Ben Andrews (Australian National University, Canberra)

Title: (Two)-convex cores and hypersurface flows in Riemannian manifolds

Abstract: In general, mean curvature flow of Riemannian hypersurfaces does not preserve convexity, but there are flows that do

including the flow by the harmonic mean curvature. I will discuss a general result (joint work with Xuzhong Chen and Guoyi Xu)

on the long-term behaviour of such flows, and also a generalization to the two-convex setting which builds on work of Simon

Brendle and Gerhard Huisken.

Bing-Long Chen (Sun Yat-Sen University, Guangzhou)

Title: Local regularity of the Einstein equations under Ricci curvature and Lie derivative conditions

Abstract: We will survey some Bernstein type theorems of stationary solutions of the Einstein equations, and derive one a priori

estimate for general solutions of the Einstein equations under only Ricci curvature and Lie derivative bounds.

Shibing Chen (University of Science and Technology of China, Hefei)

Title: The optimal partial transport problem

Abstract: We will discuss the regularity theory of optimal partial transportation. Specifically, we will first introduce the interior

C#,%, C&,% regularity theories. The former was established by Luis Caffarelli and Robert McCann, while the latter was developed

by Jiakun Liu, Xu-Jia Wang, and myself. Then, I will present my recent work with Jiakun Liu and Xianduo Wang on the global

regularity of the free boundary.

Serena Dipierro (University of Western Australia, Perth)

Title: Boundary behavior of solutions to fractional elliptic problems

Abstract: Solutions of nonlocal equations typically depend rather significantly on their values outside of a given region of

interest and, in this sense, it is often convenient to assume "global" conditions to deduce "local" results. In this talk, we present

instead a Hopf Lemma for solutions to some integro-differential equations that does not assume any global condition on the sign

of the solutions. We also show that non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary.

Jixiang Fu (Fudan University, Shanghai)

Title: On higher direct images of pluricanonical bundles

Abstract: Given a fibration between two projective manifolds, we discuss the effective generation of the higher direct images of

its pluricanonical bundles. Our results are related to two questions proposed by Popa-Schnell. This is a joint work with Jingcao

Wu.

Genggeng Huang (Fudan University, Shanghai)

Title: Free boundaries for elliptic and parabolic Monge-Ampère equations

Abstract: In this talk, we will talk about two free boundary problems for elliptic and parabolic Monge-Ampère equations. We

will show the free boundaries of these problems are smooth. The talk is based on the joint work with Tang Lan, Prof. Wang Xu-

Jia and Zhou Yang.

Kwok Kun Kwong (University of Wollongong, Wollongong)

Title: Weighted geometric inequalities and the Weinstock inequality for the first Steklov eigenvalue

Abstract: I will present some weighted geometric inequalities that involve three geometric quantities for hypersurfaces in

certain warped product manifolds. One family of inequalities concerns the k-th boundary momentum, area, and weighted

volume, with applications to the Weinstock inequality for Steklov eigenvalues. Specifically, I will show that the Weinstock

inequality holds for star-shaped mean convex domains in the Euclidean space of any dimension. This resolves a conjecture by

Bucur, Ferone, Nitsch, and Trombetti. One main ingredient is the use of inverse curvature type flows, with the key being the

identification of monotone quantities involving two or more quantities along the flow. If time permits, I will describe a second

class of sharp geometric inequalities involving a weighted curvature integral and two quermassintegrals in space forms.

Additionally, I will illustrate a novel connection between a special case of these inequalities and the 𝐿& distance between a

convex body and its Steiner ball, resulting in a stability result for both this inequality and the isoperimetric inequality. This is

joint work with Yong Wei.

