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Xiaoxiang Chai (POSTECH)
[Talk 1]
Title: Band width estimates (I)
Abstract: As is well known, a positive Ricci curvature bound implies a diameter bound for a closed manifold. When the Ricci curvature is replaced by the scalar curvature, such a diameter bound is not possible. However, a bound on the size of the manifold along a certain direction is possible if we also impose a special topology. This is known as Gromov's band width estimate. There are various proofs of the band width estimate, and we focus the prescribed mean curvature surface (or ( mu )-bubble) approach to the band width estimate of Gromov. The first lecture will introduce some basics of hypersurfaces in a Riemannian manifold, in particular, first variation and second variation of ( mu )-bubbles, torical scalar curvature rigidity and foliation construction. The key results of the first lecture were due to Schoen-Yau, Bray-Brendle-Neves, Gromov and J. Zhu.
[Talk 2]
Title: Band width estimates (II)
Abstract: We introduce the weighted ( mu )-bubble and apply the technique to study band width estimate under a spectral condition. If time permits, we will also discuss some Llarull type theorems, and spacetime settings of the band width estimates.
Seunghyeok Kim (Hanyang University)
[Talk 1]
Title: Compactness of the $Q$-curvature problem (1)
Abstract: In the first talk, I will explain my recent work with Dr. Liuwei Gong and Prof. Juncheng Wei (CUHK), which settles the question of the $C^4$-compactness for the solution set of the $Q$-curvature problem on a smooth compact Riemannian manifold for $5 le n le 24$ under the validity of the positive mass theorem.
For over a decade, an example of an $L^{infty}$-unbounded sequence of solutions for $n ge 25$ has been known (Wei and Zhao, 2013).
We will explore the differential geometric and historical backgrounds, precisely state the result, and examine its potential applications and impacts.
[Talk 2]
Title: Compactness of the $Q$-curvature problem (2)
Abstract: In the second talk, I will explain the strategy for proving the $C^4$-compactness for the solution set of the $Q$-curvature problem on a smooth compact Riemannian manifold for $5 le n le 24$.
After outlining Schoen's (1988) approach within the framework of the $C^2(M)$-compactness theorem for the Yamabe problem, I will discuss the fundamental obstacles in applying this strategy to the $Q$-curvature problem.
Finally, I explain the principal observation in my joint work with Dr. Liuwei Gong and Prof. Juncheng Wei (CUHK):
A linearized equation associated with the $Q$-curvature problem can be transformed into an overdetermined linear system, which admits a nontrivial solution due to an unexpected algebraic structure of the Paneitz operator.
Dongyeong Ko (Rutgers University)
[Talk 1]
Title: Scalar curvature comparison and rigidity of 3-dimensional weakly convex domains
Abstract: I will introduce Gromov's dihedral rigidity conjecture on scalar-mean curvature rigidity on manifolds with boundary. Then I will discuss a recent comparison and rigidity result of scalar curvature and scaled mean curvature on the boundary for weakly convex domain in Euclidean space, which is a joint work with Xuan Yao. This result is a smooth analog of Gromov's dihedral conjecture. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles.
[Talk 2]
Title: Equivariant Simon-Smith min-max theory and Minimal hypersurfaces with arbitrarily large Betti number on high dimensional spheres
Abstract: In this lecture, I will give an overview of Simon-Smith min-max theory which produces topologically controlled minimal surfaces, and explain a new equivariant min-max construction to construct minimal hypersurfaces. Also, I discuss a min-max construction of sequences of minimal hypersurfaces with arbitrarily large Betti number on high dimensional spheres in dimension 4 to 7 as an application.
Man-Chun Lee (CUHK)
[Talk 1]
Title: Introduction: Kahler-Ricci flow on manifolds of general type
Abstract: Since the work of Yau, existence of canonical metrics on complex manifolds has been an important question. It was discovered by Cao that the Ricci flow provides a natural deformation path to the unique Calabi-Yau metric whenever it exists. In the first talk, we will discuss Cao’s proof to the existence of Kahler-Einstein metric on manifolds with ample canonical line bundle
[Talk 2]
Title: Kahler-Ricci flow on Calabi-Yau manifolds
Abstract: In the second part of the talk, we will discuss compact Kahler manifolds with vanishing first Chern class. We will discuss the zero-order estimate of Yau and discuss how it leads to the existence of Ricci flat metrics
[Talk 3]
Title: Kahler-Ricci flow on non-compact manifolds
Abstract: In the third part of this talk, we will switch our attention to non-compact manifolds. Motivated by a conjecture of Yau, we will discuss how Ricci flow can be used to study non-compact Kahler manifolds with positive curvature.
Sanghoon Lee (KIAS)
Title: Type-I Blowup Solutions for Yang-Mills Flow
Abstract: In this talk, we construct an infinite-dimensional family of solutions for the Yang-Mills flow on R^n times SO(n)$ for $5 leq n leq 9$, which converge to $SO(n)$-equivariant homothetically shrinking solitons, modulo the gauge group. The method relies on the spectral analytic approach.
John Man Shun Ma (SUSTech)
[Talk 1]
Title: Introduction to geometric flows
Abstract: In this talk, we give an introduction to several geometric flows and describe some of its basic properties. We also describe the existence and uniqueness results to geometric flows on compact manifolds.
[Talk 2]
Title: Existence results for non-compact geometric flows
Abstract: In this second talk, we describe several existence results in non-compact geometric flows, particularly the Mean curvature flow.
[Talk 3]
Title: Uniqueness and backward uniqueness in non-compact ge- ometric flow
Abstract: In the third talk, we discuss the uniqueness and backward uniqueness problems in geometric flows. First, we use an energy method to prove the uniqueness of geometric flows. Second, we prove the backward uniqueness for some geometric flows via an ODE-PDE inequality.
Jiewon Park (KAIST)
Title: Some monotonicity formulas and their stability
Abstract: Numerous monotonicity formulas involving the Green function have been discovered in recent years. They are important tools leading to new breakthroughs and novel proofs of significant theorems. In this talk we prove a new family of monotonicity formulas for the Green function on manifolds supporting a certain functional inequality, which we call Property (HG), as well as rigidity theorems in the case of equality. We also discuss explicit examples of manifolds which has Property (HG). Time permitting, we will also discuss a related forthcoming joint work on Christine Breiner on quantitative stability of such monotonicity formulas as well as how the techniques work on RCD spaces, a generalization of manifolds with Ricci curvature lower bounds.
Wei-Bo Su (NCTS)
[Talk 1]
Title: Special Lagrangian submanifolds and Lagrangian mean curvature flow
Abstract: Special Lagrangian submanifolds are volume-minimizing submanifolds in Calabi–Yau manifolds, which play a vital role in String Theory, particularly in Mirror Symmetry. In this introductory lecture, I will discuss the background and definition of special Lagrangian submanifolds and the approach to establishing the existence theory of special Lagrangians using mean curvature flow.
[Talk 2]
Title: Thomas—Yau—Joyce program and singularities in Lagrangian mean curvature flow
Abstract: The Thomas–Yau–Joyce program aims to achieve the canonical decomposition of Lagrangian submanifolds through mean curvature flow with surgeries. Therefore, a deep understanding of singularities is required. I will start with the Thomas–Yau conjecture and explain Joyce’s surgery approach. In the second half, I will discuss some recent developments in singularity formation in Lagrangian mean curvature flow.