Teaching

In fall 2025, I am teaching MATH40043.01/MATH60075 “Geometric group theory”. The topic of the course will be “CAT(0) cube complexes”. We will follow the book [Sch23] closely at least at the beginning. Depends on the speed, we may focus on special cube complexes at the end.

Time: 9:55 – 12:30 every Monday at H6504.

TA: Josiah Oh， email: josiahoh@fudan.edu.cn;

Office hours: TBA

Homework is due one week after it is assigned and please submit on elearning.

Schedule:

Week 1: Introduction; Basics of Metric spaces;

Week 2: Group action, groups as metric spaces;

Week 3: CAT(k) spaces.

Week 4: CAT(0) spaces: angles, flat cones

Week 5: National holiday:)

Week 6: Flat cones, polyhedral complexes.

Week 7: M_k polyhedral complexes

Week 8: CAT(0) cube complexes, Gromov’s link condition

Week 9: RAAGs

Week 10: Hyperplanes

Week 11: Half-space systems

Week 12: Roller duality

Week 13: Coxeter groups

Week 14: Walls in Coxeter groups

Week 15: Cubulating Coxeter groups

Some videos:

CAT(0) cube complexes and group theory by Michah Sageev

CAT(0) Cube Complexes by Daniel T. Wise

Workshop: 3-Manifolds, Artin Groups and Cubical Geometry

References:

[Ag13] Ian Agol, The virtual Haken conjecture. With an appendix by Agol, Daniel Groves, and Jason Manning. Doc. Math. 18 (2013), 1045–1087.

[Ag14] Ian Agol, Virtual properties of 3-manifolds. Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. 1, 141–170, Kyung Moon Sa, Seoul, 2014.

[BH99] Martin R. Bridson, André Haefliger, Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. xxii+643 pp.

[HW08] Frédéric Haglund, Daniel T. Wise, Special cube complexes. Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620.

[Sag14] Michah Sageev, CAT(0) cube complexes and groups. Geometric group theory, 7–54, IAS/Park City Math. Ser., 21, Amer. Math. Soc., Providence, RI, 2014.

[Sch23] Petra Schwer, CAT(0) cube complexes—an introduction. Lecture Notes in Mathematics, 2324. Springer, Cham, [2023], ©2023.

[Wise12] Daniel T. Wise, From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry. CBMS Regional Conference Series in Mathematics, 117. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2012. xiv+141 pp.

[Wise14] Daniel T. Wise, The cubical route to understanding groups. Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, 1075–1099, Kyung Moon Sa, Seoul, 2014.

[Wise21] Daniel T. Wise, The structure of groups with a quasiconvex hierarchy. Annals of Mathematics Studies, 209. Princeton University Press, Princeton, NJ, [2021], ©2021. x+357 pp.

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Past Teaching at Fudan

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In fall 2024, I am teaching MATH130150 “Differential geometry II”/ SCMS817003 “Advanced topics in Geometric group theory”. The topic of the course will be “hyperbolic geometry”. We will discuss hyperbolic geometry in dimension two for around 8 weeks, then move to higher dimensions.

Time: 9:55 – 12:30 every Monday at H2203.

Office hours

TA: Josiah Oh， email: josiahoh@fudan.edu.cn;

Office hours: Wednesday 1pm at SCMS

Homework is due one week after it is assigned and please submit on elearning.

Schedule:

Week 1, Sep. 2, Introduction, hyperbolic plane.

Week 2, Sep. 9, geodeiscs/distance in hyperbolic plane.

Week 3, Sep. 14 circles/disks in hyperbolic plane.

Weel 4, Sep. 23, Horocycle, hypercycle.

Week 5, Sep. 30， boundary of hyperbolic plane, classification of isometries.

Week 6, Oct. 12, classification of isometries continued.

Week 7, Oct. 19, Isometry group of hyperbolic plane as mobious transformation.

Week 8, Oct. 26, Poincare disk model, cross rations.

Week 9, Nov. 4, trignomotry on hyperbolic plane; models for high dimensional hyperbolic spaces.

Week 10, Nov. 11, isometry groups of high dimensional hyperbolic spaces, conformal diffeomorphisms.

Week 11, Nov. 18, geodesics, hyperbolic subspaces.

Week 12, Nov. 25, boundary of high dimensional hyperbolic spaces, classfication of isometries.

Week 13, Dec. 2, flat, spherical and hyperbolic manfolds.

Week 14, Dec. 9, Mostow rigidity I.

Week 15, Dec. 16, Mostow rigidity II.

References:

[And05] Anderson, James W. Hyperbolic geometry. Second edition. Springer Undergraduate Mathematics Series. Springer-Verlag London, Ltd., London, 2005. xii+276 pp.

