The spectrum of discrete Laplacian plays
an important role in graph theory, machine learning and image processing.
Likewise, one can define 1-Laplacian of a graph and study its spectrum. From
the point of view of calculus of variation, the eigenvalue problem is closely
related to the critical point of a corresponding Lipschitz function on a
piecewise linear manifold. Using the critical point theory for Lipschitz
functions, Kung-Ching Chang obtained that the eigenvectors of 1-Laplacian
coincide with the critical points of such Lipschitz function [Chang, J. Graph Theory (2016)]. Unlike the
linear operator, the set of eigenvectors with the same eigenvalue may be very
large. To understand this set, He developed a nodal domain decomposition and
proved the related Courant nodal domain theorem [Chang, J. Graph Theory (2016); Chang-Shao-
Zhang, Adv. Math. (2017)]. Recently, Chang and his collaborators introduced the signless 1-Laplacian of graphs
and discovered new phenomena on structure of eigenvectors for 1-Laplacian. They
proved that the second eigenvalue of 1-Laplacian equals to the Cheeger constant
and the corresponding eigenvector provides the Cheeger cut, while the first
eigenvalue of signless 1-Laplacian equals to the dual Cheeger constant [Chang, J. Graph Theory (2016);
Chang-Shao-Zhang, Adv. Math. (2017); Chang-Shao-Zhang, arXiv (2016)]. Comparing with the
Laplacian, these equalities give better estimates.
Another progress is made in the study of
the Lovasz extension. They established a set-pair Lovasz extension to construct
equivalent continuous optimization problems for graph cut including the maxcut,
Cheeger cut, dual Cheeger cut, max 3-cut, anti-Cheeger cut problems, etc [Chang-Shao-Zhang, arXiv (2016); Chang-Shao- Zhang-Zhang, preprint (2017)]. In particular, the
related eigenvalue problem for maxcut involving set-valued functions in
nonlinear systems was studied since the continuous optimization for maxcut is
very important and useful in solving the maxcut [Chang-Shao-Zhang-Zhang, preprint (2017)].
2.
Self-similar Solutions of Curve Shortening Problem (JIANG Meiyue )
The generalized curve-shortening flow has
been extensively studied in the literature. Self-similar solutions of the flow
are solutions whose shapes change homothetically during
the evolution. Such solutions are important in understanding the long time
behaviors and the structure of singularities. A case which is called the affine curve shortening problem, has special interest in
geometry. In terms of the support function, self-similar solutions reduce to an
important 2-th order equation which also appears in image processing,
2-dimensional L^p-Minkowski problem, etc. Such equation with special parameter
function is invariant with respect to affine transformations of R^2, which
leads to some special features of the problem, for instance, there are
obstructions for the solvability, and there is no a priori estimates for the
solutions without additional assumption. This problem has been studied for the
pi-periodic function. However, the method does not work for 2pi-periodic
functions, which is more natural from the geometric point of view.
Meiyue Jiang and his collaborators
developed a new approach for the a priori estimate and existence of solutions,
and proved the existence of solutions of the anisotropic affine curve
shortening problem under some basic conditions [Jiang- Wang-Wei, Calc. Var. Partial Differential Equations (2011); Jiang-Wei, Discrete Contin. Dyn.
Syst. (2016)].
3.
Harmonic Analysis and Its Applications in PDE (TANG Lin)
Lin Tang obtained some important results on pseudo-differential
operators, Schrodinger equations and parabolic Monge-Ampère equations.
Tang [Tang-Zhang, J. Funct. Anal. (2016)] obtained weighted norm inequalities for
pseudo- differential operators with smooth symbols and their commutators and
introduced a new non-double weighted class, which generalizes essentially the
classical Munckenhoupt weighted class, which also give a direction for studying
pseudo- differential operators in the future.
Lin Tang and his collaborators [Pan-Tang, J. Funct. Anal. (2016)] established the
L^p boundedness of Schrodinger equations with discontinuous coefficients by
using new method.
Parabolic Monge-Ampère equations have some
important applications in Ricci flow. Tang [Tang, J. Differential Equations (2013)] obtained the interior regularity
for two type parabolic Monge-Ampère equations with VMO type data. In addition, Tang [Tang, J. Differential Equations (2015)] also solved the boundary regularity problem on parabolic Monge-Ampère
equations.
4. Solutions
of L2 Extension Problem
and Strong Openness Conjecture (GUAN Qi-an )
Qi'an Guan and Xiangyu Zhou (joint with Prof. Xiangyu Zhou) gave a solution of an L2 extension problem with an
optimal estimate. As applications, we give proofs of a conjecture of Suita on
the equality condition in Suita’s conjecture, the so-called L-conjecture, and
the extended Suita conjecture. As other applications, they gave affirmative
answer to a question by Ohsawa about limiting case for the extension operators
between the weighted Bergman spaces, and they presented a relation of our result to
Berndtsson’s important result on log-plurisubharmonicity of the Bergman kernel.
These works were called “Guan-Zhou Method” as the title of a subsection in Ohsawa’s book “L2 Approaches in Several Complex Variables. Springer
Japan, 2015” and “Their work gave the author a decisive impetus to start
writing a survey to cover these remarkable achievements” in the preface of the book.
Guan and Zhou gave a proof of Demailly's strong openness conjecture. As
direct consequences, they solved that: a conjecture about minimal singularities
and linear system posed by Demailly-Ein-Lazarsfeld, a conjecture about Proper
modications, multiplier ideal sheaves and Lelong numbers posed by
Boucksom-Favre-Jonsson, establisheh a Nadel type vanishing theorem on compact Kähler
manifolds and multiplier ideal sheaves is essentially with analytic weights,
etc. Their proof of Demailly’s strong openness conjecture was called “important
features” in Demailly’s paper “On the cohomology of pseudoeffective line
bundles”, “opened the door to new types of approximation techniques” in the
recently published survey “L2 estimate for the d-bar operator” and “among the greatest achievements ‘in the
intersection of’ complex analysis and algebraic geometry in recent years” in
AMS MathReview (MR3418526).
Continuing their solution of Demailly’s strong openness
conjecture, they discuss conditions to guarantee the effectiveness of the
conjecture and establish such an effectiveness result. They give a lower
semicontinuity property of plurisubharmonic functions with a multiplier. They
also proved two conjectures of Demailly-Kollár and Jonsson-Mustata
respectively.
Recently, as an application of the strong openness
conjecture, they characterized the multiplier ideal sheaves with weights of Lelong
number one, which was called “a basic result on Lelong numbers” in the Ohsawa’s
book “L2 Approaches in
Several Complex Variables. Springer Japan, 2015”, and “an important theorem” in
AMS MathReview (MR3406538).
