Image reconstruction via discrete curvatures
Duan, Yuping
Tianjin University
Abstract：The curvature regularities are well-known for providing strong priors in the continuity of edges, which have been applied to a wide range of applications in image processing and computer vision. However, these models are usually non-convex, non-smooth and highly non-linear, the first- order optimal condition of which are high-order partial differential equations. Thus, the numerical computation is extremely challenging. In this paper, we propose to estimate the discrete curvatures, i.e., mean curvature and Gaussian curvature, in the local neighborhood according to differential geometry theory. By minimizing certain functions of curvatures on all level curves of an image, it yields a kind of weighted total variation minimization problem, which can be efficiently solved by the proximal alternating direction method of multipliers (ADMM). Numerical experiments are implemented to demonstrate the effectiveness and superiority of our proposed variational models for different image reconstruction tasks.
Fukaya category， Landau-Ginzburg model and related open problems
Fan, Huijun
Peking University
Abstract：Landau-Ginzburg models appear naturally in the study of mirror symmetry and become a corner theory of the global mirror symmetry. The closed string version corresponds to the Grmov-Witten theory, FJRW theory, VHS and etc. In this lecture, I will explain the category theories of LG model and their important role in homological mirror symmetry conjecture and the recent work by Gaitto-Moore-Witten. I will list some main questions in this field.
临床医学中的几何问题
Kong, Dexing
Peking University
Abstract：TBA
Volumes and Chern-Simons invariants for hyperbolic 3-manifolds
Jinsung Park
Korea Institute for Advanced Study
Abstract：In the works of Neumann, Garoufalidis-Thurston-Zickert, the volume and Chern-Simons invariant for SL(n,C)-representation have been extensively studied, in particular, the regulator expression (in terms of Rogers dilogarithm functions) of these invariants was obtained for complete hyperbolic 3-manifolds of finite volume. In this talk, I will explain a generalization of such a result for incomplete hyperbolic 3-manifolds of finite volume. This is a joint work with Seokbeom Yoon.
Non-Euclidean data processing method based on information geometry
Sun, Huafei
Beijing Institute of Technology
Abstract：In this talk, first, the background of information geometry will be introduced briefly and then some approaches for data processing will be presented under information geometric frame.
Isoparametric polynomials and Hilbert 17th problem
Tang, Zizhou
Nankai University
Abstract：Hilbert’s 17th problem asks that whether every nonnegative polynomial
can be a sum of squares of rational functions. It has been answered affirmatively
by Artin. However, as to the question whether a given nonnegative polynomial is a
sum of squares of polynomials is still a central question in real algebraic geometry.
In this paper, we solve this question completely for the nonnegative polynomials
(associated with isoparametric polynomials, initiated by E. Cartan) which define the
focal submanifolds of the corresponding isoparametric hypersurfaces.
On signed Euler characteristic
Wang, Hongyu
Yangzhou University
Abstract：In this talk, we discuss the Chern-Hopf conjecture. Let M be a closed
symplectic manifold of dimension 2n with non-ellipticity. We can define an
almost Kahler structure on M by using the given symplectic form. Using
Darboux coordinate charts, we deform the given almost Kahler structure
to obtain a homotopy equivalent measurable Kahler structure on the universal
covering of M. Analogous to Teleman's L2-Hodge decomposition on
PL manifolds or Lipschitz Riemannian manifolds, we give a L2-Hodge decomposition
theorem on the universal covering of M w.r.t. the measurable
Kahler metric.
Fukaya category， Landau-Ginzburg model and related open problems
Xu, Hongwei
Zhejiang University
Abstract：Mean Curvature Flow and Sphere Theorem for Submanifolds
Abstract: In this talk, we will focus on the mean curvature flow theory with sphere theorems, and discuss the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that our main theorem is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for certain submanifolds with positive Ricci curvature. Finally, we present a list of unsolved problems in this area. This is joint work with Dr. Li Lei.
RC-positive, vanishing theorem and rigidity of harmonic maps
Yang, Xiaokui
Tsinghua University
Abstract：In this presentation, we will discuss some recent progress on the geometry of compact manifolds with RC-positive tangent bundles, including an affirmative answer to a long-standing open problem of S.T. Yau on rational connectedness of compact Kahler manifolds with positive holomorphic sectional curvature, and new Liouville type theorems for holomorphic maps and harmonic maps. Several open problems related to the theory of RC-positivity will also be discussed.