[BP92] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry. Universitext. Springer-Verlag, Berlin, 1992. xiv+330 pp.

[Bon09] Bonahon, Francis Low-dimensional geometry. From Euclidean surfaces to hyperbolic knots. Student Mathematical Library, 49. IAS/Park City Mathematical Subseries. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2009. xvi+384 pp.

[RR95] Ramsay, Arlan; Richtmyer, Robert D. Introduction to hyperbolic geometry. Universitext. Springer-Verlag, New York, 1995. xii+287 pp.

[Xu] Introduction to hyperbolic surfaces, webpage

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In spring 2024, I am teaching MATH820074 “Homological algebra and geometric applications” joint with Jianchao Wu. The topic of the course will be “l2 invariants of groups“. We use [Ka19] as the main resource of the lecture. You are also welcome to take a look at Lück’s book [Lü02] and Pansu’s notes [Pan96].

Time: 9:55 – 12:30 every Monday at H2210.

Office hours

TA: Josiah Oh， email: josiahoh@fudan.edu.cn;

Office hours: 14:00-15:00 Monday.

Homework is due one week after it is assigned and please submit on elearning.

Schedule:

Week 1, Feb. 26, introduction of l2-invariants and review on Hillbert modules.

References:

[Ka19] H. Kammeyer, Introduction to ℓ2-invariants. Lecture Notes in Mathematics, 2247. Springer, Cham, 2019. viii+181 pp.

[Lü02] W. Lück, L2-Invariants: Theory and Applications to Geometry and K-Theory, Results in Mathematics and Related Areas (3) (Springer-Verlag, Berlin, 2002). 

[Pan96]P . Pansu, Introduction to L2 Betti numbers. Riemannian geometry (Waterloo, ON, 1993), 53–86,
Fields Inst. Monogr., 4, Amer. Math. Soc., Providence, RI, 1996.

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In Fall 2023, I am teaching SCMS817004 “Homology of grups”. The topic will be bounded cohomology. The lecture will follow closely to the book [Fr17]. We are planing to cover the majority part of the book. It is going to be fun!

Time: Monday 9:55 – 12:30, HGX104

Homework is due one week after it is assigned and shall be submited on elearning.

TA: Josiah Oh， email: josiahoh@fudan.edu.cn;

Office hours: Friday 2pm-3pm, SCMS 339.

Final exam will be oral, it will take place in my office on Friday Jan. 5th. 2024.

Schedule:

Week 1, Sep. 4, review on cohomology of groups. Homework 1.

Week 2, Sep. 11, definition of bounded cohomology. I will be traveling this week. Instead, Josiah will give the lecture.

Week 3, Sep. 18, central extensions, quasimorphisms. Homework 2.

Week 4, Sep. 25, bounded cohomology in degree 2. Homework 3

Week 5, Oct. 9, Amenability. Homework 4

Week 6, Oct. 16, Charaterization of amenability by bounded cohomology. Homework 5

Week 7, Oct. 23, relative injectivity. Homework 6

Week 8, Oct. 30， normed resolution. Homework 7: prove Theorem 4.17 in detail.

Week 9, Nov. 6, ameanble spaces. Homework 8

Week 10 Nov. 13, bounded cohomology of topological spaces. Homework 9

Week 11 Nov. 20, relative bounded cohomology. Homework 10: provide more details for Remark 5.16.

Week 12 Nov.27, l1 homology and duality. Homework 11:Provide more details for the proof of Theorem 6.5 (4) implies (3).

Week 13 Dec. 4th, Gromov’s equivalence theorem. Homework 12.

Week 14 Dec. 11th, Simplicial volume I. Homework 13.

Week 15 Dec. 18th. Simplicial volume II.

References:

[Fr17] R. Frigerio, Bounded cohomology of discrete groups. Mathematical Surveys and Monographs, 227. American Mathematical Society, Providence, RI, 2017. xvi+193 pp.

[Gr82] M. Gromov, Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. No. 56 (1982), 5–99 (1983).

[Mo06] N. Monod, An invitation to bounded cohomology. International Congress of Mathematicians. Vol. II, 1183–1211, Eur. Math. Soc., Zürich, 2006.

[CFHM23] Bounded Cohomology and Simplicial Volume, London Mathematical Society Lecture Note Series 479, ed. C. Campagnolo, F. Fournier-Facio, N. Heuer and M. Moraschini, Cambrige University Press, 2023.

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Spring 2023

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In Spring 2023, I am teaching SCMS817001 “Geometric group theory” joint with Yulan Qing. The lecture will follow closely the book [Lo17].

