Program
 Geometric Analysis Seminar Student No.： 80 Time： Thu 15:30-16:30, 2018.3.22/ Fri 11:00-12:00, 2018.3.23 Instructor： Xu Guoyi, Zhang Yingying Place： Lecture Hall, Jin Chun Yuan West Bldg. Starting Date： 2018-3-17 Ending Date： 2018-7-1

2018-3-22

Speaker: Renjie Feng [Peiking University]
Time: March 22nd 15:30-16:30

Location: Lecture Hall, Jin Chun Yuan West Bldg.
Title: Spectrum of SYK model
Abstract: The SYK model is a random matrix model arising from condensed matter theory in statistics physics and black hole theory in high energy physics.  In this talk, we will first review some elementary results in random matrix theory, then we will introduce the SYK model. I will explain the spectral properties of the random matrix of SYK model,  such as the global density where a phase transition is observed, the central limit theorem of the linear statistics and the concentration of measure theory. In particular, we will derive the large deviation principle when the number of interaction of fermions is 2.

2018-3-23

Speaker: Bernhard Hanke [University of Augsburg]

Time: March 23nd 11:00-12:00

Location: Lecture Hall, Jin Chun Yuan West Bldg.
Title: $\Gamma$-structures and symmetric spaces
Abstract:

$\Gamma$-structures are  weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting $\Gamma$-structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of $\Gamma$-structures on manifolds which are nilpotent in the sense of homotopy theory. This includes homogeneous spaces with connected isotropy groups.

Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products  given  by geodesic symmetries define $\Gamma$-structures. This extends work of Albers, Frauenfelder and Solomon on $\Gamma$-structures on Lagrangian Grassmannians.

2018-4-12

Speaker: Shouhei Honda  (Tohoku University)
Time: April 12nd 15:30-16:30

Location: Lecture Hall, Jin Chun Yuan West Building

Title: Weyl's law on metric measure spaces with Ricci bounds from below.
Abstract: In this talk we give a necessary and sufficient condition for Weyl's law on a metric measure space with Ricci bounds from below. This is a joint work with Luigi Ambrosio and David Tewodrose (Scuola Normale Superiore).

2018-4-16

Speaker: Shouhei Honda  (Tohoku University)
Time: April 16th 15:30-16:30

Location: Lecture Hall, Jin Chun Yuan West Building

Title: Local spectral convergence in metric measure spaces with Ricci bounds from below.
Abstract: In this talk we give a necessary and sufficient condition for local spectral convergence with respect to measured Gromov-Hausdorff convergence. By using this, we give an affirmative answer in a stronger form to a question on harmonic functions on Alexandrov spaces, raised by Anton Petrunin. This is a joint work with Luigi Ambrosio (Scuola Normale Superiore).

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2018-3-17&18

March 17th (Saturday)

Morning
9:30-10:30am: Reiko Miyaoka (Tohoku University)
10:30-10:45am: break
10:45-11:45am: Linfeng Zhou (East China Normal University)
11:45-1:30pm: Lunch

Afternoon
1:30-2:30pm: Reiko Miyaoka (Tohoku University)
2:30-2:45pm: break
2:45-3:45pm: Shicheng Xu (Capital Normal University)
3:45-4:00pm: break
4:00-5:00pm: Keita Kunikawa(Tohoku University)

March 18th (Sunday)

Morning
9:30-10:30am: Linfeng Zhou (East China Normal University)
10:30-10:45am: break
10:45-11:45am: Shicheng Xu (Capital Normal University)

Titles and Abstracts

Reiko Miyaoka (Tohoku University)
Title:  Exceptional values of  the Gauss map of complete minimal surfaces
Abstract:  Let M be a complete minimal surface in R^3. The Gauss map G  is a holomorphic map into CP^1=S^2.  The celebrated theorem of Fujimoto says that G omits at most 4 points of CP^1, and this result is sharp. On the other hand, M is called algebraic when its total curvature is finite. In this case, M is conformally a punctured Riemann surface, and the  so-called Weierstrass data is extendable beyond the punctures.  Osserman proved that the exceptional values are at most 3 for algebraic  minimal surfaces, and conjectured that it should be at most 2.  However, this problem is still open.

In my first talk, I introduce the minimal surface theory, and give many  examples with two exceptional values as well as how we construct them. In the second talk, introducing the Nevanlinna theory, we state a trial to  attack this conjecture by extending the Nevanlinna theory.

Linfeng Zhou (East China Normal University)
Title:  The isoperimetric problem in 2-dim Finsler space forms
Abstract:  In this talk, we will give an introduction of Finsler geometry and the isoperimetric problem in Finsler manifold.  By using the variational calculus theory, some recent joint work with Mengqing Zhan will be discussed.

Shicheng Xu (Capital Normal University)
Title:  Ricci curvature, diameter and Gap Vanishing Volume Entropy
Abstract:  In this talk we consider a conjecture about a gap phenomena on volume entropy: given n, d>0, there exists a constant \epsilon(n,d)>0 such that if a compact Riemannian n-manifold M satisfies that Ricci curvature >=-(n-1) and diameter <=d, then the volume entropy h(M)<\epsilon(n,d) implies that h(M)=0. It can be viewed as an quantitative version of a revised Milnor's problem:  whether a finitely presented group of non-exponential growth implies polynomial growth? We prove that a positive answer of Milnor's problem above implies the gap phenomena, and prove some partial results under some additional assumptions.

Keita Kunikawa(Tohoku University)
Title:  Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow
Abstract:  In this talk, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kahler- Einstein manifolds to more general Kahler manifolds including Fano manifolds by using the methodology proposed by T. Behrndt. We first consider a weighted measure on a Lagrangian manifold in such a Kahler manifold and investigate the variational problem of the Lagrangian for the weighted volume under Hamiltonian deformations. We call a stationary point and a local minimizer of the weighted volume f-minimal and Hamiltonian f-stable. We show such examples naturally appear in toric Fano manifolds. Moreover, we consider the generalized Lagrangian mean curvature flow which is introduced by Behrndt and also by Smoczyk-Wang. We generalize the result by H. Li, and show that if the initial Lagrangian is a small Hamiltonian deformation of an f-minimal and Hamiltonian f-stable Lagrangian, then the generalized MCF converges to an f-minimal one. This is a joint work with Toru Kajigaya.