Student low-dimensional topology seminar, Fall 2025

Meeting time: Wednesdays 3-4pm @ 732 Evans
Organizers: Qiuyu Ren and Audrey Rosevear (email us if you'd like to be added to our mailing list)
Sign up here if you want to give a talk.

Date	Speaker	Title	References
3 Sep 2025	Qiuyu Ren	(Rational) linking numbers and linking forms
10 Sep 2025	Qiuyu Ren	Linking form and Hantzschte theorem
17 Sep 2025	Audrey Rosevear	Introduction to Anosov Flows and the Finiteness Conjecture	[Zun25] [FH19]
1 Oct 2025	Benjamin Daniels	Unasked questions in annular Khovanov homology	[BN05] [GLW18] [GN13] [Mar23] [DZ25] [HMR24] [Hub17] [Weh10]
8 Oct 2025	Gabriel Beiner	The Atiyah--Floer conjecture for admissible bundles	[Ati88] [DFL21]
15 Oct 2025	Qiuyu Ren	Ribbon concordance and simplicial volume	[Gor81] [CG83] [Koc06] [Gro82]
29 Oct 2025	Audrey Rosevear	Finite depth foliations, periodic orbits, and endperiodic maps
5 Nov 2025	Lina Liu	Orbifolds + Morse homology	[CH13]
19 Nov 2025	Qiuyu Ren	Fraction Dehn twist coefficients	[FLP79, Exposé 5-9, 11] [HKM07] [KR13]
3 Dec 2025	Audrey Rosevear	How to prove the Thurston-Nielsen classification

References

[Ati88] M. Atiyah, New invariants of 3- and 4-dimensional manifolds, Proceedings of Symposia in Pure Mathematics 48 (1988): 285-299.
[BN05] D. Bar-Natan, Khovanov's homology for tangles and cobordisms, Geometry & Topology 9.3 (2005): 1443-1499.
[CG83] A. J. Casson and C. McA Gordon, A loop theorem for duality spaces and fibred ribbon knots, Inventiones mathematicae 74.1 (1983): 119-137.
[CH13] C.-H. Cho and H. Hong, Orbifold Morse-Smale-Witten complex, arXiv preprint arXiv:1103.5528v3.
[DFL21] A. Daemi, K. Fukaya, M. Lipyanskiy, Lagrangians, SO(3)-instantons and the Atiyah--Floer Conjecture, arXiv preprint arXiv:2109.07032.
[DZ25] B. Daniels, and M. Zhang, On the Categorified Wrapping Number Conjecture, arXiv preprint arXiv:2501.02623 (2025).
[FH19] T. Fisher, B. Hasselblatt, Hyperbolic flows, Zurich Lectures in Advanced Mathematics (2019).
[FLP79] A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces, volume 66 of Astérisque, Société Mathématique de France, Paris (1979).
[Gor81] C. McA Gordon, Ribbon concordance of knots in the 3-sphere, Mathematische Annalen 257.2 (1981): 157-170.
[GLW18] E. Grigsby, A. Licata, and S. Wehrli, Annular Khovanov homology and knotted Schur-Weyl representations, Compositio Mathematica 154.3 (2018): 459-502.
[GN13] E. Grigsby, and Y. Ni, Sutured Khovanov homology distinguishes braids from other tangles, arXiv preprint arXiv:1305.2183 (2013).
[Gro82] M. Gromov, Volume and bounded cohomology, Publications Mathématiques de l'IHÉS 56 (1982): 5-99.
[HMR24] K. Hendricks, C.-Y. Mak, and S. Raghunath, Symplectic annular Khovanov homology and fixed point localizations, arXiv preprint arXiv:2408.06453 (2024).
[HKM07] K. Honda, W. Kazez, and G. Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Inventiones mathematicae 169.2 (2007): 427-449.
[Hub17] D. Hubbard, On sutured Khovanov homology and axis-preserving mutations, Journal of Knot Theory and its Ramifications 26.04 (2017): 1750017.
[KR13] W. Kazez, and R. Roberts, Fractional Dehn twists in knot theory and contact topology, Algebraic & Geometric Topology 13.6 (2013): 3603-3637.
[Koc06] D. H. Kochloukova, On a conjecture of E. Rapaport Strasser about knot-like groups and its pro-p version, Journal of Pure and Applied Algebra 204.3 (2006): 536-554.
[Mar23] G. Martin. Annular Khovanov homology and meridional disks, Journal of Knot Theory and Its Ramifications 32.02 (2023): 2250088.
[Weh10] S. Wehrli, Mutation invariance of Khovanov homology over F2, Quantum Topol 1.2 (2010): 111-128.
[Zun25] J. Zung, Pseudo-Anosov representatives of stable Hamiltonian structures, Journal of Fixed Point Theory and Applications 27.4 (2025): 1-29.

