The seminar usually takes place in R515, Cosmology Building at 4-5PM every Thursday, unless noted otherwise.
Organizers: Siao-Hao Guo 郭孝豪 (NTU), Ulrich Menne 孟悟理 (NTNU), Chung-Jun Tsai 蔡忠潤 (NTU)
Chung-Ming Pan 潘仲銘 (Institut de Mathématiques de Toulouse): Singular Gauduchon metrics
Abstract: Gauduchon metrics are useful generalizations of Kähler metrics in non-Kähler geometry. In this talk, we will present the work in which we obtain the existence of Gauduchon metrics on compact hermitian varieties admitting a smoothing. This result generalizes Gauduchon’s theorem which says that on a compact complex manifold, one can find a Gauduchon metric in every conformal class of hermitian metrics. If time permits, we will explain an application that gives a partial solution to a conjecture proposed by Di Nezza-Guedj-Guenancia.
Kuan-Hui Lee 李冠輝 (University of California, Irvine): Generalized Ricci Flow
Abstract: The generalized Ricci flow is a new field emerging from mathematics physics and generalized geometry. It can be viewed as a Ricci flow with a torsion part. In this talk, I will give a brief introduction of the generalized Ricci flow and study its connection with the Ricci flow. If time permits, I will also demonstrate the latest result and some open problems.
Beomjun Choi 최범준 (POSTECH): Liouville theorem for surfaces translating by sub-affine-critical powers of Gauss curvature
We classify the translators to the flows by sub-affine-critical powers of Gauss curvature in $\mathbb{R}^3$. If $\alpha$ denotes the power, this is a Liouville theorem for degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^{2-\frac{1}{2\alpha}}$ for $02$. More precisely, these curves are closed shrinking curves to the $\frac {\alpha}{1-\alpha}$-curve shortening flow that were previously classified by B. Andrews in 2003. This is a joint work with K. Choi and S. Kim.
Chris Evans (Queen Mary University of London): Lagrangian mean curvature flow in the complex projective plane
We look at Lagrangian mean curvature flow in the complex projective plane, studying in detail an equivariant example. We see two topologically distinct types of Lagrangian tori, with the flow jumping from one to the other by way of surgery at singularities , before converging to a minimal torus in infinite time. Along the way, we look at lots of interesting ideas which hopefully should have much wider application in Lagrangian mean curvature flow, particularly in positively curved spaces.
Xin Nie 聂鑫 (S.-T. Yau Center of Southeast University): Higgs bundles, minimal surfaces and pseudo-hyperbolic spaces.
We first explain (1) the link between classical Teichmüller theory and minimal surfaces in the adS space (works of Mess and Bonsante-Schlenker); (2) the theory of Higgs bundles and surface group representations ("higher Teichmüller theory"); (3) the generalization of (1) to higher Teichmüller theory (Collier-Tholozan-Toulisse). Then we discuss some recent progress of (3).
Wei-Hung Liao 廖偉宏 (National Yang Ming Chiao Tung University): Convergence Analysis of Dirichlet Energy Minimization for Spherical Conformal Parameterizations
We first derive a theoretical basis for spherical conformal parameterizations between a simply connected closed surface $\mathcal{S}$ and a unit sphere $\mathbb{S}^2$ by minimizing the Dirichlet energy on $\overline{\mathbb{C}}$ by stereographic projection. The Dirichlet energy can be rewritten as the sum of the energies associated with the southern and northern hemispheres and can be decreased under an equivalence relation by alternatingly solving the corresponding Laplacian equations. Based on this theoretical foundation, we develop a modified Dirichlet energy minimization with nonequivalence deflation for the computation of the spherical conformal parameterization between $\mathcal{S}$ and $\mathbb{S}^2$.
