Plenary talks:
Ngo Quoc Anh (Vietnam National University at Hanoi)
Modica's estimates in the higher-order setting
Abstract: Forty years ago, by Bernstein's technique via the maximum principle, Luciano Modica established a gradient estimate for bounded entire solutions to the nonlinear Poisson equation Δu = F'(u) in the Euclidean space. Remarkably, this is one of the main techniques for solving De Giorgi's conjecture for the Allen–Cahn equation and analyzing various elliptic equations. Later, it turns out that Modica's estimates for higher-order elliptic equations are also available despite the lack of maximum principles, and this leads to several interesting results in conformal geometry. This talk concerns the intriguing connection between the Modica estimate and the Q-curvature. Some simple-looking questions, yet still open, are also mentioned.
Huy Nguyen (Nguyễn Thế Huy) Queen Mary University of London
Recent Advances in High Codimension Geometry
Abstract: In this talk, we will discuss recent advances in understand the geometry of high coidmension submanifolds. In particular, we will survey how the high codimension mean curvature flow can be used to prove topological classifications of such submanifolds given a curvature condition. This involves finding precise curvature conditions preserved by the flow and obtain refined estimates on the structure of singularity formation together with a surgery procedure similar to the one used by Perelman to prove the Poincare and Geometrization conjectures.
Yohei Sakurai (Saitama University, Japan)
Almost splitting and quantitative stratification for super Ricci flow
Abstract: I will discuss almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. I will present almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. This talk is based on the joint work with Keita Kunikawa (Tokushima university).
Nguyen Van Thin (Thai Nguyen University of Education)
Some selected topics in complex geometry, arithmetic geometry and qualitative properties of the solutions to PDE involving nonlocal operators
Abstract: This talk contains three parts:
In the first part, I present some recently results about Second Main Theorem in Nevanlinna-Cartan theory
for meromorphic mappings and its application.
In the second part, I will talk about Schmidt subspace theorem in approximated diophantine in arithmetic
geometry.
In the final part, I will introduce some our recently results about existence of weak solution, normalized solutions
and blow up of solutions to some class of fractional partial differential equations.
Gabjin Yun (Myongji University, Korea)
Stress-Energy tensor of the traceless Ricci tensor
Abstract: In this talk, we are going to introduce a stress-energy tensor of the traceless Ricci tensor for a Riemannian manifold, and derive its properties and relations with the Bach tensor and the Cotton tensor. In particular, we will investigate the conservative property of the stress-energy tensor of the traceless Ricci tensor. We say the stress-energy tensor of the traceless Ricci tensor satisfies the conservation law if its divergence is vanishing. As an application, we study the harmonicity of Reimannian curvature tensor or Weyl tensor and conservation law of the stress-energy tensor of traceless Ricci tensor for a Riemannian manifold satisfying the vacuum static equation or critical point equation.
Invited talks
Tran Nguyen An, Thai Nguyen University of Education, Vietnam
On pseudo-supports and structure of rings and modules
Abstract:
Yimin Chen, Pusan National University, Korea
Volume-Preserving Mean Curvature Flow with Capillary Boundary in Hyperbolic Space
Abstract: We study capillary hypersurfaces in hyperbolic space that meet a totally geodesic hyperplane at a fixed angle. A constrained mean curvature flow is introduced which exactly preserves the enclosed volume while strictly decreasing a energy functional. First, we prove global existence and convergence of the flow. Next, we show that as time goes to infinity the evolving hypersurfaces converge smoothly to a totally umbilical “cap” with the same contact angle. Finally, we demonstrate that this cap uniquely minimizes the energy among all hypersurfaces enclosing the same volume.
Ha Tuan Dung, Hanoi Pedagogical University No.2, Vietnam
On semilinear elliptic equations on weighted manifolds with compact boundary
Abstract: In this talk, we discuss will quantitative properties of positive smooth solutions to a semilinear elliptic equation on a complete weighted manifold under the Dirichlet boundary condition. We will also present and discuss applications of these results to specific partial differential equations arising in geometry and physics.
Tran The Dung, VNU University of Science, Hanoi
Higher-order Riemannian spline interpolation problems: a unified approach via gradient flows
Abstract: This talk addresses the problems of spline interpolation on smooth Riemannian manifolds, with or without the inclusion of least-squares fitting. Our unified approach utilizes gradient flows for successively connected curves or networks, providing a novel framework for tackling these challenges. This method notably extends to the variational spline interpolation problem on Lie groups, which is frequently encountered in mechanical optimal control theory. As a result, our work contributes to both geometric control theory and statistical shape data analysis.
We rigorously prove the existence of global solutions in H\"{o}lder spaces for the gradient flow and demonstrate that the asymptotic limits of these solutions validate the existence of solutions to the variational spline interpolation problem. This constructive proof also offers insights into potential numerical schemes for finding such solutions, reinforcing the practical applicability of our approach.
