Columbia Geometric Topology Seminar

Spring 2024


Organizers: Francesco LinSiddhi Krishna, Ross Akhmechet

The GT seminar typically meets on Fridays at 2:00pm Eastern time in Room 307, Mathematics Department, Columbia University. 

Other area seminars. Our e-mail list. Archive of previous semesters

Spring 2023

Date Time (Eastern) Speaker Title

January 26

3:30pm Eastern (note unusual time!)

Bojun Zhao

Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy

February 2

3:30pm Eastern (note unusual time!) Sebastian Hurtado-Salazar

Length functions on Lie groups and lattices

February 9

2pm Eastern

Mike Willis

Spectral Jones-Wenzl projectors and differences between Khovanov homology and stable homotopy

February 16

3:30pm Eastern (note unusual time!)

Alex Zupan

The square knot bounds infinitely many non-isotopic ribbon disks

February 23

2pm Eastern

Sally Collins

Homology cobordism & smooth knot concordance

March 1

2pm Eastern

Eugen Rogozinnikov

Parametrizing spaces of positive representations

March 7 (Thursday)

3pm Eastern in Room 407 (note unusual day, time, and room!)

Francesco Costantino

Stated skein modules of 3-manifolds and TQFTs

March 8

2pm Eastern

Onkar Gujral

Supverised learning methods applied to low dimensional topology

March 15

2pm Eastern

no seminar

Spring Break

March 22

2pm Eastern

Emmanuel Wagner

From representation theory to topology: there and back again

March 29

2pm Eastern

no seminar

Simons Annual Meeting

April 5

2pm Eastern

Emily Stark

Graphically discrete groups and rigidity

April 12

2pm Eastern

Ian Zemke

L-space links and formality

April 19

2pm Eastern Henry Segerman

Avoiding inessential edges

April 26

2pm Eastern Patrick Orson

Unknotting nonorientable surfaces topologically

April 30 (Tuesday)

11am Eastern Sakura Park Bruno Martelli

Negative curvature and fibrations

May 21 (Tuesday)

11am Room 507 Pedram Hekmati

Real structures in Seiberg-Witten Floer theory



Bojun Zhao

Date:  January 26

Title: Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy

Abstract: Let f be a pseudo-Anosov homeomorphism on a compact orientable surface with nonempty boundary, such that f has a co-orientable stable foliation and reverses the co-orientation on it. Let M denote the mapping torus of f. In this talk, I will discuss some constructions of co-orientable taut foliations in those Dehn fillings of M with filling multislopes constrained by a bound from the degeneracy loci on the boundary components of M. In certain cases where M has connected boundary and is Floer simple, we can construct co-orientable taut foliations in all non-L-space Dehn fillings of M.



Sebastian Hurtado-Salazar

Date:  February 2

TitleLength functions on Lie groups and lattices

Abstract: We will discuss the notion of a length function on a group, focusing on lattices in Lie groups, such as SL_n(Z), and discuss how techniques from dynamics can help to understand some important questions about these groups.



Mike Willis

Date:  February 9

Title: Spectral Jones-Wenzl projectors and differences between Khovanov homology and stable homotopy

Abstract: Categorified Jones-Wenzl projectors P_n are widely studied, with endomorphisms related to the Khovanov homology of torus links. We will discuss the lifts of such projectors to the category of spectra, focusing on certain properties of P_n that fail to lift. These are some of the first structural differences found between Khovanov homology and Khovanov stable homotopy, and include the surprising negative answer to a question of Lawson-Lipshitz-Sarkar asking whether topological Hochschild homology for tangle spectra can be used to define a spectral invariant for links in S1xS2. This work is joint with Matt Stoffregen.



