Other area seminars. Our e-mail list. Archive of previous semesters
Columbia Geometric Topology SeminarSpring 2025 |
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Organizers: Ross Akhmechet, Deeparaj Bhat, Siddhi Krishna, Francesco Lin
The GT seminar typically meets on Fridays at 2:00pm Eastern time in Room 407, Mathematics Department, Columbia University.
Other area seminars. Our e-mail list. Archive of previous semesters
| Date | Time (Eastern) | Speaker | Title |
|---|---|---|---|
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January 24 |
2pm |
no seminar |
first week of classes |
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January 31 |
4pm in Room 407 (note unusual time!!) | Jonathan Zung (MIT) |
Expansion and torsion homology of 3-manifolds |
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February 7 |
2pm |
Juan Munoz-Echaniz (SCGP) |
Monodromy of singularities and Seiberg—Witten theory |
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February 14 |
2pm | Luya Wang (IAS) |
Algebraic torsion and symplectic linear plumbings |
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February 21 |
2pm | Seraphina Lee (UChicago) |
Lefschetz fibrations with infinitely many sections |
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February 28 |
2pm | no seminar |
|
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March 7 |
2pm | David Rose (UNC Chapel Hill) | Towards a higher TQFT from Khovanov homology |
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March 14 |
2pm | Boyu Zhang (Maryland) |
Configuration spaces of points and mapping class groups of 4-manifolds |
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March 21 |
2pm | no seminar | happy spring break! |
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March 28 |
2pm |
no seminar |
Simons annual meeting |
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April 4 |
2pm |
Mike Miller Eismeier (Vermont) |
Naturality in HF^oo |
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April 11 |
2pm |
no seminar |
|
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April 18 |
2pm |
Bena Tshishiku (Brown) | Finite group actions on 3-manifolds and Nielsen realization |
| April 25 |
2pm |
Thomas Massoni (MIT) |
Taut foliations through a contact lens
|
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May 2 |
2pm |
Ian Biringer (Boston College) |
Ranks of covers of hyperbolic 3-manifold groups |
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May 9 |
Double header! 2:30PM in Room 507 | Marco Marengon (Renyi) |
Splitting links by integer homology spheres |
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May 9 |
Double header! 3:30PM in Room 507 | Laura Wakelin (KCL) |
Every slope is non-characterising for some knot |
Name: Jonathan Zung
Title: Expansion and torsion homology of 3-manifolds
Abstract: nally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders. On the other hand, I will show that such higher expanders must be rather topologically complicated: they must have lots of torsion homology.
Name: Juan Munoz-Echaniz
Title: Monodromy of singularities and Seiberg—Witten theory
Abstract: The monodromy of a complex isolated hypersurface singularity captures geometric and topological information about how the nearby smooth fibers degenerate into the singularity. The homological monodromy—the action on the homology of the Milnor fiber—has been extensively studied ever since pioneering work of Brieskorn and Milnor. However, the monodromy diffeomorphism itself—acting on the Milnor fiber as a mapping class—is comparatively less understood. In this talk I will discuss the following result: the monodromy diffeomorphism of a weighted-homogeneous isolated hypersurface singularity of complex dimension 2 has infinite order in the smooth mapping class group of the Milnor fiber (fixing the boundary) provided the singularity is not ADE. (In turn, the monodromy of an ADE singularity has finite order in the smooth mapping class group, by a classical result of Brieskorn). The proof involves studying the Seiberg—Witten equation along the fibers of the Milnor fibration, by a combination of techniques from Floer homology, symplectic and contact geometry. This is based on joint work with Hokuto Konno, Jianfeng Lin and Anubhav Mukherjee.
Name: Luya Wang
Title: Algebraic torsion and symplectic linear plumbings
Abstract: Given a contact three-manifold, one can ask what types of symplectic four-manifolds it can bound, if any. This is related to the question whether certain complex algebraic curves can be realized with prescribed singularities. A key tool in this study is the embedded contact homology algebraic torsion, which encodes information about the Reeb dynamics of the contact manifold and the behavior of pseudoholomorphic curves in its symplectization. In this talk, I will describe this quantity and its geometric implications. We will do some calculations for the concave boundaries of symplectic linear plumbings of disk bundles and see how they agree with predictions coming from toric geometry. This is based on joint work with Aleksandra Marinkovic, Jo Nelson, Ana Rechtman, Laura Starkston, and Shira Tanny.
Name: Seraphina Lee
Title: Lefschetz fibrations with infinitely many sections
Abstract: A Lefschetz fibration M^4 \to S^2 is a generalization of a surface bundle which also allows finitely many nodal singular fibers. The Arakelov--Parshin rigidity theorem implies that nontrivial, holomorphic Lefschetz fibrations of genus g \geq 2 admit only finitely many holomorphic sections. In this talk, we will show that no such finiteness result holds for smooth or symplectic sections by giving examples of genus-g (g \geq 2) Lefschetz fibrations with infinitely many homologically distinct sections. We will also give examples with infinitely many orbits of sections under the action of fiberwise diffeomorphisms of M that preserves the set of fibers of M \to S^2. This is joint work with Carlos A. Serván.
