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

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
April 30 (Tuesday)	TBA	Bruno Martelli

Abstracts

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

Title: Length 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

Title: The 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

Title: Homology 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

Title: Supverised 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.

Columbia Geometric Topology Seminar

Spring 2023

Abstracts

Other relevant information.

Previous semesters:

Other area seminars.

Our e-mail list: You can subscribe here for announcements for this seminar, as well as occasional related seminars and events.