School of Mathematical Sciences, Peking University

Research Areas

Topology

Time: Mar 24 , 2023

Topology

1. Categorification Theory (ZHENG Hao)

In [Zheng, Acta Math. Sin. (Engl. Ser.) (2014)], Hao Zheng proposed a geometric approach to categorify the tensor products of integrable representations of symmetric quantum Kac-Moody algebras. Namely, for each such tensor product we constructed a category whose Grothendieck group can be identified with the underlying space of the modules by sending indecomposable projective modules to the canonical basis and, moreover, a functorial action of the generators of the quantum algebra on this category with specific isomorphisms categorifying the defining relations.

2. Complex manifolds (CAI JinXing)

The automorphism group of a compact complex manifold acts on its cohomology ring. Such action is faithful for Riemann surfaces with genus bigger than one, but not always faithful on algebraic surfaces or algebraic varieties of higher dimension. In [Cai-Liu-Zhang, Compos. Math. (2013)], Jin-Xing Cai and his collaborators showed the action on a complex minimal surface of general type with irregularity bigger than 2 is faithful. In [Cai, J. Algebra (2012)], Cai gave a complete classification and explicit construction of surfaces of general type with automorphism group of order 4, geometric genus at least 35 and irregularity at least 3.

3. Contact Topology (DING Fan)

In [Ding-Geiges, J. Topol. (2009)] Fai Ding and his coauthor constructed the contact analogue of Kirby calculus, namely, they gave the contact versions of various topological handle moves, including handle cancellation and the first and second Kirby moves. This continues their previous work on the surgery description of contact three-manifolds from the standard contact three-sphere.In [Ding-Geiges-van Koert, J. Lond. Math. Soc. (2) (2012)], Ding et al. gave representations of a certain class of contact 5-manifolds via 2-dimensional diagrams, and described moves on such diagrams that do not change the contact 5-manifold. As an application, they classified Stein fillable contact 5-manifolds for which the 6-dimensional Stein filling has no 3-handles, up to connect sums.In [Ding-Geiges, Compos. Math. (2010)], the authors computed the mapping class groups of S1×S2 using contact geometry, originally due to H. Gluck.

4. Knot Theory (JIANG Boju, WANG Jiajun, ZHENG Hao)

In [Jiang, Acta Math. Sin. (Engl. Ser.) (2016)], Boju Jiang gave a new skein characterization of the Conway potential function for colored links. As a special case, they gave a new skein characterization of the Alexander-Conway polynomial of knots. In [Jiang- Wang-Zheng, J. Knot Theory Ramifications (2017)], they gave a skein characterization of the HOMFLY polynomial. These characterizations do not use the smoothing of crossings.

5. Three-dimensional Topology (WANG Shicheng)

1) In [Derbez-Liu-Wang, J. Topol. (2015)], Shicheng Wang et al. considered the set of volumes of all representations from the fundamental group of a three-manifold to the isometry group of the hyperbolic three-space or the universal cover of SL(2,R). They showed that a three-manifold with positive Gromov norm or Seifert fibered piece in its JSJ decomposition has a finite cover with nontrivial representation volume to one of the two groups. They also gave examples that there are manifolds with trivial representation volumes in both cases.
2) A three-manifold M is said to dominate N if there is a nonzero degree map from M to N. In [Boileau-Rubinstein-Wang, Comment. Math. Helv. (2014)], Wang et al. showed that a closed orientable 3-manifold dominates only finitely many integral homology 3-spheres, and an integral homology 3-sphere 1-dominates at most finitely many closed 3-manifolds.
3) In [Wang-Wang-Zhang-Zimmermann, Groups Geom. Dyn. (2015); Wang-Wang- Zhang, Acta Math. Sin. (Engl. Ser.) (2016); Wang-Wang-Zhang, J. Knot Theory Ramifications (2017)], they considered the finite group action on surfaces which can extend to a group action on S3 for some embedding of the surface in S3. For each genus, Wang et al. obtained a complete list of the maximal order of such groups (finite and cyclic) and concretely constructed these actions.