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Conference on Low Dimensional Topology

Nov 13,2020 - Nov 15,2020

This 3 day conference is intended to bring together researchers in mainland China working in the field of low dimensional topology. The aim is to discuss recent progress and to end with a discussion of future directions and open questions.

Please note that the conference will not be held on the SUSTech campus but in the International Center for Mathematics. Directions can be found here.

Hotel Address:Vienna Haomian International Hotel, Nanshan District (Shenzhen Tanglang Metro Station)

Invited speakers:

Haimiao Chen |
陈海苗 |
Beijing Technology and Business University |

Liang Chen |
陈亮 |
Northeast Normal University |

Zhi Chen |
陈智 |
Hefei University of Technology |

Zhiyun Cheng |
程志云 |
Beijing Normal University |

Zhi Lv |
吕志 |
Fudan University |

Jiming Ma |
马继明 |
Fudan University |

Shicheng Wang |
王诗宬 |
Peking University |

Jie Wu |
吴杰 |
Hebei Normal University |

Yunhui Wu |
吴云辉 |
Tsinghua University |

Yi Xie |
谢羿 |
BICMR |

Jingling Yang |
杨璟玲 |
SUSTech |

Min Yan |
严民 |
SUSTech |

Tian Yin |
田垠 |
Tsinghua University |

Wenyuan Yang |
杨文元 |
Peking University |

Fei Yu |
于飞 |
Zhejiang University |

Participants:

Xiaoming Du |
杜晓明 |
South China University of Technology |

Ziqiang Feng |
封自强 |
SuSTech |

Hongyu Guo |
郭宏宇 |
Harbin Normal University |

Weili Guo |
郭威力 |
Beijing University of Chemical Technology |

Ruizhi Huang |
黄瑞芝 |
Chinese Academy of Sciences |

Siyao Liu |
刘思瑶 |
Harbin Normal University |

Ye Liu |
刘晔 |
Xi'an Jiaotong-Liverpool University |

Yi Liu |
刘毅 |
BICMR |

Huadi Qu |
曲华迪 |
SuSTech |

Zipei Nie |
聂子佩 |
Princeton University |

Zizi Wang |
王子夕 |
Fudan University |

Pecheng Xu |
徐鹏程 |
Guangdong University of Finance |

Bin Yu |
余斌 |
Tongji University |

Fan Ye |
叶帆 |
University of Cambridge |

Yuti Zhang |
张钰堤 |
Harbin Normal University |

**Title：Topology and geometry of submanifolds from duality viewpoint**

**Speaker: Liang Chen, Northeast Normal University **

Abstract：Singularity and degeneracy destroy the structure of manifolds and give rise to essential difficulties in researching the deteriorative manifolds. Thus, it is crucial to develop new methods for investigating the singular or degenerate submanifolds. By characterizing the inner connection between pseudo spheres in semi-Euclidean space, the Legendrian duality is an effective method developed by Chen and Izumiya in 2009 for studying the submanifolds in non-flat space. Specially, the Legendrian duality is applicable for studying the singular or degenerate submanifolds. In this talk, using the singularity theory of mappings, we investigate the geometrical properties of the (singular or degenerate) submanifolds immersed in non-flat space from the viewpoint of duality. The part results in this talk is a joint work with Haibo Yu.

**Title: Cablings, Homotopy Groups, and A Question on Virtual Braid Groups**

**Speaker: Jie Wu, Hebei Normal University**

Abstract: In this talk, we will discuss the homotopy aspects of braid groups. In the first section, we will give a review on the classical connections between braids and homotopy theory. Then we talk the connections between Brunnian braids and homotopy groups in the second section. In the last section, we will discuss the theory of classical and virtual Brunnian braids.

**Title: On links with Khovanov homology of small rank**

**Speakers: Yi Xie, BICMR**

Abstract: In this talk I will discuss a classification result on the links whose Khovanov homology has rank no greater than 8, where the coefficient ring is Z/2. Moreover, I will show that Khovanov homology detects all links in the Thistlethwaite Link Table whose Khovanov homology has rank no greater than 12. This is joint work with Zhenkun Li and Boyu Zhang.

