Program
3rd YMSC Postdoc Workshop
Student No.:100
Time:Saturday 8:20-17:20 November 25, Sunday 8:20-12:10 November 26
Instructor:  []
Place:Lecture Hall, Floor 3, Jin Chun Yuan West Building
Starting Date:2017-11-25
Ending Date:2017-11-26

Schedule of the 3rd YMSC Postdoc Workshop
November 25(Saturday) 2017
Lecture Hall, Floor 3, Jin Chun Yuan West Building

08:30-09:00

Jinzhao Pan( 潘锦钊)

On the full Birch and Swinnerton-Dyer conjecture of CM elliptic curve over Q at potentially supersingular prime

09:05-09:35

Cindy Tsang( 曾善谊)

Counting binary quartic forms with small Galois group

09:40-10:10

Duo Li(李铎)

Strictly nef vector bundles

10:10-10:30

Tea break

10:30-11:00

Jingyue Chen( 陈靖越)

Differential Zeros of Certain Special Functions

11:05-11:35

Chen He(贺琛)

Localization of certain torus actions on odd-dimensional manifolds

11:40-12:10

Jiongyue Li (李冏玥)

Bifurcation Results on Compact Spin Manifolds

12:10-14:00

Lunch break

14:00-14:30

Yunlong Zang(臧云龙)

On Scalar Wave Equation in Extreme Kerr Geometry

14:35-15:05

IDELON-RITON Guillaume

About some scattering properties of the massive Dirac equation on the Schwarzchild-Anti-de Sitter spacetime

15:10-15:40

Xixia Ma(马西霞)

Existence of Solution of Navier-Stokes Equations With Nonhomogeneous Boundary Conditions

15:40-16:00

Tea break

16:00-16:30

Shuang Liu( 刘双)

Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs

16:35-17:05

Yong Han(韩勇)

Conformal restriction measure on simple loops

 

 

Schedule of the 3rd YMSC Postdoc Workshop
November 26(Sunday) 2017
Lecture Hall, Floor 3, Jin Chun Yuan West Building

08:30-09:00

Shaochuang Huang(黄少创)

Some aspects on short-time existence of Ricci flow on non-compact manifold

09:05-09:35

Yingxiang Hu( 胡鹰翔)

Willmore inequality on hypersurfaces in hyperbolic space

09:40-10:10

Yi Huang( 黄意)

How rare are embedded ideal n-simplices in finite volume hyperbolic (n+1)-manifolds?

10:10-10:30

Tea break

10:30-11:00

Kun Tian(田昆)

Some mathematical representations of biological sequences with their applications.

11:05-11:35

Kowshik Bettadapura

Complex Supergeometry and Obstruction Theory

11:40-12:10

Yajie Zhang( 张雅杰)

Parabolic Transmission Problems on Polygonal Domains and Application to Finite Element Method

12:10

Lunch

 

 

 

 

TITLES AND ABSTRACTS

Speaker: Kowshik Bettadapura
Title: Complex Supergeometry and Obstruction Theory
Abstract: In similarity with algebraic geometry, complex supergeometry is a geometry built on super-commutative (as opposed to commutative) algebras. Obstruction theory in complex supergeometry concerns the classification problem of complex superspaces. In this talk I will review some of the rudiments of obstruction theory; present a general classification and, if time permits, comment on the key motivation driving my research: obstruction theory of the supermoduli space of curves.

 

Speaker: Jingyue Chen
Title: Di↵erential Zeros of Certain Special Functions
Abstract: We study the zero loci of local systems of the form c⇧, where ⇧ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space X, and c is a  given di↵erential operator on the space of sections V- = r(X, K—1). We give several di↵erent descriptions of the zero locus of c⇧. As applications, we prove that the locus is algebraic and  in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus. This is joint work with A. Huang, B.H. Lian and S.-T. Yau.

