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09:30 | Reception and Information |
Olivier Biquard | Guofang Wei | Jeffrey Viaclovsky | Akito Futaki | Yongbin Ruan | Tuschmann Wilderich |
10:30 | Coffee Break | ||||||
11:00 | Yusheng Wang | Yunhui Wu | Taiji Marugame | Yu Li | Adachi Masanori | Chengjian Yao | |
12:00 | Lunch | ||||||
14:00 | Aeryeong Seo | Hao Fang | Free Discussions |
Gang Liu | Satoshi Nakamura | Jian Wang | Departure |
15:00 | Coffee Break | Coffee Break | |||||
15:30 | Yang Li | Yohsuke Imagi | Mijia Lai | Ruobing Zhang | Wei Sun | ||
We construct new asymptotically conical Ricci flat Kähler metrics with singular cone at infinity, by looking at rank 2 complex symmetric spaces. In general there are two possible asymptotic cones, which can be understood in terms of the wonderful compactification of De Concini-Procesi, but there may be obstructions to the existence of the corresponding Ricci flat Kähler metric. Joint work with Thibaut Delcroix.
We discuss some constant curvature problems for conic 4-manifolds, which have been extensively studied in the smooth case. Partially inspired by stability studies in Kaehler geometry, we propose the proper analogue conditions in conformal geometry and provide some existence and non-existence results after discussing the corresponding PDEs. Parts of works are joint with Biao Ma and Wei Wei, respectively.
We set up Donaldson-Fujiki picture for constant Cahen-Gutt momentum in the theory of deformation quantization. We also see an intersection formula for K-stability in this case.
In this talk, I will introduce a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two distinct surfaces with Gaussian curvature less than 1, which are also conformal to the Euclideanunit disk with the same conformal factor on the boundary, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to show the best constantof a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. Other applications of this inequality include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on theplane, classification of solutions for mean field equations on flat tori and the standard sphere, etc. The resolution of several open problems in these areas will be presented. Some generalizations of the inequality to include singular terms or moregeneral surfaces will also be presented.
Special Lagrangians are a certain class of area-minimizing Lagrangian submanifolds of Calabi-Yau manifolds (of complex dimension greater than two, in the case of primary interest). Geometric Measure Theory contains a natural notion of possibly-singular special Lagrangians, the moduli spaces of which are automatically compact; but essentially nothing is known globally about the moduli spaces. There are however some local results about the structure of moduli spaces, and an interaction between them and the Fukaya categories of ambient Calabi-Yau manifolds.
I will first report recent works on Obata equation for manifolds with boundary. Then I discuss two related applications: one is in the setting of conformally compact Einstein manifolds, the other is about the rigidity for a warped product manifold.
On collapsing K3 surfaces, the Ooguri-Vafa metric is the local model for the neighbourhood of the singular SYZ fibre. It contains within itself a region modelled on the famous Taub-NUT metric. I will give a sketchy report on my recent work generalising these constructions to complex dimension 3, which is expected to be relevant for the SYZ conjecture on 3-folds.
In this talk, we show that any asymptotically flat manifold with controlled holonomy admits a refined torus fibration over an ALE manifold. In addition, we prove that any complete asymptotically flat Ricci-flat metric on $\mathbb R^4$ must be isometric to the Euclidean or the Taub-NUT metric, provided that the tangent cone at infinity is not $\mathbb R \times \mathbb R_+$. This is a joint work with Xiuxiong Chen.
A fundamental result of Donaldson-Sun states that non-collapsed Gromov-Hausdorff limits of polarized K\"ahler manifolds, with 2-sided Ricci curvature bounds, are normal projective varieties. We extend their approach to the setting where only a lower bound for the Ricci curvature is assumed. More precisely, we show that non-collapsed Gromov-Hausdorff limits of polarized K\"ahler manifolds, with Ricci curvature bounded below, are normal projective varieties. In addition the metric singularities are precisely given by a countable union of analytic subvarieties. This is a joint work with Gabor Szekelyhidi.
Chains are natural family of curves on a CR manifold. They satisfy a second order ODE and play a role of geodesics in CR geometry. In this talk, we show that chains can be characterized as geodesics of a certain Kropina metric (a singular Finsler metric). As an application, we reprove and generalize some important facts on chains: (i) Chains determine the CR structure up to conjugacy. (ii) Two nearby points can be joined by a chain. This is joint work with J.-H. Cheng, V. S. Matveev, and R. Montgomery.
In geometric analysis on complex manifolds, we often extract plurisubharmonic potential functions from geometric conditions that we handle. The typical example is Oka's lemma (1953) that states plurisubharmonicity of the logarithm of the reciprocal of the distance to pseudoconvex hypersurfaces in Euclidean spaces, which is an important step in solving classical Levi's problem.
Diederich and Fornaess (1977) showed that any smoothly bounded pseudoconvex domain in a Stein manifold admits bounded plurisubharmonic exhaustions. Roughly speaking, the Diederich-Fornaess index of the domain is the supremum of the Hoelder exponents of these exhaustions and measures how well the pseudoconvex boundary can be approximated by strictly pseudoconvex hypersurfaces from inside of the domain.
In this talk, after reviewing recent development, in particular, by Bingyuan Liu, around this index, we shall explain formulae that express this index and Steinness index, which is a similar index for such approximations from outside of the domain.
