Schedule
Schedule, Fall 2020:
September 15th: 12:00 ET Hannah Turner (UT Austin): Cyclic symmetries and L-spaces
12:30 ET Maria Trnkova (UC Davis):Geodesics of hyperbolic 3-manifolds
A recording of these talks is available here.
September 22nd: 12:00 ET Boyu Zhang (Princeton): Equivariant Cerf theory and perturbative SU(n) Casson invariants
12:30 ET Brandon Bavier (MSU): Alternating Knots on Surfaces
A recording of these talks is available here.
September 29th: 12:00 ET Jose Perea (MSU): DREiMac: Dimensionality Reduction with Eilenberg-MacLane Coordinates
12:30 ET Onkar Singh Gujral: The Khovanov Functor and Cobordisms between Split Links
A recording of these talks is available here.
October 6th: 12:00 ET Marco Moraschini (Regensburg) : Stable integral simplicial volume (slides)
12:30 ET Kuldeep Saha (IISERB Bhopal): Application of open book and Lefschetz fibration to optimal embedding of 3 and 4 manifolds
A recording of these talks is available here.
October 13th: 12:00 ET Andy Putman (Notre Dame): The topology of the mapping class group of a surface
12:30 ET Sanjay Kumar (MSU): Fundamental shadow links realized as links in S^3
A recording of these talks is available here.
October 20th: 12:00 ET Fraser Binns (Boston College): Link Detection Results for Knot Floer Homology
12:30 ET Ken Baker (Miami): Exceptional surgeries and surfaces
A recording of these talks is available here.
Abstracts, Fall 2020:
Fraser Binns: A basic question in knot theory is "how can we distinguish links?". In this talk I will address this question, discussing a number of detection results for knot Floer homology, an invariant defined using symplectic topology. This is work in progress, joint with Gage Martin.
Ken Baker: An exceptional surgery is a Dehn surgery on a hyperbolic knot between non-hyperbolic 3-manifolds. As the non-hyperbolicity of 3-manifolds is typically heralded by surfaces of non-negative Euler characteristic, one strategy for constructing exceptional surgeries is through finding ways of producing such surfaces by Dehn surgery. To that end, we'll consider doing surface-framed surgeries on knots in surfaces of nearly non-negative Euler characteristic. This perspective allows us to not only recover but also unify, generalize, and explain many exceptional surgeries.
Boyu Zhang: In 1985, Casson introduced an invariant for integer homology 3-spheres by counting the SU(2) representations of the fundamental group. Boden-Herald generalized the Casson invariant to SU(3) using gauge-theoretic techniques. In this talk, I will review the history of the construction of generalized Casson invariants, and present a construction of SU(n) Casson invariant for all n. This is joint work with Shaoyun Bai.
Brandon Bavier: A goal of knot theory is to relate diagramatic properties of the knot to geometric properties of the knot complement. When we are working with hyperbolic alternating knots, we can find two-sided bounds of both the volume and the cusp volume based on the twist number. We can generalize the lower bound of the cusp volume to a wider class of knots, called weakly generealized alternating knots. In this talk, we will define both the cusp volume and weakly generalized alternating knots, and use this particular proof generalization to show how other such relations might generalize.
Jose Perea: One salient question in Topological Data Analysis is how to use signatures like persistent (co)homology to infer spaces parametrizing a given data set. This is relevant in nonlinear dimensionality reduction, as nontrivial data topology can prevent accurate descriptions with low-dimensional Euclidean coordinates. I will describe in this talk how one can use persistent cohomology together with Eilenberg-MacLane spaces for dimensionality reduction purposes.
Onkar Singh Gujral: A cobordism between links induces a map between the Khovanov homologies of the links. The Khovanov map induced by a split cobordism is determined entirely by the individual components of the cobordism. What about non-split cobordisms between split links? For such cobordisms, we will show that the induced Khovanov map does not "see" linking information between distinct components. In particular, closed components can be "pulled off". Our main tool will come from lifting Batson and Seed's link invariant to cobordisms and showing that it cannot "see" linking information. This result is used to show that a strongly homotopy ribbon concordance induces an injective map on Khovanov Homology, generalising a result of Levine and Zemke. Joint work with Adam Levine.
Marco Moraschini: Simplicial volume is a homotopy invariant introduced by Gromov in his seminal paper "Volume and bounded cohomology", which measures the complexity of a compact manifold in terms of its (real) singular chains. Despite its definition is purely algebraic, remarkably it encodes some information about the geometry of manifolds. For instance, the celebrated Proportionality Theorem by Thurston and Gromov shows that in the case of hyperbolic manifolds the simplicial volume is proportional to the Riemannian volume. Unfortunately, computing the simplicial volume for arbitrary manifolds is rather hard in general, because you have to deal with singular chains with *real* coefficients.