Time: 9:55 – 12:30 every Tuesday at H5306.

TA: Josiah Oh， email: josiahoh@fudan.edu.cn

Office hours: Monday 2pm-3pm at SCMS 339.

Homework is due one week after it is assigned and please submit it on elearning.

Final project 1: How to prove a group is free?

Schedule:

Week 1, Feb. 21, Groups, generators and relations. Homework: [Lo17], 2.E.5 and 2.E.21.

Week 2, Feb. 28, New groups from old. Homework: [Lo17], 2.E.27 and 2.E.31.

Week 3, Mar. 7., Cayley graphs. Homework: [Lo17], 3.E.7 and 3.E.17.

Week 4, Mar. 14., Embed countable groups into 2-generated groups. Homework: [Lo17] 3.E.21, 3.E.25.

Week 5, Mar. 21., Group actions. Homework: [Lo17] 4.E.1 and 4.E.7.

Week 6, Mar. 28, free groups and actions on trees. Homework: [Lo17] 4.E.9 and 4.E.14.

Week 7, Detect free subgroups. Homework: [Lo17] 4.E.16 and 4.E.18.

Week 12, hyperbolic groups, Homework: [Lo17] 7.E.6 and 7.E.7.

The rest of the semester will be taught be Yulan.

References:

[BH99] Martin R. Bridson, André Haefliger, Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. xxii+643 pp.

[DK18] Cornelia Druţu, Michael Kapovich, Geometric group theory. With an appendix by Bogdan Nica. American Mathematical Society Colloquium Publications, 63. American Mathematical Society, Providence, RI, 2018. xx+819 pp.

[dlH00] Pierre de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. vi+310 pp.

[Lo17] Clara Löh, Geometric group theory. An introduction. Universitext. Springer, Cham, 2017. xi+389 pp.

[LS77] Roger C. Lyndon, Paul E. Schupp, Combinatorial group theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. Springer-Verlag, Berlin-New York, 1977. xiv+339 pp.

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spring 2022

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In spring 2022, I am teaching MATH820074 “Homological algebra and geometric applications”. The main topic of the course will be “Cohomology of groups“. We use [Br82] as the main resource of the lecture. [Bi81] and [Gr70] are also useful. Here is a quote from Robert Bieri on “How to Learn Mathematics” which could be equally useful.

Time: 9:55 – 12:30 every Monday at Tencent meeting, id: 818-5846-9116.

Office hours: By appointment.

Homework is due one week after it is assigned and please submit on elearning.

Final project: Please write down whatever you know about this problem.

Schedule:

Week 1, Feb. 21st, Review of some homological algebra I. Note 1, Homework 1

Week 2, Feb. 28th, Review of some homological algebra II. Note 2, Homework 2: [Br82], p.12, Exe 1.

Week 3, Mar. 7th, Free resolution. Note 3, Homework 3: [Br82], p.17, Exe 3.

Week 4, Mar. 14th, Projectives, co-invariants and Definition of H_ast(G), Note 4, Homework 4

Week 5, Mar. 21st, Homology of a group with constant coefficients， Note 5, Homework 5: [Br82], p.36, Exe 1.

Week 6, Mar. 28th, Functoriality, homology of amalgamated free product, Note 6, Homework 6: [Br82], p.52, Exe 3.

Week 7, Apr. 2nd, (Co)Homology of groups with Coefficients I. Note 7 Homework 7: [Br82] p.56, Exe 1 and 2 (for exe 1, you need to first prove that a Z-module is flat if and only if it is torsionfree).

Week 8, Apr. 11th, (Co)Homology of groups with Coefficients II. Note 8, Homework 8: [Br82], p.67, Exe.

Week 9, Apr. 18th, (Co)Homology of groups with Coefficients III, Note 9, Homework 9.

Week 10, (Co)Homology of groups with Coefficients IV. Note 10, Homework 10, [Br82], p.83 Exe 1.

Week 11, May 3rd, self reading.

Work 12, May 9th, low dimensional cohomology and group extension. Note 11, Homework 11.

Week 13, May 14th, self reading.

Week 14, May 16th, Spectral sequences I Note 12.

Week 15, May 23th, Spectral sequences II. Note 13.

References:

[Bi81] Robert Bieri, Homological dimension of discrete groups. Second edition. Queen Mary College Mathematics Notes. Queen Mary College, Department of Pure Mathematics, London, 1981. iv+198 pp.

[Br82] Kenneth S. Brown, Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin, 1982. x+306 pp.

[Gr70] Karl W. Gruenberg, Cohomological topics in group theory. Lecture Notes in Mathematics, Vol. 143 Springer-Verlag, Berlin-New York 1970 xiv+275 pp.