海岸城市气象和海洋中的数学问题及解决方案
Zhengan Yao
Sun Yat-sen University
Abstract：海岸城市的气象具有特殊的性质，它与大陆和海洋的气候变化都息息相关，主要是与其空气流体以及洋流的相应性质相关，并涉及其大规模数据的计算等。
Deformable registration frameworks based on functions of bounded deformation
Xiaoping Yang
Nanjin University
Abstract：Deformable image registration is a widely used technique in the field of computer vision and medical image processing. Basically, the task of deformable image registration is to find the displacement field between the moving image and the fixed image. Many variational models are proposed for deformable image registration, under the assumption that the displacement field is continuous and smooth. However, displacement fields may be discontinuous, especially for medical images with intensity inhomogeneity, pathological tissues, or heavy noises. In the mathematical theory of elastoplasticity, when the displacement fields are possibly discontinuous, a suitable framework for describing the displacement fields is the space of functions of bounded deformation (BD). Inspired by this, we propose some novel deformable registration models, called the BD and BGD (bounded generalized deformation) models, which allow discontinuities of displacement fields in images. The BD and BGD models are formulated in variational frameworks by supposing the displacement fields to be functions of BD or BGD. The existence of solutions of these models is proven. Numerical experiments on 2D images show that the BD and BGD models outperform the classical demons model, the log-domain diffeomorphic demons model, and the state-of-the-art vectorial total variation model. Numerical experiments on two public 3D databases show that the target registration error of the BD model is competitive compared with more than ten other models. This is joint work with Ziwei Nie, Chen Li and Hairong Liu.
The Fixed Trust Region Method for Image Recovery
Tieyong Zeng
The Chinese University of Hong Kong
Abstract：In mathematical optimization, a trust region is the subset of the region of the objective function that is approximated using a model function (often a quadratic). If an adequate model of the objective function is found within the trust region, then the region is expanded; conversely, if the approximation is poor, then the region is contracted. Trust-region methods are also known as restricted-step methods. In this talk, we illustrate how to use a fixed trust region method for image recovery.
Enriques 2n-folds and analytic torsion
Ken-Ichi Yoshikawa
Kyoto University
Abstract：In this talk, a compact Kaehler manifold of even dimension is called simple Enriques if it is not simply connected and its universal covering is either Calabi-Yau or hyperkaehler. These manifolds were introduced and studied independently by Boissiere-NieperWisskirchen-Sarti and Oguiso-Schroer. We introduce a holomorphic torsion invariant of simple Enriques 2n-folds and study the corresponding function on the moduli space of such manifolds. In the talk, we report its basic properties such as the strong plurisubharmonicity and the automorphy, as well as possible (conjectural) applications. If time allows, we will also give an explicit formula for the invariant as an automorphic function on the moduli space in some cases.
Positive scalar curvature on noncomapct foliations
Zhang, Weiping
Nankai University
Abstract：正数量曲率度量的存在性是微分几何中的重要研究课题之一。本演讲将介绍叶状流形（foliation）上的正数量曲率问题的研究进展，特别是新近与苏广想合作得到的非紧叶状流形的结果。
BCOV invariant and birational equivalence
Zhang, Yeping
KIAS
Abstract：Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called BCOV invariant. Now we consider a pair $(X,Y)$,
where $X$ is a Kaehler manifold and $Y\subseteq X$ is a canonical divisor.
In this talk, we extend the BCOV invariant to such pairs.
The extended BCOV invariant is well-behaved under birational equivalence.
We expect that these considerations may eventually lead to a positive answer to Yoshikawa's conjecture
that the BCOV invariant for Calabi-Yau threefold is a birational invariant.
Singular Kaehler-Einstein metrics on $Q$-Fano compactifications of Lie group
Zhu, Xiaohua
Peking University
Abstract：We will give a criterion for the existence of singular Kaehler-Einstein metrics on $Q$-Fano compactifications of Lie group. In particular, we will classify $SO(4)$-Fano compactifications with singular Kaehler-Einstein metrics.