Alex Zupan

Date:  February 16

TitleThe square knot bounds infinitely many non-isotopic ribbon disks

Abstract: A knot K in S^3 is (smoothly) slice if K is the boundary of a properly embedded disk D in B^4, and K is ribbon if this disk can be realized without any local maxima with respect to the radial Morse function on B^4.  In dimension three, a knot K with nice topology – that is, a fibered knot – bounds a unique fiber surface up to isotopy.  Thus, it is natural to wonder whether this sort of simplicity could extend to the set of ribbon disks for K, arguably the simplest class of surfaces bounded by a knot in B^4.  Surprisingly, we demonstrate that the square knot, one of the two non-trivial ribbon knots with the lowest crossing number, bounds infinitely many distinct ribbon disks up to isotopy.  This is joint work with Jeffrey Meier.



Sally Collins

Date:  February 23

TitleHomology cobordism & smooth knot concordance

Abstract: The 0-surgeries of two knots K1 and K2 are homology cobordant rel meridians if there exists an integer homology cobordism X between them such that the two positive knot meridians are in the same homology class of X. It is a natural question to ask: if two knots have 0-surgeries related in this sense, must they be smoothly concordant? We give a pair of rationally slice knots as counterexample, and along the way expand upon an involutive knot Floer homology technique for obstructing torsion in the smooth concordance group first introduced by Hom, Kang, Park, and Stoffregen. No previous knowledge of Heegaard Floer theory will be assumed.



Eugen Rogozinnikov

Date:  March 1

Title:Parametrizing spaces of positive representations

Abstract: Higher Techmüller theory deals with spaces of representations of the fundamental group of a surface into a reductive Lie group $G$, modulo the conjugation, especially with the connected components (called higher Teichmüller spaces) that consist entirely of injective representations with discrete image.

In the last two decades in works of Fock, Goncharov, Burger, Iozzi, Guichard, Wienhard, and others researchers, it was discovered that the most interesting higher Teichmüller spaces are emerging from the groups $G$ having a positive structure, i.e. certain submonoid $G_+$ with no invertible non-unit elements. Some of these submonoids have been known since 1930’s as totally positive matrices and then generalized by Lustzig for split real Lie groups. However it left out a large class of non-split reductive Lie groups such as $SO(p,q)$. O. Guichard and A. Wienhard filled this gap in 2018 by introducing the Theta-positivity, which also includes submonoids $SO(p,q)_+$ sitting in a unipotent group of $SO(p,q)$ and $Sp(2n,R)_+$ which is the set of upper uni-triangular block 2x2-matrices with a symmetric positive definite matrix in the upper right corner.
In my talk, I introduce the Theta-positivity for Lie groups and explain how the spaces of positive representations of the fundamental group of a punctured surface into a Lie group with a positive structure can be parametrized, and how we can describe the topology of these spaces using this parametrization. This is a joint work with O. Guichard and A. Wienhard.


Francesco Costantino

Date: March 7

Title:Stated skein modules of 3-manifolds and TQFTs

Abstract:  After reviewing the definition of stated skein modules for surfaces and 3 manifolds, I will detail how this recent notion allows to relate topological constructions (related to cut and paste techniques) to algebraic ones (braided tensor products of algebra objects in braided categories for instance). I will explain how the stated skein algebra of some special surfaces provides a topological description for some notable algebras (e.g. the quantised functions ring $O_q(\mathfrak{sl_2})$ or its ``transmutation’’ BSL_2(q)). Then I will describe how stated skein moduli of 3-manifolds fit into a TQFT framework albeit a non completely standard one. If time permits I will also discuss some unexpected non injectivity results in dimension 3. (Joint work with Thang Le)



Onkar Gujral

Date: March 8

TitleSupverised learning methods applied to low dimensional topology

Abstract:  In this talk we will survey recent work applying neural networks to knot theory. We will start with the necessary background on neural networks and then describe various knot invariants and properties that people have been able to predict with them. These include quasipositivity, the slice genus, the tau-invariant and unknottedness. We will also describe some work in which authors have used invariants like the Khovanov polynomial and Jones polynomial to predict invariants like the slice genus and s-invariant. 