**Title****：Distinguishing 4-dimensional geometries via profinite completions**

**Speaker: Jiming Ma,Fudan University**

Abstract：It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except for geometries $ \mathbb{H}^{4}$, $\mathbb{H}^{2}_{\mathbb{C}}$ and $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some closed orientable 4-dimensional Seifert manifolds with general fibre $T^2$ are associated with one. For a Seifert 4-manifold $M$, we show that whether $M$ is geometric could be detected by the profinite completion of its fundamental group. This is joint work with Zixi Wang.

** **

**Title: Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus**

**Speaker: Yunhui Wu, Tsinghua University**

Abstract: In this article we study the first eigenvalues of closed hyperbolic surfaces for large genus. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than $\frac{L_1(X_g)}{g^2}$ up to a uniform positive constant multiplication. Where $L_1(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$. Moreover,we also show that this new lower bound is optimal as $g \to \infty$. This is a joint work with Yuhao Xue.

**Title: Some recent progress of virtual knots**

**Speaker:Zhiyun Cheng, Beijing Normal University**

Abstract: Virtual knot theory studies the embeddings of S^1 in thickened surfaces up to stable equivalence, which can be regarded as a generalization of classical knot theory. In this talk, I will briefly discuss some recent virtual knots invariants derived from chord index.

** **

**Title: Croke-Kleiner admissible groups: Property (QT) and quasiconvexity**

**Speaker:Wenyuan Yang, Peking University**

Abstract: Croke-Kleiner admissible groups firstly introduced by Croke-Kleiner belong to a particular class of graph of groups which generalize fundamental groups of 3--dimensional graph manifolds. In this talk, we will present a characterization of strongly quasiconvex subgroups by the finiteness of height in a CKA group. With further assumption on the vertex groups, we show that a CKA group satisfies a property (QT) introduced by Bestvina-Bromberg-Fujiwara that the group admits an equivariant quasi-isometric embedding into a finite product of quasi-trees. This is a joint work with Hoang Nguyen.

**Title: Extending periodic maps on surfaces over 4-sphere**

**Speaker: Wang Shicheng, Peking University**

Abstract: Let $F_g$ be the closed orientable surface of gunus $g$, and $w_g$ be a periodical map of the maximum order on $F_g$. We show that (1) for each $g$, $w_g$ is periodically extendable over $S^4$ for some non-smooth embedding $e: F_g\to S^4$, and not periodically extendable over $S^4$ for any smooth embedding $e: F_g\to S^4$. (2) for $g=4k+1, 4k+2$, $w_g$ is not extendable over $S^4$ for any smooth embedding $e: F_g\to S^4$; and for $g=4k, 4k+3$, $w_g$ is extendable over $S^4$ for some smooth embedding $e: F_g\to S^4$ .

This is a joint work with Wang Zhongzi.

We thank Ding Fang, Liu Yi and Wang Chao for mathematics they taught us.

**Title: A presentation of the fundamental groups of three manifolds from certain braid group actions**

**Speaker: Zhi Chen, Hefei University of Technology**

Abstract: There is a way to present the knot groups from the natural action of braid groups on free groups. We give a very similar way to presenting the fundamental groups of three manifolds, by using another action of the braid groups. Possibly we also investigate applications of that presentation.

** **

**Title: On the structure of the Kauﬀman bracket skein algebras of planar surfaces**

**Speaker: Haimiao Chen, Beijing Technology and Business University**

Abstract:Let R be a ring containing an invertible element A. For a sur-face Σ, its Kauﬀman bracket skein algebra S(Σ; R) is the R-module generated by framed links L in Σ × [0, 1], modulo some local relation-s; the multiplication L1 · L2 is defined by stacking L1 over L2. For each n ≥ 5, we give a presentation for S(Σ0,n; Q(A)) by generators and relations. When g ≥ 1, some partial results on the structure of S(Σg,n; Q(A)) are obtained.