The situation appears easier as soon as we consider integral coefficients instead of real ones. In this new setting, it is convenient to work with a gradient version of the integral simplicial volume, called stable integral simplicial volume: We take the infimum of the normalised integral simplicial volumes of all finite coverings. Then, it is natural to ask when the simplicial volume of a manifold agrees with the stable integral simplicial volume. In this talk, we will show that in dimension 2 and 3 the ordinary simplicial volume of aspherical manifolds agrees with the stable integral simplicial volume. Finally, we will discuss how this problem relates to the following question by Gromov: Does the Euler characteristic vanish for an oriented closed connected aspherical manifold with zero simplicial volume? This is a joint work with Daniel Fauser, Clara Loeh and José Pedro Quintanilha.
Sanjay Kumar: In this talk, I will discuss two conjectures which relate quantum topology and hyperbolic geometry. Chen and Yang conjectured that the asymptotics of the Turaev-Viro invariants determine the hyperbolic volume of the $3$-manifold, and Andersen, Masbaum, and Ueno (AMU) conjectured for a surface that the Nielsen-Thurston classification of the mapping class predicts the order of the quantum representations. For a manifold $M$ constructed as the mapping tori of an element $f$ in the mapping class group, Detcherry and Kalfagianni showed that if $M$ satisfies the Chen-Yang volume conjecture then $f$ satisfies the AMU conjecture. Using techniques from Turaev's shadow theory, I will construct infinite families of links in $S^3$ with complement homeomorphic to the complement of links in connected sums of $S^2 \times S^1$ in a family shown to satisfy the Chen-Yang volume conjecture known as the fundamental shadow links. These link complements in $S^3$ can be realized as the mapping tori for explicit elements $f$ in the mapping class group of a surface implying that $f$ satisfies the AMU conjecture.
Kuldeep Saha: The embedding of closed orientable $n$-manifolds in $\mathbb{R}^{2n-1}$ is a well known fact. We will see how elementary techniques using open book decompositions and Lefschetz fibrations can be used to give a more visual proof of this fact for $n = 3,4$. On the way we also mention some open problems regarding embeddings in the category of open books.
Hannah Turner: Whether a manifold with "simple" Heegaard Floer homology (an L-space) also has "simple" topology or geometry is an interesting open problem. A more modest goal is to classify L-spaces among families 3-manifolds. I'll survey how to exploit cyclic symmetries of manifolds to investigate the complexity of Heegaard Floer homology. Some of this is joint work with Ahmad Issa.
Maria Trnkova: A computer program "SnapPea" and its descendant “SnapPy” compute many invariants of a hyperbolic 3-manifold M. Some of their results can be rigorous but some not. In this talk we will discuss computation of geodesics length and will mention a number of applications when it is crucial to know the precise length spectrum up to some cut off.
C.Hodgson and J.Weeks introduced a length spectrum algorithm implemented in SnapPea. The algorithm uses a tiling of the universal cover by translations of a Dirichlet domain of M by elements of a fundamental group. In theory the algorithm is rigorous but in practice its output does not guaranty the correct result. It requires to use the exact data for the Dirichlet domain which is available only in some special cases. We show that under some assumptions on M an approximate Dirichlet domain can work equally well as the exact Dirichlet domain. Our result explains the empirical fact that the program "SnapPea" works surprisingly well despite it does not use exact data.
Schedule, Spring 2020:
April 14th: 12:00 ET Yu Pan (MIT): Augmentations and exact Lagrangian surfaces.
12:30 ET Nick Salter (Columbia): Framed mapping class groups, or the topology of families of flat surfaces
A recording of these talks is available here
April 21st: 12:00 ET Allison Miller (Rice): Satellite Operators on Knot Concordance. (notes)
12:30 ET Irving Dai (MIT): Cobordism questions and Heegaard Floer homology
A recording of these talks is available here
April 28th: 12:00 ET Maggie Miller (Princeton): Codimension-2 knots in 4-manifolds
12:30 ET Marc Kegel (Humboldt): Contact surgery numbers
A recording of these talks is available here. We are sorry to admit that a particularly goonish Trends organizer forgot to start the recording promptly, so the excellent first four minutes of Maggie's talk are lost to time.