Emmanuel Wagner

Date: March 22

Title: From representation theory to topology: there and back again

Abstract: In this talk, I will explain how foam evaluation allows to see an sl(2) action on the equivariant gl(n) Khovanov-Rozansky link homologies. We will also see how to extend the previous action functorially. Joint work with You Qi, Louis-Hadrien Robert and Joshua Sussan.

Emily Stark

Date: April 5

Title: Graphically discrete groups and rigidity

Abstract: Rigidity problems in geometric group theory frequently have the following form: if two finitely generated groups share a geometric structure, do they share algebraic structure? The work of Papasoglu--Whyte demonstrates that infinite-ended groups are quasi-isometrically flexible; our results show that if you assume a common geometric model, then there is often rigidity. To do this, we introduce the notion of a graphically discrete group, which imposes a discreteness criterion on the group's lattice envelopes. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds; free groups are non-examples. We will discuss new examples and rigidity phenomena for free products of graphically discrete groups.This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

Ian Zemke

Date: April 12

Title: L-space links and formality

Abstract: A well-known result of Ozsvath and Szabo describes the knot Floer complex of an L-space knot in terms of its Alexander polynomial. The family of L-space knots includes all algebraic knots. The situation of L-space links is comparably less understood. We will describe a conjectural description of the link Floer complex, as well as proof of this conjecture for plumbed L-space links (this family includes all algebraic links) as well as 2-component L-space links. The case of plumbed L-space links is joint with M. Borodzik and B. Liu, and the case of 2-component L-space links is joint work in progress with D. Chen and H. Zhou. The case of 2-component L-space links has applications towards computations of certain satellite operators in Heegaard Floer theory.



Henry Segerman

Date: April 19

Title: Avoiding inessential edges

Abstract: Results of Matveev, Piergallini, and Amendola show that any two triangulations of a three-manifold with the same number of vertices are related to each other by a sequence of local combinatorial moves (namely, 2-3 and 3-2 moves). For some applications however, we need our triangulations to have certain properties, for example that all edges are essential. (An edge is inessential if both ends are incident to a single vertex, into which the edge can be homotoped.) We show that if the universal cover of the manifold has infinitely many boundary components, then the set of essential ideal triangulations is connected under 2-3, 3-2, 0-2, and 2-0 moves. Our results have applications in veering triangulations and in quantum invariants such as the 1-loop invariant. This is joint work with Tejas Kalelkar and Saul Schleimer.



Patrick Orson

Date: April 26

TitleUnknotting nonorientable surfaces topologically

Abstract: Knot invariants are typically used to give a negative answer to the question of when two embeddings are ambiently isotopic, and rarely to give a positive answer. An exception is the celebrated result of Freedman and Quinn that if the complement of a 2-sphere embedded in the 4-sphere has cyclic fundamental group then that 2-sphere is topologically unknotted. We recently showed that the analogous result for closed nonorientable surfaces in the 4-sphere is also true (in most cases). This talk will describe this recent work and highlight some key ideas from the proof. This is joint work with Anthony Conway and Mark Powell.



Bruno Martelli

Date: April 30

Title: Negative curvature and fibrations

Abstract: One of the most intriguing aspects in low-dimensional topology is the existence, discovered by Jorgensen in the late 70s, of hyperbolic 3-manifolds that fiber over the circle. In this talk we will review some aspects of this beautiful theory, with the notable contributions of Thurston, and more recently of Agol and Wise. Then we will show that this phenomenon is not restricted to dimension 3, by exhibiting some examples in dimension 5 (in collaboration with Italiano and Migliorini). We will also discuss the dimensions n > 3 in general.


Pedram Hekmati

Date: May 21

Title: Real structures in Seiberg-Witten Floer theory

Abstract: Involutions play a special role in Seiberg-Witten Floer theory as they can couple to the intrinsic charge conjugation symmetry and yield different flavours of equivariant Floer cohomology. In this talk, I will outline how this leads to three distinct series of Floer theoretic invariants for rational homology spheres and discuss some applications to knot concordance invariants and non-smoothable involutions on 4-manifolds with boundary. This is joint work with David Baraglia.


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