Past semesters

Spring 2025

Organizers: Qiuyu Ren and Audrey Rosevear.

Date	Speaker	Title	References
24 Jan 2025	Qiuyu Ren	Killing Calegari's homotopy 4-spheres	[Cal09] [CK24] [AC65]
31 Jan 2025	Qiuyu Ren	Continuation
7 Feb 2025	Audrey Rosevear	Left orderable groups	[Cal07] Chapter 2, [BRW05], [Zha22], [Li24]
14 Feb 2025	Gabriel Beiner	Structures on manifolds and the triangulation conjecture	[FQ14] Chapter 8, [GS80], [Hir63], [Man13]
21 Feb 2025	Nancy Eagles	Dehn surgery and non-separating two-spheres	[HL22] [Gab87] [Ni13]
7 Mar 2025	Jackson Koos Davis	Akbulut's cork and an explicit exotic R^4	[Akb91] [GS23]
14 Mar 2025	Qiuyu Ren	Convex surface boundary is balanced
21 Mar 2025	Robert Bowden	Skein algebras and their applications	[Lic97] [Jon87] [Wit89]
11 Apr 2025	Gabriel Beiner	The Bauer--Furuta invariant and 10/8	[BF04] [Fur01] [HLS+22]
18 Apr 2025	Qiuyu Ren	Sutured manifolds and the guts of link complements	[Gab87] [AZ22]
25 Apr 2025	Audrey Rosevear	Holonomy and the structure of foliations	[Cal07] Chapter 2,4, [CC00] Chapter 2,4,8

References

[AC65] J. J. Andrews and M. L. Curtis, Free groups and handlebodies. Proceedings of the American Mathematical Society 16.2 (1965): 192-195.
[Akb91] S. Akbulut, A fake compact contractible 4-manifold. Journal of Differential Geometry 33.2 (1991): 335-356.
[AZ22] I. Agol and Y. Zhang, Guts in sutured decompositions and the Thurston norm. arXiv preprint arXiv:2203.12095 (2022).
[BF04] S. Bauer and M. Furuta, A stable cohomotopy refinement of Seiber--Witten invariants: I. Invent. Math. 155, 1--19 (2004).
[BRW05] S. Boyer, D. Rolfsen, and B. Wiest, Orderable 3-manifold groups. Annales de l'institut Fourier, Vol. 55, No. 1, 2005.
[Cal07] D. Calegari, Foliations and the geometry of 3-manifolds. Clarendon press, 2007.
[Cal09] D. Calegari, 4-spheres from fibered knots. https://lamington.wordpress.com/2009/11/09/4-spheres-from-fibered-knots/, 2009.
[CC00] A. Candel and L, Conlon, Foliations I. Vol. 23. American Mathematical Soc., 2000.
[CK24] J. C. Cha and M. H. Kim, Calegari's homotopy 4-spheres from fibered knots are standard. arXiv preprint arXiv:2411.10051 (2024).
[Gab87] D. Gabai, Foliations and the topology of 3-manifolds. III. J. Differential Geom., 26(3):479–536, 1987.
[FQ14] M. Freedman and F. Quinn, The Topology of Four-Manifolds. Princeton Mathematical Series No. 39, Princeton University Press, 2014.
[Fur01] M. Furuta, Monopole Equation and the 11/8-conjecture. Math. Res. Lett. 8, 279--291 (2001).
[GS80] D. Galewski and R. J. Stern, Classification of Simplicial Triangulations of Topological Manifolds. Ann. Math. 111, no. 1 (1980), 1-34.
[GS23] R. Gompf, and A. Stipsicz, 4-manifolds and Kirby calculus. Vol. 20. American Mathematical Society, 2023.
[Hir63] M. W. Hirsch, Obstruction theories for smoothing manifolds and maps. Bull. Amer. Math. Soc. 69 (1963), 352-356.
[HL22] J. Hom and T. Lidman, Dehn Surgery and Nonseparating Two-Spheres. Open Book Series, vol. 5, no. 1, Oct. 2022, pp. 145-153.
[HLS+22] M. Hopkins, J. Lin, X. D. Shi, and Z. Xu, Intersection forms of spin 4-manifolds and the Pin(2)-equivariant Mahowald invariant. Comm. Amer. Math. Soc. 2, 22--132 (2022).
[Jon87] V. F. Jones, Hecke algebra representations of braid groups and link polynomials. New Developments in the Theory of Knots, 1987, 20-73.
[Li24] T. Li, Taut foliations of 3-manifolds with Heegaard genus 2. Duke Mathematical Journal 173.8 (2024): 1427-1475.
[Lik97] W. R. Lickorish, An introduction to knot theory. Vol. 175, Springer Science & Business Media, 1997.
[Man13] C. Manolescu, The Conley index, gauge theory, and triangulations. J. Fixed Point Theory Appl. 13 (2013), 431-457.
[Ni13] Y. Ni, Nonseparating spheres and twisted Heegaard Floer homology. Algebr. Geom. Topol., 13(2):1143-1159, 2013.
[Wit89] E. Witten, Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121.3 (1989): 351-399.
[Zha22] B. Zhao, Left orderability and taut foliations with one-sided branching. arXiv preprint arXiv:2209.04752 (2022).

