\) representing the time evolution of a classical system with \(H\) as its energy function. Now suppose that \(H\) is a convex function. What does this tell us about the dynamics of the corresponding system?
The surprising answer is: quite a lot! The goal of this lecture series will be to explore this mysterious interaction between convex and symplectic geometry. After a short introductory talk (Lecture 1), we will give an overview of the necessary background from symplectic geometry (Lecture 2-3). We will then discuss convexity and some of its implications (Lectures 4-5) and conclude with a discussion of open problems (Lecture 6).
Lecture 1: Introduction
Lecture 2: Symplectic geometry, contact geometry and Hamiltonian dynamics
Lecture 3: Rotation numbers and Maslov indices
Lecture 4: Convexity
Lecture 5: Weak versions of convexity
Lecture 6: Recent progress and open problems
Yau's solution of the Calabi conjecture was used by Demailly and Paul to understand the relationship between Kähler classes and Hodge classes, i.e., homology classes of subvarities. In this short course, we understand and extend this result to generalized Monge-Ampère equations, which include the J-equation and deformed Hermitian-Yang-Mills equation.
I will speak about (compact) special Lagrangian submanifolds with isolated conical singularities. I am planning to explain the known results and my current work in progress. In the modern point of view, the fundamental progress on special Lagrangian submanifolds with isolated conical singularities was made by Joyce around 2000. I have been studying (rather slowly) the remaining problems. On the other hand, there has been interaction with symplectic topology, going back to Thomas--Yau's proposals. But I will not discuss this in detail in these talks; I will presumably explain only those related to the study of isolated conical singularities of special Lagrangian submanifolds.
The celebrated non-abelian Hodge correspondence relates Higgs bundles over a Riemann surface and representations of the surface group into a Lie group. Higgs bundles together with harmonic metrics are equivalent to equivariant harmonic maps from the universal cover of \(X\) to the symmetric space \(GL(n,\mathbb{C})/U(n)\). We will investigate their relationships in detail. The plan of the minicourse is as follows.
Lecture 1: Briefly introduce Higgs bundles and the non-abelian Hodge correspondence. Explain Hitchin fibration and Hitchin section.
Lecture 2: Investigate the explicit relationship between Higgs bundles and harmonic maps or minimal surfaces.
Lecture 3: Provide various examples in lower rank geometry.
Lecture 4: Discuss Labourie conjecture and Dai-Li conjectures.
Area-minimizing subvarieties of codimension larger than one arise naturally in many different contexts, like special Lagrangians and associative submanifolds in special holonomy geometries. However, their behavior and regularity remain largely open questions. We have the fundamental regularity results of Almgre-De Lellis-Spadaro and Naber-Valtorta, both of which are masterpieces of tour de force hard analysis.
The groundbreaking work of Yongsheng Zhang on gluing calibrations opens up the venue for using soft methods to deal with regularity problems. Roughly speaking, if we ask for the existence of homologically area-minimizing currents with a certain property so that the property can be localized and exist on calibrated examples. Then the existence problem can be solved globally in some metric. Along this line, we prove that homologically area-minimizing currents can have fractal singular sets in general homology classes. If time permits, we will also prove that homologically area-minimizing currents in general codimension are not smoothable under perturbations of metric, without structural requirements on the currents themselves.
The theory of calibrations is powerful in creating area-minimizing (perhaps singular) submanifolds which are inevitable objects in geometric measure theory. Basic knowledge on calibrated geometries and some fundamental method of extending calibration pairs will be introduced in the first two lectures. The remaining lectures are devoted to showing that every known area-minimizing hypercone can be realized as a tangent cone at a singular point of some compact homologically area-minimizing singular hypersurface. Some contents could be prerequisites for Zhenhua's courses.
References:
1. R. Harvey and H. B. Lawson, Jr.: Calibrated geometries, Acta Math. 148 (1982) 47--157.
2. R. Harvey and H. B. Lawson, Jr.: Calibrated foliations, Amer. J. Math. 104 (1982) 607--633.
3. Y.S. Zhang: On extending calibration pairs, Adv. Math. 308 (2017) 645--670.
4. Y.S. Zhang: On realization of tangent cones of homologically area-minimizing compact singular submanifolds, JDG, 109 (2018) 177--188