**Title:Fixed Points of G-CW-complex with Prescribed Homotopy Type**

Abstract:Suppose G is a finite group, and f is a map from a finite CW-complex F to the fixed point set of a finite G-CW-complex Y. Is it possible to extend F to become the fixed point set of a finite G-CW-complex X, and extend f to become a G-map from X to Y, such that g is a homotopy equivalence after forgetting the G-action? In the problem, Y is the prescribed homotopy type of the desired G-CW-complex X.

For the case Y is a point, the problem becomes the fixed point set of contractible G-CW complex. In 1942, Smith gave the necessary homological condition for the fixed point set. In 1971, for semi-free actions by cyclic groups, Jones proved the sufficiency of Smith condition. In 1975, for actions by groups of not prime power order, Oliver gave necessary and sufficient condition in terms of the Euler characteristic of the fixed set. We extend these classical theories to general Y, and also give some examples.

Stavros Garoufalidis (ICM SUSTech)

Ingrid Irmer (ICM SUSTech)

Yi Liu (BICMR)

Jiping Zhang (ICM SUSTech)

SUSTech International Center for Mathematics

SUSTech Mathematics Department

Friday 13th of November

8:00-8:30 Registration

8:30-8:40 Welcome Speech

8:40-9:40 Shicheng Wang

*Extending periodic maps on surfaces over 4-sphere*

9:40-10:40 Min Yan

*Fixed Points of G-CW-complex with prescribed homotopy type*

* *

10:40-11:00 Tea Break

11:00-12:00 Yang Jingling

*Distance one surgeries on the lens space L(p, 1) and band surgeries on the torus knot T(2,p)*

12:00-14:00 Lunch

14:00-15:00 Jiming Ma

*Distinguishing 4-dimensional geometries via profinite completions*

15:00-15:20 Tea Break

15:20-16:20 Yunhui Wu

*Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus*

16:20-17:20 Zhi Chen

*A presentation of the fundamental groups of three manifolds from certain braid group actions*

Saturday 14th of November

8:30-9:30 Yifei Zhu

*Generalized modular forms in topology*

9:30-10:30 Jie Wu

*Cablings, homotopy groups, and a question on virtual braid groups*

* *

10:30-11:00 Conference photo +Tea break

11:00-12:00 Haimiao Chen

*On the structure of the Kau**ﬀ**man bracket skein algebras of planar surfaces*

* *

12:00-14:00 Lunch

14:00-15:00 Yi Xie

*On links with Khovanov homology of small rank*

15:00-16:00 Zhiyun Cheng

*Some recent progress of virtual knots*

16:00-17:00 Liang Chen

*Topology and geometry of submanifolds from duality viewpoint*

18:00 Conference Dinner

Sunday 15th of November

8:30-9:30 Yin Tian

*The Drinfeld center of monoidal 2-categories in 3+1D Dijkgraaf-Witten Theory*

9:30-10:30 Wenyuan Yang

*Croke-Kleiner admissible groups: Property (QT) and quasiconvexity*

* *

10:30-11:00 Tea Break

11:00 -12:00 Yi Liu

*T**o be announced*

Address: 201 Jinqizhigu Building, No.1 Tangling Road, Nanshan District, Shenzhen City, Guangdong Province.

(Please take lift No. 5 or 6（low-rise）to the 2nd floor)

Postcode: 518055

Tel: 26920864-8027

By Metro: Tanglang Station (Exit D), follow Liuxian Avenue (westbound) to Tangling Road (about 900 metres), then turn right

By Bus: Tanglang Depot North Station, Line M554, M343

Please send an email to Ms Zheng Jiayi zhengjy@mail.sustech.edu.cn

There are funds available for graduate students and postdocs to attend.

** **

Please email your CV to Ms Zheng Jiayi zhengjy@mail.sustech.edu.cn to apply.

Jiayi Zheng (SUSTech):zhengjy@mail.sustech.edu.cn