TAN, Ser Peow

Professor, Dept of Mathematics, NUS

Deputy Director, Institute for Mathematical Sciences (IMS), NUS, Singapore.

 

 

 

Education

 

Current interests (mathematical)

·         Geometric structures on manifolds

·         Circle Packings on Projective surfaces

·         Simple geodesics on surfaces and variations of McShane's identities

·         Trigonometry of hyperbolic 4 space

·         Dynamics of the mapping class group action on SL(2,C) character varieties.

 

 

Curriculum Vitae

A copy of my recent CV can be downloaded here:  CV (pdf file).

 

 

Recent Papers

 

§         Sadayoshi Kojima, Shigeru Mizushima and Ser Peow Tan, "Circle packings on surfaces with projective structures and uniformisation", Pacific J. Math. 225 (2006), no. 2, 287--300. Preliminary version ps file,  pdf file, also available at Math. ArXiv math.GT/0308147.

 

§         Sadayoshi Kojima, Shigeru Mizushima and Ser Peow Tan, "Circle packings on surfaces with projective structures: A survey",  Spaces of Kleinian groups, 337--353, London Math. Soc. Lecture Note Ser., 329, Cambridge Univ. Press, Cambridge, 2006. ps file, pdf file.

 

§         Ser Peow Tan, Yan Loi Wong and Ying Zhang, "Generalizations of McShane's identity to hyperbolic cone-surfaces",  J. Differential Geom. 72 (2006), no. 1, 73--112. ps file, pdf file , prelimary version available at math. ArXiv math.GT/0404226.

 

§         Ser Peow Tan, Yan Loi Wong and Ying Zhang, "Necessary and sufficient conditions for McShane's identity and variations",  Geom. Dedicata 119 (2006), 199--217. Preliminary version ps file, pdf file, also available at math. ArXiv math.GT/0411184.

 

§         Ser Peow Tan, Yan Loi Wong and Ying Zhang, "McShane's identity for classical Schottky groups", Pacific Journal of Math 237  (2008),  no. 1, 183--200. ps file, pdf file, available at math. ArXiv math.GT/0411628.

 

§         Ser Peow Tan, Yan Loi Wong and Ying Zhang, "Generalized Markoff maps and McShane's identity", Advances in Mathematics 217 (2008) 761-813. ps file, pdf file.

 

§         Mong Lung Lang, and Ser Peow Tan, "A simple proof of the Markoff conjecture for prime powers", Geometriae Dedicata 129 (2007) 15-22. Available at math.Arxiv math.NT/0508443, Revised version as of April 2006 here: ps file, pdf file.

 

§         Ser Peow Tan, Yan Loi Wong and Ying Zhang, "The SL(2,C) character variety of the one-holed torus",  Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 103--110 (electronic).  (This is basically a research announcement announcing the results of [4] and[6] above as well as [9] below) (preliminary version: ps file, pdf file. Available at math.Arxiv math.GT/0509033, final version: ERA.)

 

§         Ser Peow Tan, Yan Loi Wong and Ying Zhang, "End invariants of SL(2,C) characters of the one-holed torus", American Journal of Math. 130 (2008), 385-412.  ps file,   pdf file

 

§         Shawn Pheng Keong Ng and Ser Peow Tan, "The complement of the Bowditch space in the SL(2,C) character variety", Osaka Journal of Math. 44 (2007), 247-254. ps file, pdf file.

 

§         Ser Peow Tan, Yan Loi Wong and Ying Zhang, "Length series identities for surfaces, three manifolds and representation varieties", Complex Analysis and Geometry of hyperbolic spaces, (Kyoto, 2005).  RIMS Kokyuroku (2006) 151--152. pdf file.

 

 

List of Selected Publications ( click  here for full list of publications)

1.      Yoshinobu Kamishima and Ser Peow Tan, Deformation spaces on geometric structures, Advanced Studies in Pure Mathematics. 20, pp 263-299, 1992.

2.      Ser Peow Tan, Deformations of flat conformal structures on a hyperbolic 3-manifold, J. Differential Geom. 37, pp 161-176, 1993.