Irina Markina (University of Bergen): On exceptional families of measures
It is an ongoing project to study collections of measures that are negligible in a sense of ``modules". The idea originated in complex analysis as ``a conformal module of a family of curves" in looking for an invariant object under conformal transformations on the complex plane. Later the definition of the module was successfully applied to the nonlinear potential theory and quasiconformal analysis in a wider sense in Euclidean spaces. B. Fuglede, by studying the completion of functional spaces, generalized the notion of the module of a family of curves to the module of a family of measures. A collection of measures is exceptional if the corresponding module vanishes. We aim to find exceptional families of measures on Carnot groups, related to geometric objects such as "intrinsic graphs". It leads to the notion of a Grassmannian on specific Carnot groups.
This is a joint work with Bruno Franchi, University of Bologna, Italy.
Zihui Zhao (The University of Chicago): Boundary unique continuation and the estimate of the singular set
Unique continuation property is a fundamental property of harmonic functions, as well as solutions to a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes to infinite order at a point, it must be zero everywhere. In the same spirit, we can use the local growth rate of harmonic functions to deduce global information, such as estimating the size of the singular set for elliptic PDEs. This is joint work with Carlos Kenig.
Chao-Ming Lin 林朝明 (University of California, Irvine): On the convexity of general inverse $\sigma_k$ equations and some applications
In this talk, I will show my recent work on general inverse $\sigma_k$ equations and the deformed Hermitian—Yang—Mills equation (hereinafter the dHYM equation). First, I will show my recent result. This result states that if a level set of a general inverse $\sigma_k$ equation (after translation if needed) is contained in the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the Monge—Ampère equation, the J-equation, the dHYM equation, the special Lagrangian equation, etc. Second, I will introduce some semialgebraic sets and a special class of univariate polynomials and give a Positivstellensatz type result. These give a numerical criterion to verify whether the level set will be contained in the positive orthant. Last, as an application, I will prove one of the conjectures by Collins—Jacob—Yau when the dimension equals three or four. This conjecture states that under the supercritical phase assumption, if there exists a C-subsolution to the dHYM equation, then the dHYM equation is solvable.
Paul Minter (Institute of Advanced Study and Princeton University): Quantitative estimates on minimal hypersurface singular sets
Understanding the nature of singularities which arise in minimal submanifolds has a rich history. In the 1960's it was established that codimension one area minimisers are smoothly embedded away from a (generally unavoidable) set of codimension 7, referred to as the singular set. Nowadays, due to the work of N. Wickramasekera, we know that this regularity result is actually a consequence of just stability (i.e. non-negative second variation of area) as well as the absence of a certain singularity type know as a classical singularity (which includes, for example, crossings of smoothly embedded submanifolds). One can then ask if the singular set obeys more refined estimates than just a dimension bound. The work of Naber--Valtorta provides Minkowski content bounds for codimension one area minimisers, and work of A. Song shows that one can achieve measure bounds for codimension one minimal hypersurfaces which have finite index and have sufficiently small singular set a priori. In this talk, I will discuss recent work, joint with Nicolau S. Aiex and Sean McCurdy, where we improve on these results, proving a Minkowski content bound on the singular set in the finite index setting when one just rules out classical singularities. In particular, we are able to control the measure of the entire tubular neighbourhood of the singular set.
Chao-Ming Lin 林朝明 (University of California, Irvine): Introduction to Some General Inverse $\sigma_k$ Type Equations
In Kähler geometry, the existence of special Kähler metric(s) under different settings is widely considered. In this talk, we will discuss some famous settings and their backgrounds. For example, the Kähler metric satisfies the complex Monge--Ampère equation or the J-equation.
The target audience is advanced undergraduate students and grad students.
Yu-Shen Lin 林昱伸 (Boston University): A Uniqueness Theorem for 2D Complex Monge-Ampere Equations
Given a rational elliptic surface $Y$ with an anti-canonical cycle $D$, Hein solved the corresponding complex Monge-Ampere equation and proved that the complement $Y\backslash D$ admits complete Ricci-flat metrics. It is natural to ask if the cohomology class of the metric and the asymptotic behavior uniquely determine the Ricci-flat metric. I will give an affirmative answer based on a joint work with T. Collins and A. Jacob. As a consequence, the space of Ricci-flat metrics modulo gauge equivalence is finite dimensional. If time permits, I will explain the application to mirror symmetry.