Mat Langford (Australian National University, Canberra)

Title: Ancient solutions to extrinsic geometric flows

Abstract: Ancient solutions describe the asymptotic geometry of singularities which form under extrinsic geometric flows. In

this context, the positively curved ancient solutions are particularly important. This is fortunate, as these are precisely the class

of ancient solutions which appear to permit a reasonable classification (cf. Appell's theorem for positive ancient solutions to the

heat equation). However, outside of the mean curvature flow, very few classification results are as yet available (despite the

emergence of a clear conjectural picture). I shall present some recent work on the subject (with Rengaswami and Lynch).

Congming Li (Shanghai Jiaotong University, Shanghai)

Title: Some recent work on qualitative analysis of nonlinear PDEs of elliptic type

Abstract: We present some recent work on qualitative analysis of some nonlinear elliptic type PDEs. The talk is mainly on

maximum principles, Liouville type theorems and classification of solutions. We start with some recent work on steady Euler

equations and give a short introduction of some related methods, and some results with applications. Then, we study the Hardy-

Littlewood-Sobolev type system and curvatures related geometric equations. Here we mainly deal with nonlocal type.

Dongsheng Li (Xi’an Jiaotong University, Xi’an)

Title: Global solutions of obstacle problems for fully nonlinear elliptic operators

Abstract: In this talk, we will study global solutions of obstacle problems for fully nonlinear elliptic operators in 𝑅'. The elliptic

operator 𝐹 is assumed to be convex and C#,%, and the solution 𝑢 to be positive and 𝐶#,# with bounded 𝐷&𝑢. If the coincidence set

can be contained between two parallel hyperplanes of dimension 𝑛 − 1, we will show that the coincidence set is either a

bounded convex set or a cylinder with a bounded convex set as base.

Jiayu Li (University of Science and Technology of China, Hefei)

Title: On the symplectic mean curvature flows

Abstract: We will talk about the recent progress on symplectic mean curvature flows. We will prove a Bernstein type theorem

for symplectic translating soliton. This is a joint work with Professors Han Xiaoli and Sun Jun.

Qi-Rui Li (Zhejiang University, Hangzhou)

Title: Some variational problems from functionals with duality

Abstract: We will discuss a class of functionals with duality. Such functionals arise from several geometric and physical

applications including a class of prescribing Gauss curvature type problems and optimal transport problems. We will report

some new ideas used in the study of the functionals and the related variational problems. The talk is based on joint work with

Qiang Guang and Xu-Jia Wang.

Jian Lu (South China Normal University, Guangzhou)

Title: Some recent results on Minkowski type problems

Abstract: Minkowski type problems arise from modern convex geometry. In the smooth case, they are usually equivalent to

solving a class of Monge-Ampère type equations defined on the unit hypersphere. These equations could be degenerate or

singular in different conditions. We will talk about some new results on the existence, uniqueness, and non-uniqueness of

solutions to several Minkowski type problems.

Xinan Ma (University of Science and Technology of China, Hefei)

Title: Jerison-Lee identity and semi-linear subelliptic equation on C-R manifold

Abstract: D. Jerison and J. M. Lee (JAMS 1988) found a three-dimensional family of differential identities for critical exponent

equation on Heisenberg group 𝐻' by using computer. They also care about whether there exists a theoretical framework that

would predict the existence and the structure of such formulae. In this talk we answer the question with the help of dimensional

conservation and invariant tensors. Then we generalize the Jerison-Lee identities in Cauchy-Riemann (CR) manifold on

subelliptic equations. Several new types of identities on CR manifold are found and these identities are used to get the rigidity

result for a class of CR Lane-Emden equation in subcritical case, rigidity means that the subelliptic equation has no other

solution than some constant at least when a parameter is in a certain range. The rigidity result also deduces the sharp Folland-

Stein inequality on closed CR manifold. This is the joint work with Qianzhong Ou, and Tian Wu.

Hongyi Sheng (University of California, San Diego)

Title: Localized deformation of the scalar curvature and the mean curvature

Abstract: On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean

curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact

subdomains in an arbitrary Riemannian manifold with boundary, as motivated by an attempt to generalize the Riemannian

Penrose inequality in dimension 8. This result is a generalization of Corvino's result about localized scalar curvature

deformations; however, the existence part needs to be handled delicately since the problem is non-variational. We also briefly

discuss generic conditions that guarantee localized deformations, and some related geometric properties.