Our formulae involve not only Levi form but also d'Angelo 1-form and work for Takeuchi 1-convex domains in arbitrary complex manifolds. This is a joint work with Jihun Yum.
The notion of coupled K\”ahler Einstein metrics was introduced recently by Hultgren-W.Nystr\”om. In this talk, we discuss deformation of coupled K\”ahler Einstein metrics on Fano manifolds.
In particular, we obtain a necessary and sufficient condition for a given coupled K\”ahler Einstein metrics to be deformed to another one for another close decomposition of the first Chern class for a Fano manifold admitting non-trivial holomorphic vector fields.
This generalizes a result of Hultgren-W.Nystr\”om.
One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute higher genus Gromov-Witten theory of compact Calabi-Yau 3-fold. There have been a collection of remarkable conjecture from physics for so called 14 one-parameter models, simplest compact Calabi-Yau 3-folds similar to the quintic 3-folds. These conjectures were originated from universal properties of BCOV B-model. The backbone of this collection are four structural conjectures: (1) Yamaguchi-Yau finite generation; (2) Holomorphic anomaly equation; (3) Orbifold regularity and (4) Conifold gap condition.
In the talk, I will present background and our approach to the problem.
This is a joint work with F. Janda and S. Guo. Our proof is based on certain localization formula from log GLSM theory developed by Q. Chen, F. Janda and myself.
In 1985 Diederich and Ohsawa proved that every disc bundle over a compact Kahler manifold is weakly 1-complete. In this talk, under certain conditions we generalize this result to the case of fiber bundles over compact Kahler manifolds whose fibers are classical bounded symmetric domains.Moreover if the bundle is obtained by the diagonal action on the product of irreducible bounded symmetric domains of classical type, we show that it is hyperconvex.
The parabolic flow method can be applied to solving complex quotient equations on closed Kahler manifolds. As a result, we solve the complex quotient equations.
There are many interesting examples of complete non-compact Ricci-flat metrics in dimension 4, which are referred to as ALE, ALF, ALG, ALH gravitational instantons. In this talk, I will describe some examples of these geometries, and another type called ALH_b gravitational instantons. These metrics are related to K3 surfaces because they arise as bubbles for sequences of Ricci-flat metrics on K3 surfaces, and are therefore important for understanding the behavior of Calabi-Yau metrics near the boundary of the moduli space. I will also describe some aspects of these constructions.
It is not known whether a contractible 3-manifold admits a complete metric of positive scalar curvature. For example, the Whitehead manifold is a contractible 3-manifold but not homeomorphic to $R^{3}$. In this talk, I will present my proof that the Whitehead manifold does not have a complete metric with positive scalar curvature. I will further explain that a contractible genus one 3-manifold, a notion introduced by McMillan, does not admit a complete metric of positive scalar curvature.
In this talk, we will begin with the Soul Theorem and Soul Conjecture in Riemannian geometry (the conjecture was proven by Professor Perelman in 1994). Then their analogues in Alexandrov geometry will be introduced, especially for the Soul Conjecture. The conjecture in dimension less than or equal to 3 was solved by Shioya-Yamaguchi in 2002. In this talk, we will give a rough proof for the conjecture in dimension 4, which is a joint work with Professor Xiaochun Rong in 2018.
In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. In several joint works with S.Seto, L. Wang; C. He; and X. Dai, S.Seto, we prove the conjecture for the sphere. Namely for any strictly convex domain in the unit $S^n$ sphere, the gap is $\ge 3\frac{\pi^2}{D^2}$. As in B. Andrews and J. Clutterbuck's work, the key is to prove a super log-concavity of the first eigenfunction.
I will report on general results and open questions about spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds, and, in particular, present the first classes of manifolds for which these spaces have non-trivial higher rational homotopy, homology and cohomology groups. Whereas the index theoretic invariants that are commonly used to distinguish path components of spaces and moduli spaces of metrics build on the Atiyah-Patodi-Singer Index Theorem, in this context one has to invoke instead constructions that involve the Cheeger-Gromoll Splitting Theorem. If time permits, I will also relate how spaces of metrics behave under surgery constructions performed on the underlying manifolds.
We confirm a folklore conjecture suggesting that for a complete noncompact manifold of finite volume with curvature $-1 \leq K_M \leq 0$, if the universal covering space of $M$ is a visibility manifold, then the fundamental group of each end of $M$ is almost nilpotent. This is a joint work with Ran Ji.
A Symplectic Calabi-Yau 4-manifold is a symplectic 4-manifold with vanishing first Chern class. It is an open problem in symplectic topology that whether a simply-connected compact symplectic Calabi-Yau manifold must be symplectmorphic to K3 surface with one of its Kahler structures. For the important hypersymplectic manifold case, which means the manifold admits a two-sphere worth of symplectic structures, hypersymplectic flow is introduced to deform the hypersymplectic structure to hyperkahler structure. We will review the progresses towards this geometric flow, most importantly long time existence under several conditions.
This talk focuses on the collapsing of Ricci-flat Kaehler metrics in all dimensions. The first part of my talk centers around the collapsing geometry in dimension 4. We will accurately describe the moduli space structure of elliptic K3 surfaces. Also we will give both existence and classification results for K3 surfaces with 3-dimensional nilpotent collapsing fibers. Next, we will introduce a large family of entirely new examples of higher dimensional collapsed Calabi-Yau spaces. This class of examples accurately describe the complex structures degeneration for the first time in the literature.
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