Name: David Rose
Title: Towards a higher TQFT from Khovanov homology
Abstract: An original aim of the categorification program is the construction of a 4d TQFT that gives a "higher" analogue of the 3d TQFTs associated with root of unity evaluations of the Jones polynomial (Turaev-Viro and Witten-Reshetikhin-Turaev theories). While this remains an open problem, I will discuss the first steps in a program to build such a 4d TQFT using structures underlying Khovanov homology. Specifically, I will present recent work which defines dg category-valued invariants of surfaces. Surprisingly, although we do not work at a root of unity, a twisted Hom-pairing on our invariants categorifies a canonical pairing of spin networks in the state spaces of the TV/WRT theories. Time permitting, I'll also discuss work in progress that gives a 3-manifold invariant which categorifies the "stable" Turaev-Viro invariant. (This is all joint work with Matt Hogancamp and Paul Wedrich.)
Name: Boyu Zhang
Title: Configuration spaces of points and mapping class groups of 4-manifolds
Abstract: In this talk, I will present several results on mapping class groups of 4-manifolds proved using configuration spaces of points. In particular, we show that if Sigma is a closed surface with a positive genus, then the mapping class group of S^2 times Sigma is not finitely generated. We also show that Gabai’s lightbulb theorem does not hold for closed surfaces without the pi_1 negligibility condition, which answered a question of Gabai. This talk is based on joint works with Jianfeng Lin and Yi Xie.
Name: Mike Miller Eismeier
Title: Naturality in HF^oo
Abstract: The Heegaard Floer group HF^oo(Y, s_0) is known to depend only on the triple cup product of Y and c_1(s_0). In particular, it appears to carry purely cohomological information. A first approximation is given by the 'cup homology' HC(Y), the E^4 page of a spectral sequence to HF^oo. It is conjectured that this spectral sequence collapses. Millions of computer examples in the range 9 <= b_1(Y) <= 14 give that in all cases, HF^oo(Y) is isomorphic to HC(Y) as an abelian group --- a stronger result than we should expect just from knowing that the spectral sequence collapses. One might expect there is a good reason these are isomorphic. I will prove that there is not by showing these functors are not naturally isomorphic over F_2. Along the way, we introduce a 'Lefschetz decomposition' of exterior algebras over Z. This work is joint with Analisa Faulkner Valiente.
Name: Bena Tshishiku
Title: Finite group actions on 3-manifolds and Nielsen realization
Abstract: For a manifold M, the Nielsen realization problem asks when a group of symmetries of the fundamental group π_1(M) is realized by a group of homeomorphisms of M. This problem was posed in different forms by Nielsen (1930s) and Thurston (1970s) and has connections to geometry, dynamics, and rigidity. This talk will focus on recent work and open problems related to Nielsen realization for 3-manifolds.
Name: Thomas Massoni
Title: Taut foliations through a contact lens
Abstract: In the late '90s, Eliashberg and Thurston established a remarkable connection between foliations and contact structures in dimension three: any co-oriented, aspherical foliation on a closed, oriented 3-manifold can be approximated by both positive and negative contact structures. Additionally, if the foliation is taut then its contact approximations are tight. In this talk, I will present a converse result on constructing taut foliations from suitable pairs of contact structures. While taut foliations are rather rigid objects, this viewpoint reveals some degree of flexibility and offers a new perspective on the L-space conjecture. A key ingredient is a generalization of a result of Burago and Ivanov on the construction of 'branching' foliations tangent to continuous plane fields, which might be of independent interest.
Name: Ian Biringer
Title: Ranks of covers of hyperbolic 3-manifold groups
Abstract: Let M be a closed hyperbolic 3-manifold and let M <- M_1 <- M_2 <- … be a tower of finite covers such that the fundamental group of each M_i can be generated by some fixed number of elements, independent of i. For example, if M fibers over the circle, a tower of finite covers of the circle gives a tower of covers of M as above. We show this is essentially the only way to construct such towers.
Name: Marco Marengon
Title: Splitting links by integer homology spheres
Abstract: We show that there exist two embedded 2-spheres in the 4-sphere which are separated by an integer homology 3-sphere Y (i.e., a manifold with the same integer homology of the 3-sphere), but not by the standard 3-sphere.
Name: Laura Wakelin
Title: Every slope is non-characterising for some knot
Abstract: For any fixed rational number p/q, Dehn surgery gives a map from the set of knots in the 3-sphere to the set of closed orientable 3-manifolds. In 1978, Gordon conjectured that these maps are never injective. I will explain how this can be seen for |p|=1 via JSJ decompositions and for |q|=1 via RBG links, before going on to discuss recent joint work with Kyle Hayden and Lisa Piccirillo in which we prove the general case of this conjecture