 

Speaker: IDELON-RITON Guillaume
Title: About some scattering properties of the massive Dirac equation on the Schwarzchild- Anti-de Sitter spacetime
Abstract: This talk will first introduce the Schwarzschild-Anti-de Sitter spacetime and some of its geometrical properties that are relevant for the analysis of the massive Dirac fields. After giving a short discussion of the Cauchy problem, I will give some results about the existence and asymptotic completeness of the wave operators with some ideas of the proof. A direct consequence of this is the existence of the asymptotic velocity operator. The second part of the talk will then provide a lower bound of the local energy decay for these fields which is obtained by means of the construction of quasimodes. If time is left, I will introduce the concept of resonances and show that they exist in this setting.

 

Speaker: Yong Han
Title: Conformalrestriction measure on simple loops
Abstract: In this talk, I will show the conformal restrictionprobability measure on simple loop space can be characterized by a parameter and construct it by two di↵erent ways. Some properties will also be issued.

 

Speaker: Chen He
Title: Localization of certain torus actions on odd-dimensional manifolds
Abstract: Let torus T act on a compact smooth manifold M, if the equivariant cohomology HT⇤(M) is a free module of HT⇤(pt), then according to the Chang-Skjelbred Lemma, HT⇤(M) can be localized to the 1-skeleton M1 consisting of fixed points and 1-dimensional orbits. Goresky, Kottwitz and MacPherson considered the even-dimensional case where M is an algebraic mani- fold and M1 is 2-dimensional, and introduced a graphic description of its equivariant cohomology. In this talk, we will deal with the odd-dimensional case.

 

Speaker: Yingxiang Hu
Title: Willmore inequality on hypersurfaces in hyperbolic space
Abstract: In this talk, I will prove a geometric inequality for star-shaped and mean-convexhypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a gen- eralization of Willmore inequality for closed surfaces in hyperbolic 3-space.

 

Speaker: Shaochuang Huang
Title: Some aspects on short-time existence of Ricci flow on non-compact manifold
Abstract: In this talk, I will survey some short-time existence results of Ricci flow without assuming the initial metric has bounded curvature. I will sketch an example how to produce a Ricci flow on non-compact manifold by pseudo-locality.

 

Speaker: Yi Huang
Title: How rare are embedded ideal n-simplices in finite volume hyperbolic (n+1)-manifolds?
Abstract: Spoiler: quite rare. Specifically, we show that the collection of embedded ideal n-simplices in any finite volume hyperbolic (n+1)-manifold occupy a set of zero volume. This  is a generalization of the Birman-Series geodesic sparsity theorem.

 

Speaker: Duo Li
Title: Strictly nef vector bundles
Abstract: We investigate properties of strictly nef vector bundles over projective varieties. We show that on elliptic curves, strictly nef vector bundles are ample, whereas there exist Hermitian flat and strictly nef vector bundles on any smooth curve with genus g 2.  We  also establish  that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety Xn(n3) has strictly nef ⇤2TX, then it is isomorphic to Pn or quadric Qn.

 

Speaker: Jiongyue Li
Title: Bifurcation Results on Compact Spin Manifolds
Abstract: The bifurcation phenomena describe the distribution structure of solutions to an equation in some proper function space. We analyze how bifurcation occurs near the trivial solutions of nonlinear Dirac equations on compact spin manifolds.

 

Speaker: Shuang Liu
Title: Volume doubling, Poincar inequality and Gaussian heat kernel estimate for non-negatively curved graphs
Abstract: In this talk, I will present some recent developments of volume doubling property, Gaussian heat kernel estimate and then Poincar inequality on graphs, under the assumption of the curvature-dimension inequality CDE’(n,0), which can be considered as a notion of curvature for graphs. By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property. From this we can prove the Gaussian estimate for heat kernel, and then the Poincar inequality and the Harnack inequality.

 

Speaker: Xixia Ma
Title: Existence of Solution of Navier-Stokes Equations With Nonhomogeneous Boundary Con- ditions
Abstract: We will recallsome new results on existence of solution of steady Navier-Stokes equations with nonhomogeneous boundary conditionand the related open problem. Finally, we prove existence of the steady Magnetohydrodynamic equations in a two-dimensional Bounded domain with nonhomogeneous boundary value.