Fall 2024

Organizers: Qiuyu Ren and Audrey Rosevear

Date	Speaker	Title	References
9 Sep 2024	Audrey Rosevear	Product-covered flows	[Cal07] Chapter 6.8
16 Sep 2024	Audrey Rosevear	Applications of product-covered flows
23 Sep 2024	Qiuyu Ren	Prime and JSJ decompositions of 3-manifolds	[Mar16] Chapter 9-12
30 Sep 2024	Qiuyu Ren	JSJ decompositions of knot complements	[Bud06] Section 2, first half of Section 4
14 Oct 2024	Qiuyu Ren	Spine or no spine? Dimension 3 vs 4 (5) vs beyond	[FKL24]
21 Oct 2024	Nancy Eagles	Knot surgeries in contact topology	[Gei08] Chapter 2,4,6, [Hon00], [DiG04]
28 Oct 2024	Gabriel Beiner	Minimal genus problems in 4-manifolds	[Law99], [GoS99] Chapter 2, [OzS00]
4 Nov 2024	Audrey Rosevear	When the adjunction inequality isn't sharp	[Kro99] [Neg16]
18 Nov 2024	Nancy Eagles	Exposition of the Arnold chord conjecture in 3-dimensions	[Arn86] [HuT11]
25 Nov 2024	Jianru Duan	The multivariable Alexander polynomial and Thurston norm	[McM02]
9 Dec 2024	Galen Liang	The Bennequin inequality	[Etn05]

References

[Arn86] V. I. Arnold, The first steps of symplectic topology. Russian Math. Surveys 41 (1986), no. 6, 1–21. MR0890489.
[Bud06] R. Budney, JSJ-decompositions of knot and link complements in the 3-sphere. L'enseignement Mathématique (2) 52 (2006), 319-359.
[Cal07] D. Calegari, Foliations and the geometry of 3-manifolds. Clarendon press, 2007.
[DiG04] F. Ding, H. Geiges, A Legendrian surgery presentation of contact 3-manifolds. Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 3. Cambridge University Press, 2004.
[Etn05] J. Etnyre, Legendrian and transverse knots. Handbook of knot theory, Elsevier Science, 2005, 105-185.
[FKL24] M. Freedman, V. Krushkal, T. Lidman, Spineless 5-manifolds and the deformation conjecture. ArXiv preprint arXiv:2401.03498v3.
[Gei08] H. Geiges, An introduction to contact topology. Cambridge Studies in Advanced Mathematics, 109, Cambridge Univ. Press, Cambridge, 2008. MR2397738.
[GoS99] R. Gompf, and A. Stipsicz, 4-manifolds and Kirby calculus. Vol. 20. American Mathematical Society, Providence, 1999.
[Hon00] K. Honda, On the classification of tight contact structures: I. Geometry and Topology 4.309 (2000): 368.
[HuT11] M. Hutchings, C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions, I. Math. Res. Lett. 18 (2011) 295-313. MR2784673.
[Kro99] P. Kronheimer, Minimal genus in S^1\times M^3. Inventiones mathematicae 135 (1999): 45-61.
[Law97] T. Lawson, The minimal genus problem. Expositiones Mathematicae 15 (1997): 385-432.
[Mar16] B. Martelli, An introduction to geometric topology. ArXiv preprint arXiv:1610.02592 (2016).
[McM02] C. T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology. Annales Scientifiques de l'École Normale Supérieure, (4) 35 (2002), no. 2, 153-171.
[Neg16] M. Nagel, Minimal genus in circle bundles over 3-manifolds. Journal of Topology 9.3 (2016): 747-773.
[OzS00] P. Ozsváth, Z. Szabó, The symplectic Thom conjecture. Annals of Mathematics (2000): 93-124.