May 5th: 12:00 ET Kyle Hayden (Columbia): A softer side of complex curves
12:30 ET Dror Bar-Natan (Toronto): Over then under tangles (talk site)
A recording of these talks is available here
May 12th: 12:00 ET Alexander Rasmussen (Yale): Analogs of the curve graph for infinite type surfaces
12:30 ET Will Rushworth (McMaster): Ascent concordance
A recording of these talks is available here
May 19th: 12:00 ET Oguz Savk (Bogazici): Brieskorn spheres and homology cobordism (slides)
12:30 ET Puttipong Pongtanapaisan (Iowa): Meridional rank and bridge number of knotted surfaces
A recording of Oguz's talk is available here and of Puttipong's talk is available here
May 26th: 12:00 ET Artem Kotelskiy (Indiana University):
Heegaard Floer and Khovanov theories through the lens of immersed curve invariants I (slides)
12:30 ET Claudius Zibrowius (UBC/Regensburg): (slides)
Heegaard Floer and Khovanov theories through the lens of immersed curve invariants II
A recording of these talks is available here
June 16th: 11:50 ET Katherine Raoux (MSU): Minimal Genus Problems and Knot Floer homology
12:25 ET Priyam Patel (Utah): Infinite-type surfaces
A recording of these talks is available here.
June 23rd: 11:50 ET Gage Martin (Boston College): Khovanov homology and link detection
12:25 ET Sam Ballas (Florida State): Gluing equations for projective structures on 3-manifolds
A recording of these talks is available here.
June 30th: 11:50 ET Scott Taylor (Colby College): Making nonadditive invariants additive
12:25 ET Rylee Lyman (Rutgers): Outer Automorphisms of Free Groups: Topology of Graphs
A recording of these talks is available here
July 7th: 11:50 ET Anthony Conway (MPIM): Algebraic knots in concordance
12:25 ET Sami Douba (McGill): Embeddability of 3-manifold groups in compact Lie groups
A recording of these talks is available here
July 14th: 11:50 ET Rhea Palak Bakshi (George Washington):
Skein Modules and Framing Changes of Links in 3-Manifolds.
12:25 ET Josh Greene (Boston College): The rectangular peg problem
July 21st 11:50 ET Dongsoo Lee: On the kernel of the zero-surgery homomorphism from knot concordance
12:25 ET Donghao Wang: Monopole floer homology for 3-manifolds with torus boundary
July 22-August 31 Summer break!
Titles and abstracts Spring 2020:
Rhea Palak Bakshi: We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$-manifold is when the manifold admits a properly embedded non-separating $S^2$. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough's work on the mapping class groups of $3$-manifolds. We also express our results in the language of skein modules. In particular, we relate our results to the framing skein module and the Kauffman bracket skein module.
Josh Greene: I will discuss the context and solution of the rectangular peg problem: for every smooth Jordan curve and rectangle in the Euclidean plane, one can place four points on the curve at the vertices of a rectangle similar to the one given. The solution involves symplectic geometry in a surprising way. ‘Joint work with Andrew Lobb.
Dongsoo Lee: Kawauchi defined a group structure on the set of homology S1×S2’s under an equivalence relation called H^{~}-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero- surgery. It is natural to ask whether the zero-surgery homomorphism is injective. We show that this question has a negative answer in the smooth category. Indeed, using knot concordance invariants derived from knot Floer homology we show that the kernel of the zero-surgery homomorphism contains an infinite rank subgroup.
DongHao Wang: The monopole Floer homology of closed 3-manifolds was defined by Kronheimer-Mrowka and has greatly influenced the study of 3-manifold topology. In this talk, we will generalize their construction and define the monopole Floer homology for any 3-manifolds with torus boundary. Its Euler charateristic recovers the Milnor-Turaev torsion invariant by a classical theorem of Meng-Taubes. We will also explain a (3+1) TQFT property and a Thurston-norm detection result
Rylee Lyman: Outer Automorphisms of Free Groups: Topology of Graphs
In this talk I will say hello from dimension one. The study of outer automorphisms of free groups has benefited from a deep analogy with mapping class groups of surfaces. Indeed, there is a sense in which the outer automorphism group of a free group is like the mapping class group of a graph. I will explore this analogy and introduce some of the common methods for studying outer automorphisms of free groups. Every effort will be made to keep the talk focused on topological methods.
Scott Taylor: Making nonadditive invariants additive
Several classical knot and 3-manifold invariants can be defined using Morse functions and/or handle structures. Some of these invariants are additive under connected sum and some are not. I’ll quickly recall the definitions and basic properties of some of these invariants and then will discuss a recent framework that (at least partially) unifies them. This framework gives rise to new families of knot, spatial graph, and 3-manifold invariants that are closely related to the classical invariants but which are additive under connected sum. Much of this is joint work with Maggy Tomova.