3.      Ser Peow Tan, Complex Fenchel-Nielsen Coordinates for Quasi-Fuchsian Structures, International Journal of Mathematics, 5-2, pp 239-251, 1994.

4.      Ser Peow Tan, Branched $CP^1$-structures on surfaces with prescribed real holonomy, Mathematische Annalen, 300, pp 649-667, 1994.

5.      Mong Lung Lang, Chong-Hai Lim and Ser Peow Tan, Independent generators for Congruence subgroups of the Hecke Groups, Mathematische Zeitschrift, 220, pp 569-594, 1995.

6.      Mong Lung Lang, Chong-Hai Lim and Ser Peow Tan, Principal congruence subgroups of the Hecke groups. J. Number Theory 85 (2000), no. 2, 220--230.

7.      Sadayoshi Kojima, Shigeru Mizushima and Ser Peow Tan, Circle packings on surfaces with projective structures. J. Differential Geom.  63 (2003), no. 3, 349—397.

8.      Ser Peow Tan, Yan Loi Wong and Ying Zhang, Generalizations of McShane's identity to hyperbolic cone-surfaces, J. Differential Geom.  72 (2006), no. 1, 73--112.

9.      Ser Peow Tan, Yan Loi Wong and Ying Zhang, Generalized Markoff maps and McShane's identity. Adv. Math.  217  (2008),  no. 2, 761--813.

10.  Ser Peow Tan, Yan Loi Wong and Ying Zhang, End invariants for ${\rm SL}(2,\mathbf C)$ characters of the one-holed torus.  Amer. J. Math.  130  (2008),  no. 2, 385--412.

 

Collaborators (Past and Present)

 

 

PhD Thesis of my student Ying Zhang

My student Ying Zhang successfully defended his PhD thesis in July 2004. The title is "Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane's Identity". You can download the pdf file here.

 

 

Gallery of Images

These pictures arise from from recent joint work with Yasushi Yamashita where we study  the dynamics of the action of the modular group PSL(2,Z) on the (relative) SL(2,C) character varieties of a one-holed torus (based on previous work with Y.L. Wong, Y. Zhang and S.P.K. Ng). The relative character varieties are determined by fixing the trace of the boundary curve on the torus. The pictures are produced by a computer program created by Yasushi Yamashita. Colored regions represent characters where the action of the modular group is proper and the black region represents characters where the action is not proper. When the action is proper, the orbit is controlled by a finite sub-tree of a infinite trivalent tree, the changes in the colors represent changes in the complexity of this finite tree. The size of the tree grows to infinity as one approaches the boundary of the colored regions.

 

 

This series of pictures shows how the Maskit slice deforms as the boundary trace changes from -2+0i to -2+4i. The modular group acts properly discontinuously on the colored regions and the action is not proper on the black region.

 

 

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This series of pictures shows how the Maskit slice deforms as the boundary trace changes from -2+0i to -6+0i.

 

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This third series can be considered as deformations of the Riley slice  again as boundary trace changes from -2+0i to -6+0i.

 

 

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This animation shows various slices of the relative SL(2,C) character varieties of a one holed torus, as the boundary trace changes from -2 to -6. The trace of a generator is fixed at 2+6i, the colored region represents characters for which the modular group acts properly, the black region represents characters for which the action of the mapping class group is not proper. The initial picture below corresponds (apparently, assuming a conjecture of B. Bowditch) to a slice of quasi-fuchsian space.

 

Click to see Animation  

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Below are some interesting miscellaneous pictures obtained by varying parameters etc in the program of Yamashita:

 

 

 

 

 

 

 

 

 

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Tan Ser Peow
Department of Mathematics
National University of Singapore
2, Science Drive 2, Singapore 117543
Republic of Singapore

mattansp  (at) nus (dot) edu (dot) sg

(65) 6516-6160 (office)
(65) 6779-5452 (fax)

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Tan Ser Peow <mattansp (at) nus (dot) edu (dot) sg>

 

 

 

 

 

 


Last Modified: 4 Mar 2009.

 

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