 

Speaker: Jinzhao Pan
Title: On the full Birch and Swinnerton-Dyer conjecture of CM elliptic curve over Q at poten- tially supersingular prime
Abstract: We first introduce the BSD conjecture, and some known results related to our work, then we focus on our case: let E be a CM elliptic curve defined over Q of analytic rank 1 with potentially supersingular reduction at a prime p 5, such that there is a quadratic twist A of E over Q which has good supersingular reduction at p. I will explain the ongoing work in proving the full BSD conjecture of such E at p and the partial results.

 

Speaker: Kun Tian
Title: Some mathematical representations of biological sequences with their applications.
Abstract: Sequence analysis has become one of the most active and important areas of bioin- formatics as the tools for getting biological sequences increase. Analyzing the morphological structures from a large number of sequences, obtaining the information such as homology, evolu- tionary relationship and evolutionary history of di↵erent species, and inferring their development ancestors have become important issues. However, due to the huge size and high complexity of the data, it will lead to a lot of time to compute the solving process if there is no e↵ective algorithm, even becoming an impossible diffi cult problem to work out. In this talk, we introduce two new methods to solve the above problem. The first method is the natural vector method, which is a whole-genome, non-aligned and non-parametric rapid representation for sequences. Natural vector reflects the distribution of nucleotides or amino acids in gene sequence or protein sequence, which contains the total numbers, the average positions and the high order central moments of nucleotides or amino acids. The second one is the Yau-Hausdor↵ method, which takes all possible translations and rotations into consideration to achieve the best match of graphical curves of two DNA or protein sequences. The Yau-Hausdor↵ methodcan be used for measuring the similarity of sequences based on two important tools: the Yau-Hausdor↵ distance and graphical representation of sequences. The complexity of this method is lower than that of any other two dimensional minimum Hausdor↵ algorithm. These methods has been applied to build a variety of genomic databases for predicting and classifying new sequences quickly and precisely, which provides more accurate description of the evolution relationships between species.

 

Speaker: Cindy Tsang
Title: Counting binary quartic forms with small Galois group
Abstract: Counting the number of GL2(Z)-equivalence classes of integral and irreducible binary forms of a fixed degree d is a classical problem in analytic number theory. The cases d = 2, 3 were known, and the case d = 4 was considered recently by M. Bhargava and A. Shankar. In this joint project with Stanley Xiao, we are also interested in the case d = 4 but only those forms whose Galois group is small (by which we mean it is isomorphic to a subgroup of the dihedral group of order eight). It turns out that such forms may be parametrized by integral binary quadratic forms. I will explain this in more detail and give some counting theorems we were able to prove.

 

Speaker: Yunlong Zang
Title: On Scalar Wave Equation in Extreme Kerr Geometry
Abstract: We consider the Cauchy problem for the massless scalar wave equation in Extreme Kerr geometry for smooth initial data compactly supported outside the event horizon. Firstly, we derive an integral representation of the solution as an infinite sum over the angular momen- tum modes, each of which is an integral of the energy variable ! on the real axis. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. Furthermore, based on this integral representation, we prove that the solution decays pointwise in time in L1.

 

Speaker: Yajie Zhang
Title: Parabolic Transmission Problems on Polygonal Domains and Application to Finite Ele- ment Method
Abstract: We study theoretical and practical issues of the second-order parabolic equation ut + Lu = f , where L = div(A) is a second-order operator with piecewise smooth coeffi  cient matrix A, with possibly jump discontinuities across a finite number of curves, called the interface. We analyze the problem on polygonal domains. Under some additional condition we establish well-posedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed
on adjacent sides of the polygonal domain, or when the interfaces are not smooth, we fail to acquire well-posedness on weighted Sobolev space but we are able to obtain the decomposition u = ureg + ws, into a function ureg with better decay at the vertices and a function ws that is locally constant near the vertices, thus proving well-posedness in an augmented space. Based on the theoretical analysis we are able to implement a certain Finite Element scheme with improved graded meshes, which can recover the rate of convergence for piecewise polynomials of degree m 2' 1. Three numerical tests are included in the last.