Organizer：Akito Futaki

Speakers：李子鈺，Anton Iliashenko，Alejandro Gil García，邓嘉龙

Time：Sat.,13:00-17:30 ,Sept.14, 2024

Venue：Jingzhai 静斋 304

13:00 - 14:00

李子鈺 Li Ziyu (YMSC)

Title:

Random zeros and random critical points on compact Kahler manifolds

Abstract:

We will introduce some results of the statistical behavior of random zeros and random critical points of random polynomials (or holomorphic sections) when $N \rightarrow \infty$ , and how they depend on the underlying geometry.

14:10 - 15:10

Anton Iliashenko (BIMSA)

Title:

A special class of k-harmonic maps inducing calibrated fibrations

Abstract:

We consider two special classes of k-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map u:(L^k,g)→(M^n,h) where k≤n and α is a calibration k-form on M. Away from the critical set, the image is an α-calibrated submanifold of M. These were previously studied by Cheng-Karigiannis-Madnick when α was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map u:(M^n,h)→(L^k,g) where n≥k and α is a calibration (n−k)-form on M. Away from the critical set, the fibres u^{−1}{u(x)} are α-calibrated submanifolds of M. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where (M,h) are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the G_2 version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study.

15:20 - 16:20

Alejandro Gil García (BIMSA)

Title:

Curvature, symmetries and hypersurfaces of supergravity c-map spaces

Abstract:

Supergravity c-map spaces are a class of quaternionic Kähler manifolds of negative scalar curvature arising from supergravity and string theory. In this talk we review some recent results about the structure of these spaces. First we show that any deformed supergravity c-map space is not locally homogeneous. As a corollary, we obtain that for any homogeneous projective special Kähler manifold, its corresponding deformed supergravity c-map space is complete and of cohomogeneity one. Then we exhibit a group of isometries acting effectively on a subclass of supergravity c-map spaces. Finally, we study the geometry of a hypersurface of the one-loop deformation of the Hermitian symmetric spaces of non-compact type dual to complex Grassmannians of 2-planes. This is a joint work with V. Cortés, M. Röser, A. Saha and D. Thung.

16:30 - 17:30

邓嘉龙 Deng Jialong (YMSC)

Title:

Revisiting Non-Embedding Theorems

Abstract:

As this is a learning seminar, I will provide a brief historical overview of non-embedding theorems. Specifically, I will focus on Nikolayevsky's proof demonstrating that a non-simply connected complete hyperbolic n-manifold cannot be isometrically immersed in a (2n-1)-dimensional Euclidean space.