# Columbia Geometry and Topology Seminar

Spring 2022

Organizers: Kyle Hayden, Siddhi Krishna
The GT seminar typically meets on Fridays at 2:00pm Eastern time via the Zoom link above. (The password is `math').

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

## Fall 2021

Date Time (Eastern) Speaker Title

January 28

2pm Eastern

Oliver Singh

Pseudo-isotopies and diffeomorphisms of 4-manifolds

February 4

2pm Eastern

Markov chains on groups and quasi-isometries

February 11

2pm Eastern

Slicing knots in definite 4-manifolds

February 18

2pm Eastern

March 4

2pm Eastern

Counterexamples in 4-manifold topology

March 11

2pm Eastern

Taut foliations of 3-manifolds with Heegaard genus two

March 25

2pm Eastern

Oyku Yurttas

Curves, braids and crosscap transpositions

April 1

2pm Eastern

Equivariant knots and knot Floer homology

April 1

4:30pm Eastern (Special Bonus Seminar!)

Mathematics Hall, Room 312 (in-person)

Special Lagrangians from the perspective of Morse theory
April 5

12pm Eastern (Special Bonus Seminar!)

Uris Hall, Room 331 (in-person)

Marithania Silvero Khovanov homology of 4-braids in polynomial time
April 5

1pm Eastern (Special Bonus Seminar!)

Uris Hall, Room 331 (in-person)

Józef Henryk Przytycki Khovanov homology of 4-braids in polynomial time: Independence complexes of circle graphs

April 8

2pm Eastern

Hermitian Lie groups as symplectic groups over noncommutative algebras

April 15

2pm Eastern "Representations are sheaves" for Legendrian 2-weaves
April 22 2pm Eastern Feride Ceren Köse On the amphichirality of symmetric unions
April 22

4:45pm Eastern (Special Bonus Seminar!)

Room TBD (in-person)

Marco Golla Symplectic fillings of divisorial contact structures
April 29 2pm Eastern Alexandra Edletzberger Quasi-Isometries for certain Right-Angled Coxeter groups

## Abstracts

Oliver Singh

Title: Pseudo-isotopies and diffeomorphisms of 4-manifolds

Abstract: I will talk about pseudo-isotopy, a notion important for understanding self-diffeomorphisms of manifolds up to isotopy. Pseudo-isotopies of manifolds in dimensions 5 and up were understood in the 70s by work of Cerf for simply connected manifolds, and by Hatcher and Wagoner in the non-simply connected case, using invariants from algebraic K-theory. Quinn later proved Cerf’s result topologically in dimension 4, leading to a classification of self-homeomorphisms of simply connected 4-manifolds up to isotopy. I will talk about my work on what Hatcher and Wagoner’s K-theoretic invariants can say about pseudo-isotopies of non-simply connected 4-manifolds, and how they can be used to construct diffeomorphisms of certain 4-manifolds which are pseudo-isotopic but not isotopic to the identity.

Alessandro Sisto

Title: Markov chains on groups and quasi-isometries

Abstract: Random walks on groups provide a model for a "generic" element of a group, and they're very interesting and very well-studied. In geometric group theory it is natural to consider quasi-isometric groups, but unfortunately random walks are not compatible with quasi-isometries, in the sense that they cannot be "pushed forward" via quasi-isometries in any meaningful sense. To resolve this, in this talk I will propose the study of more general Markov processes on groups that are indeed "quasi-isometry compatible", and present the first results about them. In particular, I will discuss a central limit theorem for random walks  whose proof exploits this perspective of pushing forward Markov chains.

Based on joint work with Antoine Goldsborough.

Allison N. Miller

Title: Slicing knots in definite 4-manifolds

Abstract: A knot is called "slice" if it bounds an embedded disc in the 4-ball. A natural extension of this idea is to think about knots that bound embedded discs in other simple 4-manifolds. We'll talk about some constructions and obstructions in the specific case of connected sums of the complex projective plane. Tools include Donaldson's theorem on the intersection form of smooth definite 4-manifolds and Freedman's result that knots with trivial Alexander polynomial are topologically slice.

Diego Santoro

Abstract: The L-space conjecture predicts strong connections among properties relating the Heegaard Floer homology, foliations and the fundamental group of an irreducible rational homology 3-sphere. I will introduce this conjecture and present some results concerning the structure of the set of the L-space surgery slopes for links with unknotted components and linking number zero. For what concerns foliations, I will discuss the existence of taut foliations on the Dehn fillings of some k-holed torus bundles over the circle. These results will be used to study the L-space conjecture for the rational homology spheres arising as Dehn surgery on the Whitehead link.

Arunima Ray

Title: Counterexamples in 4-manifold topology

Abstract: I will discuss the relationships among a variety of equivalence relations on 4-manifolds, such as diffeomorphism, homeomorphism, h-cobordism, and homotopy equivalence, with the goal of organising a zoo of counterexamples and discovering unanswered questions. There will be a flowchart and a table. The talk is based on an upcoming, partly survey paper with Daniel Kasprowski and Mark Powell.

Tao Li

Title: Taut foliations of 3-manifolds with Heegaard genus two

Abstract: Let M be a closed, orientable, and irreducible 3-manifold with Heegaard genus two. We prove that if the fundamental group of M is left-orderable then M admits a co-orientable taut foliation.

