Student low-dimensional topology seminar, Spring 2025

Meeting time: Fridays 1-2pm @ 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
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
2 May 2025

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).

Past semesters

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.