Yuguang Shi (Peking University, Beijing)

Title: Non-compact manifolds with positive scalar curvature

Abstract: The talk consists of two parts. In the first part of the talk, I will discuss a kind of open manifolds carries no complete

positive scalar curvature metric, and in the second part of the talk, I will discuss Llarull type theorems on complete manifolds

with positive scalar curvature. The talk based on my recent joint works with T.Hao, Y.Sun, R.Wu, J.Wang and J.Zhu.

Gang Tian (Peking University, Beijing)

Title: Bounding curvature for Kähler-Ricci flow at finite time

Abstract: Kähler-Ricci flow may develop singularity at finite time. In order to continue the flow across singularity, we need to

understand geometry of metrics along the flow. Bounding curvature becomes a crucial method. A famous estimate of Perelman

gave a very important curvature bound for manifolds with positive first Chern class. In this talk, I will report on some recent

progress on bounding curvature for general Kähler manifolds and its application to giving new understanding of finite time

singularity.

Enrico Valdinoci (University of Western Australia, Perth)

Title: Sheet happens (but only as the root of 1-s)

Abstract: We discuss the regularity properties of two-dimensional stable s-minimal surfaces, presenting a robust C&,% estimate

and an optimal sheet separation bound, according to which the distance between different connected components of the surface

must be at least the square root of 1-s.

Glen Wheeler (University of Wollongong, Wollongong)

Title: Convergence of solutions to a convective Cahn-Hilliard type equation of the sixth order in case of small deposition rates

Abstract: In 2003, Savina et al. considered a surface-diffusion based process describing the formation of quantum dots (and

their faceting) on a growing crystalline hypersurface. We consider a thin-film approximation, which is a sixth-order convective

Cahn-Hilliard equation with spatio-temporary chaos called Kardar-Parisi-Zhang instability. This is due to the presence of a

destabilising term of the form δ|∇h|& that destroys the gradient-flow structure. Numerical experiments of Korzec in 2012

indicated that for δ small enough but still positive, trajectories stabilise. In this talk I present joint work with Piotr Rybka

(Warsaw U) that proves this. We perform an analysis of equilibria, linearisations, and spectral properties. This facilitates

application of an abstract result of Carvalho-Langa-Robinson and a convergence theorem due to Hale-Raugel, which are the

decisive ingredients for the proof.

Wenjiao Yan (Beijing Normal University, Beijing)

Title: Results related with complex structures on 𝑆"

Abstract: It is a longstanding problem that whether there exists a complex structure on the 6-dimensional sphere? Many famous

mathematicians have made efforts on this problem, such as Hopf, Wen-tsun Wu, Borel, Serre, LeBrun, Shiing-Shen Chern, Atiyah,

etc. This talk consists of two parts. (i) Taking advantage of isoparametric theory, we construct almost complex and complex

structures on certain isoparametric hypersurfaces in the unit sphere. As a consequence, there is a closed 8-dimensional manifold

𝑁( such that there exists a complex structure on 𝑆" Å~ 𝑁(. (ii) As a generalization of LeBrun's result, we prove that there is no

almost complex structure on the standard 𝑆"(1) which is compatible with the standard metric, such that the length of Nijenhuis

tensor is smaller than a certain constant everywhere. This talk is based on joint works with Professor Zizhou Tang.

Xiaoping Yang (Nanjing University, Nanjing)

Title: Measure upper bounds of nodal sets of solutions to Dirichlet problem of Schrödinger equations

Abstract: In this talk, we will discuss measure upper bounds of nodal sets of solutions to Dirichlet problem of Schrödinger

equations. With help of introducing a lift, a variant frequency function and a doubling index, establishing some elliptic estimates,

monotonicity properties and doubling estimates, and developing a continually dividing iteration procedure, we control the

measure upper bounds by the doubling index and obtain the bounds. This is a joint work with Hairong Liu and Long Tian.