Nguyen Thi Anh Hang, Thai Nguyen University of Education, Vietnam
Flexibility of affine cones over complete intersections of three quadrics
Abstract:
Đoàn Tùng Lâm, John Hopkins University, USA
Topological Triviality of Cones with Small Entropy
Abstract: In this talk, I will give you a brief introduction to mean curvature flow. Then we will show that regular hypercones in $\mathbb{R}^{n+1}$ with small entropy have simple topology. As a result, the link of the given cone is topologically an $(n-1)-$sphere. If time permitted, we will briefly prove that the space of those links can be deformation retracted onto the space of equatorial spheres.
Nguyen Tien Manh, VNU University of Science, Hanoi
On the Parabolic frequency under the general geometric flow
Abstract: In 1979, Almgren introduced a frequency functional to study the regularity of harmonic functions on R n , a concept that has played a key role in understanding behavior of solutions to a PDE. Inspired by Almgren’s work, Poon generalized this idea to the heat equation and proved the monotonicity of this parabolic frequency. Recently, Colding-Minicozzi provided a new method to prove Poon’s monotonicity results on general manifolds without any further assumption on the curvatures. Following Colding-Minicozzi’s method, this talk concerns results on the parabolic frequency functionals under the general geometric flow.
Nguyen Van Hoang, FPT University, Vietnam
The sharp Hardy-Adams inequality in the hyperbolic space
Abstract: The Hardy-Moser-Trudinger inequality was proved by Wang and Ye (Adv. Math. 230 (2012), no. 1, 294-320). This inequality combines the sharp Hardy and Moser-Trudinger inequality in the two dimensional unit ball. It was proved in higher dimension by Nguyen (Trans. Amer. Math. Soc 377 (2024), no. 4, 2297-2315). An analogue of the Hardy-Moser-Trudinger inequality for higher order derivatives, i.e. the Hardy-Adams inequality, was established by Lu and Yang (Adv. Math. 319 (2017), 567-598) and by Li, Lu and Yang (Adv. Math. 333 (2018) 350-385) in the even dimension. In this talk, we present the sharp Hardy--Adams inequality in arbitrary dimension. Our approach is based on the non-increasing symmetric rearrangement technique together with some sharp Hardy-Sobolev inequalities in unit ball.
Tran Thanh Hung, Texas Tech Univ, USA
Ricci Solitons and Isoparametric Functions
Abstract: In this talk, we will describe a specific connection between Ricci solitons and isoparametric functions. The former comes from the theory of Ricci flows, initiated by R. Hamilton in the 80s and played a key role in Perelman's resolution of the Poincare conjecture, one of the seven Millennium Prize Problems. The latter was motivated by questions in geometric optics and the classification in an ambient round sphere, formulated as Question 34 in S.T. Yau's list, is remarkably deep. The intersection of these theories is sufficiently rich that gives insight into a fundamental conjecture towards generalizing Perelman's work to higher dimensions. We'll consider both the complex and real setups.
Seungsu Hwang, Chung-Ang University, Korea
Conformal vector field and Einstein type manifolds
Abstract: We discuss an Einstein-type manifolds admitting a non-trivial conformal vector field. An Einstein-type manifold is a generalization of the critical point equation, vacuum static spaces, and $V$-static spaces. In this talk we discuss a rigidity result of a compact Einstein-type manifold admitting a non-trivial conformal vector field.
Jihyeon Lee, IBS-CGP, Pohang, Korea
The first eigenvalue estimates on λ-hypersurfaces
Abstract: In recent decades, various eigenvalue estimates have been established for minimal-type hypersurfaces under geometric constrains. The foundational result was given by Choi and Wang in 1983, who proved a lower bound for the first eigenvalue of closed minimal surfaces in complete Riemannian manifolds with positive Ricci curvature. This result was later generalized by Cheng–Mejia–Zhou and Ding–Xin to the settings of closed f-minimal surfaces and closed self-shrinkers, respectively. Beyond closed cases, Brendle and Tsiamis considered complete non-compact self-shrinkers to obtain the lower bound 1/4 for the first eigenvalue. Very recently, Conrado and Zhou also contributed to this direction by obtaining eigenvalue estimates for f-minimal hypersurfaces in gradient shrinking Ricci solitons. In this talk, I will present a further generalization to λ-hypersurfaces in Rn+1. Assuming the condition |λ| ≤1/2-1/(2n), we show that the first eigenvalue is bounded below by 1/4-λ^2/2 . This estimate recovers the result of Brendle–Tsiamis in the special case λ=0. Since self-shrinkers and λ-hypersurfaces can be viewed as minimal and CMC hypersurfaces in a smooth metric measure space with weight function |x|^2/4, our result provides a natural extension of eigenvalue estimates from the minimal to the CMC setting.