Oyku Yurttas

Title: Curves, braids and crosscap transpositions

Abstract: Multicurves (systems of mutually disjoint essential simple closed curves) have played a central role in the study of mapping class groups of surfaces since the work of Dehn. Such systems are usually described combinatorially using techniques such as the Dehn-Thurston coordinate system or train tracks. In the case where the surface is an n-punctured disk D_n multicurves are beautifully described by their Dynnikov coordinates,  collection of 2n-4 linear combinations of intersection numbers with the 3n-5 edges of a near-triangulation of D_n.  The Mapping Class Group of D_n is canonically isomorphic to Artin's braid group modulo its center. The action of Artin's braid generators on the set of Dynnikov coordinates is given by so-called update rules. In this talk we survey Dynnikov coordinates and update rules, and then provide natural analogues of these tools for an n--punctured non--orientable surface N_{k,n} of genus k. Namely, we introduce  generalized Dynnikov coordinates for multicurves in N_{k,n}, and then describe the action of crosscap transpositions in terms of these coordinates.

Irving Dai

Title: Equivariant knots and knot Floer homology

Abstract: We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly non-equivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden. This is joint work with Abhishek Mallick and Matthew Stoffregen.

Emily Windes

Title: Special Lagrangians from the perspective of Morse theory

Abstract: In this talk, we consider a Lagrange multipliers problem where the constraint is a section of a bundle E->M. We relate the Morse homology of a function restricted to the zero set of the section to the Morse homology of the associated Lagrange function on the total space E^*.  Then we discuss a similar, infinite-dimensional Lagrange multipliers problem appearing in Donaldson and Segal’s paper "Gauge Theory in Higher Dimensions II". The long term goal is to apply Floer theory to a functional whose critical points are a generalization of three-dimensional, special Lagrangian submanifolds.

Marithania Silvero and Josef Przytycki (Double Header)

Title: Khovanov homology of 4-braids in polynomial time

Abstract: Khovanov homology is a link invariant which generalizes Jones polynomial. In general, computing Jones polynomial (so also Khovanov homology) is NP-hard. However, if we consider a closed braid of a fixed number of strands, it is well-known that all classical quantum invariants (in particular Jones polynomial) can be computed in polynomial time. We conjecture that the complexity of computing Khovanov homology of a closed braid of fixed number of strands is polynomial with respect to the number of crossings.

In this talk we show some advances on the conjecture, showing that the result holds when considering extreme Khovanov homology of closed braids of at most 4 strands. As a consequence, we get an obstruction for a link to have braid index 4 in terms of it extreme Khovanov homology.

Eugen Rogozinnikov

Title: Hermitian Lie groups as symplectic groups over noncommutative algebras

Abstract: In my talk, I introduce the symplectic group $\Sp_2(A,\sigma)$ over a noncommutative algebra $A$ with an anti-involution $\sigma$ and show that many classical Lie groups can be seen in this way. Of particular interest will be the classical Hermitian Lie groups of tube type and their complexifications. For these groups, I construct different models of the symmetric space in terms of the group $\Sp_2(A,\sigma)$. We obtain generalizations of several models of the hyperbolic plane and the three-dimensional hyperbolic space. This is a joint work with D. Alessandrini, A. Berenstein, V. Retakh and A. Wienhard.

Kevin Sackel

Title: "Representations are sheaves" for Legendrian 2-weaves

Abstract: Given a trivalent plane graph embedded in the 2-sphere, there is a rich algebraic structure which arises from contact geometric considerations. The graph corresponds to a certain associated Legendrian surface via a construction of Treumann and Zaslow. In prior work, Casals and Murphy computed the Legendrian contact dg-algebra for (the Legendrian satellite of) this Legendrian 2-weave over commutative coefficients, the group ring of the first homology group. We extend their computation to the non-commutative setting, working over the group ring of the fundamental group. For this family of Legendrian surfaces, we further verify the conjecture that "representations are sheaves," i.e. that the moduli space of representations of this fully non-commutative dg-algebra are in bijective correspondence with a certain moduli space of constructible sheaves. Aside from the contact-geometric story, we will emphasize the extraordinarily combinatorial and computational nature of the algebraic structure one obtains.

Feride Ceren Köse

Abstract: Symmetric unions are an interesting class of knots. Although they have not been studied much for their own sake, they frequently appear in the literature. One such instance regards the question of whether there is a nontrivial knot with trivial Jones polynomial. In my talk, I will describe a class of symmetric unions, constructed by Tanaka, such that if any are amphichiral, they would have trivial Jones polynomial. Then I will show how such a knot not only answers the above question but also gives rise to a counterexample to the Cosmetic Surgery Conjecture. However, I will prove that such a knot is in fact trivial and hence cannot be used to answer any of these questions. If time permits, I will discuss how the arguments that go into this proof can be generalized to study amphichiral symmetric unions.

Marco Golla

Title: Symplectic fillings of divisorial contact structures

Abstract: If a (possibly singular) complex curve in a Kähler surface has positive self-intersection, then it has a standard symplectically concave neighbourhood, and therefore an associated divisorial contact structure. Motivated by the study of singular symplectic curves in the complex projective plane, we will discuss the existence and classification of fillings of some of these contact structures. This is based on joint work with Laura Starkston.

Alexandra Edletzberger

Title: Quasi-Isometries for certain Right-Angled Coxeter groups

Abstract: In the hunt for a solution to the Quasi-Isometry Problem of right-angled Coxeter groups (RACGs), we use a quasi-isometry invariant that is obtained by a certain splitting of the groups. In this talk, we introduce this splitting, the so-called Graph of Cylinders, and give its construction for a large family of (in particular non-hyperbolic) RACGs, that can be read off the presentation graph. This means, that we can distinguish certain RACGs up to quasi-isometry just by comparing the information their presentation graphs provide. Often, even more is possible: Under one additional assumption, by this comparison RACGs can also be identified as quasi-isometric.