Min Zhang (Zhejiang University, Hangzhou)

Title: Practical Impacts of Optimal Mass Transport: From Brain Imaging Analysis to Advanced Image Compression Techniques

Abstract: The application of Optimal Mass Transport theory moves beyond the traditional realms and has started to influence

diverse areas such as computer graphics, machine learning (for example, in generative models), economics (resource

allocation), etc. As computational methods continue to improve, the feasibility of employing OMT in practice increases, making it

an ever more potent tool in both theoretical explorations and concrete applications. This presentation aims to explore the

practical impact of optimal mass transport, beginning with a concise overview of its theoretical underpinnings. Subsequently, I

will pivot to its practical implementations, with a particular focus on the analysis of brain imaging and image compression.

Weiping Zhang (Nankai University, Tianjin)

Title: Deformations of Dirac operators

Abstract: We will describe some applications of deformed Dirac operators in geometry and topology.

Xi Zhang (Nanjing University of Science and Technology, Nanjing)

Title: On the existence of harmonic metrics on non-Hermitian Yang-Mills bundles

Abstract: In this talk, I will talk about the non-Hermitian Yang-Mills (NHYM for short) bundles over compact Kähler manifolds.

We will show that the existence of harmonic metrics is equivalent to the semi-simplicity of NHYM bundles. This work is joint

with Dr. Changpeng Pan and Zhenghan Shen.

Zhou Zhang (University of Sydney, Sydney)

Title: Mean curvature flow in almost Fuchsian manifold and generalisations

Abstract: In an earlier work, we studied the mean curvature flow in Fuchsian manifold. The preservation of graphic property of

the evolving surface plays a pivotal role. The more general case of almost Fuchsian manifold is now the focus. The metric is still

explicit by Uhlenbeck and slightly more complicated than the wrapped product metric for the Fuchsian case.

We have arrived at a similar conclusion for the mean curvature flow and are adapting it for the modified mean curvature flow.

The program provides a way of constructing constant mean curvature surfaces and even a foliation as conjectured from

topological point of view for 3-fold.

This is based on joint works with Longzhi Lin and Zheng Huang.

Bin Zhou (Peking University, Beijing)

Title: On singular Abreu equations

Abstract: In this talk, we report the recent progress on the solvability of singular Abreu equations which arise in the

approximation of convex functionals subject to a convexity constraint. In dimension two, the singular Abreu equations can be

solved by regularity of Monge-Ampère equation and the special algebraic structure. In higher dimensions, we solve the singular

Abreu equations by transforming into linearized Monge-Ampère equations with drifts. We establish global Hölder estimates for

the linearized Monge-Ampère equation with drifts under suitable hypotheses, and then use them to the regularity and

solvability of the second boundary value problem for singular Abreu equations in all dimensions. Many cases with general righthand

side will also be discussed.

Xin Zhou (Cornell University, Ithaca)

Title: Existence of four minimal spheres in 𝑆! with a bumpy metric

Abstract: In 1982, S. T. Yau conjectured that there exists at least four embedded minimal 2-spheres in the 3-sphere with an

arbitrary metric. In this talk, we will show that this conjecture holds true for bumpy metrics and metrics with positive Ricci

curvature. This is a joint work with Zhichao Wang (Fudan University).

Xiaohua Zhu (Peking University, Beijing)

Title: Complete steady Ricci solitons with positive curvature away from a compact set

Abstract: To characterize a complete steady Ricci soliton, one of main steps is to classify ancient solutions as the blow-down

models of soliton. In this talk, we will discuss and classify a class of steady Ricci solitons with positive curvature away from a

compact set which admit a split shrinking Ricci soliton of codimensional 1 as a blow-down model.