Spring 2024 (past name: student 3-manifold seminar)

Organizers: Ethan Dlugie and Qiuyu Ren

Date	Speaker	Title
24 Jan 2024		Organizational meeting
31 Jan 2024	Ethan Dlugie	The Burau representation and shapes of polyhedra
7 Feb 2024	Qiuyu Ren	Generalized Poincaré conjectures and Milnor's exotic 7-spheres
14 Feb 2024	Audrey Rosevear	The Giroux correspondence
21 Feb 2024	Ethan Dlugie	Isogenies, ping-pong, and Banach-Tarski
28 Feb 2024	Gabriel Beiner	The Casson invariant and instanton Floer homology for homology 3-spheres
13 Mar 2024	Qiuyu Ren	Constructions of homotopy 4-spheres
3 Apr 2024	Galen Liang	Defining ECH with examples and applications I
10 Apr 2024	Matthew Palani Lideros	A Survey of quantum invariants
17 Apr 2024	Galen Liang	Defining ECH with examples and applications II
24 Apr 2024	Audrey Rosevear	Geodesic laminations and the Thurston-Nielsen classification theorem

Fall 2023

Organizers: Ethan Dlugie and Qiuyu Ren

This semester we have quantum topology as a central theme.

Date	Speaker	Title
31 Aug 2023		Organizational meeting
7 Sep 2023	Ethan Dlugie	Knots and braids, in preparation for quantum invariants
14 Sep 2023	Qiuyu Ren	Jones polynomial as a trace
21 Sep 2023	Nancy Eagles	Unknotting knots and the Jones polynomial
28 Sep 2023	John Nolan	The Jones polynomial for a torus knot
5 Oct 2023	John Lentfer	Hecke algebra representations and Ocneanu’s trace
12 Oct 2023	Elliot Kienzle	Gauge theory for knots: three perspectives on the linking number
19 Oct 2023	Elliot Kienzle	Topological quantum field theories and the Jones Polynomial
26 Oct 2023	Larsen Linov	How to make a rubber band bracelet
2 Nov 2023	Gabriel Beiner	Chern-Simons theory and the Wess-Zumino-Witten model
9 Nov 2023	Matthew Palani Lideros	Quantum representations of mapping class groups
16 Nov 2023	Qiuyu Ren	Witten-Reshetikhin-Turaev invariants via skein theory, colored Jones polynomial, and related conjectures
30 Nov 2023	Qiuyu Ren	Witten-Reshetikhin-Turaev invariants via skein theory, colored Jones polynomial, and related conjectures II

References: (* indicates expository papers or lecture notes)

Jones polynomial:
V. Jones, Jones polynomial for dummies*
V. Jones, A polynomial invariant for knots via von Neumann algebras
V. Jones, Hecke algebra representations of braid groups and link polynomials
S. Sawin, Links, Quantum Groups, and TQFT's*
(Witten-)Reshetikhin-Turaev invariants:
N. Yu. Reshetikhin, V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups
R. C. Kirby, P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C)
E. Witten, Quantum field theory and the Jones polynomial
Colored Jones polynomial:
N. Yu. Reshetikhin, V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups
S. Garoufalidis, T. T. Lê, The colored Jones function is q-holonomic
Volume conjecture:
H. Murakami, An Introduction to the Volume Conjecture*
A blog post* by D. Moskovich
R. M. Kashaev, A link invariant from quantum dilogarithm
H. Murakami and J. Murakami, The colored Jones polynomials and the simplicial volume of a knot
Q. Chen, T. Yang, Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants
(Kauffman bracket) skein modules of 3-manifolds:
J. H. Przytycki, Fundamentals of Kauffman bracket skein modules*
J. H. Przytycki, Skein modules of 3-manifolds
Skein TQFT:
C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological quantum field theories derived from the Kauffman bracket
C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Three-manifold invariants derived from the Kauffman bracket
General/Miscellancy:
L. H. Kauffman, S. L. Lins, Temperley–Lieb recoupling theory and invariants of 3-manifolds*
T. Ohtsuki, Quantum invariants. A study of knots, 3-manifolds, and their sets*
L. Kauffman, Quantum topology and quantum computing*
E. Witten, Quantum field theory and the Jones polynomial
M. F. Atiyah, The geometry and physics of knots*
S. Sawin, Links, Quantum Groups, and TQFT's*
B. Bakalov, A. Kirillov jun., Lectures on tensor categories and modular functors*

Ancient semesters

Fall 2021 - Spring 2023 talk titles can be found in our current signup sheet on the top of the page. Spring 2019 - Spring 2020 talks can be found here.