Sanghun Lee, Pusan National University, Korea
Stable capillary hypersurfaces and rigidity in (weighted) Riemannian manifolds
Abstract: In this talk, we investigate rigidity phenomena for stable capillary hypersurfaces embedded in (weighted) Riemannian manifolds, focusing on their relationship with scalar curvature. We begin by discussing rigidity in 3-dimensional Riemannian manifolds. Next, we explore topological or geometric invariants in higher dimensions, delving into the rigidity of high-dimensional Riemannian manifolds. Finally, we extend the rigidity results for three-dimensional Riemannian manifolds to the setting of three-dimensional weighted Riemannian manifolds. This is joint work with Sangwoo Park and Juncheol Pyo.
Yong Luo, Chongqing University of Technology, China
Universal inequalities for eigenvalues of the Dirichlet Laplacian and the clamped plate problem
Abstract: In this talk we introduce some new universal inequalities for eigenvalues of the Dirichlet Laplacian and the clamped plate problem defined on bounded domains $\Omega \subset M^n$, when $M^n$ is a submanifold of the Euclidean space or $M^n$ is the hyperbolic space. This is based on joint works with Xianjing Zheng.
Eungmo Nam, KIAS, Korea
Half-space type theorem of the mean curvature flow
Abstract: In this talk, using a paraboloid as a geometric barrier, we give a new non-existence theorem for translators of the mean curvature flow in Euclidean space.
Jiewon Park, KAIST, Korea
Quantitative Rigidity for Ricci Curvature
Abstract: On manifolds with nonnegative Ricci curvature, we establish a quantitative relationship between the pinching of a monotone functional defined by Colding and the distance to the nearest cone. Moreover, we demonstrate that the pinching quantitatively controls almost splitting. This is joint work with Christine Breiner.
Juncheol Pyo, Pusan National University, Korea
Translating Solitons for the Mean Curvature Flow
Abstract: Translating solitons and self-shrinkers are solitons for the mean curvature flow (MCF). They serve not only as blow-up models of singularities of the MCF, but also as minimal surfaces in certain Riemannian manifolds. In this talk, we compare properties of minimal surfaces and MCF solitons, particularly with respect to Bernstein-type theorems and properness. More precisely, we present rigidity results for graphical translators that move in non-vertical directions. In addition, we introduce sufficient conditions for the properness of translating solitons.This is based on joint work with Daehwan Kim, Yuan Shyong Ooi, and John Ma.
Keomkyo Seo, Sookmyung Women’s University, Korea
Overdetermined boundary value problems in a Riemannian manifold
Abstract: Serrin’s overdetermined problem is a famous result in mathematics that deals with the uniqueness and symmetry of solutions to certain boundary value problems. It is called "overdetermined" because it has more boundary conditions than usually required to determine a solution, which leads to strong restrictions on the shape of the domain. In this talk, we discuss overdetermined boundary value problems in a Riemannian manifold and discuss a Serrin-type symmetry result to the solution to an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature and it will be discussed about an overdetermined problems with a nonconstant Neumann boundary condition in a warped product manifold.
Nguyen Tien Tai, VNU University of Science, Hanoi
Radial symmetry of solutions to higher (fractional) order elliptic equation
Abstract: Let $s$ be positive and $n \geq 2$ be an integer such that $n >2s$. In this paper, we are concerned with positive distributional solution to the following elliptic equation
(-\Delta)^s u= f(|x|, u) \quad\text{in }\mathbb R^n\setminus\{\0\}.
By imposing some suitable conditions on $f$, we obtain the radially symmetry property of positive distributional solutions by using the method of moving spheres in integral form, extending the previous result of Jin, Li and Xu '08 to the fractional setting. This is my joint work with Quỳnh Lê.
Pham Quynh Trang, Thai Nguyen University of Education, Vietnam
Concentration phenomena for anisotropic fractional Choquard equations and potential competition
Abstract: In this talk, we study the existence of weak solution to a anisotropic fractional Choquard equation in whole space. The nonlinear function has subcritical growth. Two potential functions are continuous and satisfy some natural assumptions. Using Variational methods, we show that the existence of solution and concentration of maximum point of solution. Furthermore, we also get the semiclassical state solutions. This is joint work with T. V. Nguyen.
Abhitosh Upadhyay, Indian Institute of Technology, India
Stability and eigenvalue bounds on a smooth domain of the Euclidean space
Abstract: We will first recall some classical geometric inequalities for hypersurfaces in Euclidean space, where equality holds if and only if the hypersurface is a geodesic sphere. The upper bounds involve isoperimetric ratio and mean curvature terms. We then observe that these inequalities arise from a lower bound on the $L^2$-norm of the position vector and derive stability results from a general pinching result of the moment of inertia.
Hoang Ngoc Yen, Thai Nguyen University of Education, Vietnam
On the sectional genera and Cohen Macaulay rings
Abstract: In this talk, we provide characterizations of a Cohen-Macaulay local ring in terms of the sectional genera, the CohenMacaulay type, and the second Hilbert coefficients for certain primary ideals. We also characterize Gorenstein rings and quasi-Buchsbaum rings in terms of the sectional genera for certain primary ideals. My talk is based on joint work with S. Kumashiro and H. L. Truong.