Nick Salter: Framed mapping class groups, or the topology of families of flat surfaces
Abstract: Families of surfaces are everywhere in mathematics, not just in topology, but in algebraic geometry, complex analysis, dynamics, and even number theory. The topology of a family of surfaces is governed by a “monodromy representation” that is valued in the mapping class group. I’m interested in (a) developing tools within the mapping class group to better understand monodromy and (b) applying these tools to problems involving families of surfaces, inside and out of topology proper. The thrust of my work over the past few years has been to understand the monodromy of families of surfaces endowed with certain tangential structures (e.g. a framing, a holomorphic 1-form with prescribed zeroes, an “r-spin structure”, etc.) and to apply this to study the topology of the spaces supporting such families (strata of abelian differentials, linear systems in certain algebraic surfaces, versal deformation spaces of plane curve singularities). This represents joint work with Aaron Calderon and Pablo Portilla Cuadrado.
Yu Pan: Augmentations and exact Lagrangian surfaces.
Augmentations are some algebraic invariants of Legendrian that are tightly related to both embedded and immersed exact Lagrangian fillings. We will talk about various relations between embedded and immersed exact Lagrangian surfaces using tools related to augmentations.
Allison Miller: Satellite Operators on Knot Concordance.
Abstract: We'll start by talking a little about why 4-manifold topology is interesting and unusual, and why knot theory offers powerful tools to help us better understand it. Next, I'll sketch the very basics of knot concordance, focusing on geometrically nice operators coming from the classical satellite construction. I'll go on to state some results and open questions in this area, and then close by discussing recent work (joint with P. Feller and J. Pinzon-Caicedo) on how various 4-dimensional measures of knot complexity change under satelliting.
Irving Dai: Cobordism questions and Heegaard Floer homology
Since its inception, Floer theory has provided a powerful tool for studying 3- and 4-manifolds. Motivated by connections with smooth 4-manifold topology, we give a brief survey of some questions and results regarding the homology cobordism group and discuss some recent applications to the theory of corks. We give a brief overview of how Heegaard Floer homology can be used to approach these topics.
Maggie Miller: Codimension-2 knots in 4-manifolds
Just as classical knots (circles in 3-manifolds) are useful in the study of 3-dimensional topology, understanding knotted surfaces is useful in the study of 4-dimensional topology. Any 4-manifold arises from sums of basic 4-manifolds via surgery on tori (Iwase); certain surgeries on 2-spheres and tori can produce exotic 4-manifolds, and the complexity of an h-cobordism of 4-manifolds can be described by counting intersections of 2-spheres (Morgan--Szabo). However, many theorems about classical knots fail or remain unknown in dimension four.
I will discuss some of these interesting phenomena and big open questions about surfaces in dimension-4, and describe some of my previous/current work in this area (especially joint work with Mark Hughes and Seungwon Kim proving an analogue of the Reidemeister theorem for surfaces in arbitrary 4-manifolds).
Marc Kegel: Contact surgery numbers
The surgery number of a 3-manifold M is the minimal number of components in a surgery description of M. Computing surgery numbers is in general a difficult task and is only done in a few cases.
In this talk, I want to report on the same question for contact manifolds. In particular, we will study a method to compute contact surgery numbers for contact structures on some Brieskorn spheres. This talk is based on joint work with Sinem Onaran.
Kyle Hayden: A softer side of complex curves
There is a rich, symbiotic relationship between knot theory and the study of complex curves, spanning from Wirtinger's work on knot groups and algebraic curves in the 1890's, to Gong's recent calculations of the Kronheimer-Mrowka concordance invariant. I'll offer a topological perspective on complex curves using the important class of "quasipositive braids", which naturally arise as cross-sections of complex curves. Then I’ll describe recent work that uses this softer perspective to construct pairs of holomorphic disks in the 4-ball that are “smoothly exotic”, i.e. isotopic through ambient homeomorphisms but not through diffeomorphisms. I'll close with some open questions about knots and complex curves.
Dror Bar-Natan: Over then under tangles
Brilliant wrong ideas should not be buried and forgotten. Instead, they should be mined for the gold that lies underneath the layer of wrong. In this paper we explain how "over then under tangles" lead to an easy classification of knots, and under the surface, also to some valid mathematics: an easy classification of braids and virtual braids, an understanding of the Drinfel'd double procedure in quantum algebra, and more.
Alexander Rasmussen: Analogs of the curve graph for infinite type surfaces
The mapping class group of a finite type surface admits a non-proper action on a graph known as the curve graph. Hyperbolicity of this graph and deeper study of this action have been responsible for most of the advances on the geometry of the mapping class group as well as many advances on its algebra. In this talk we will discuss how to adapt this machinery to study mapping class groups of infinite type surfaces, as well as some of the challenges and open problems that remain.
Will Rushworth: Ascent concordance
In the field of knot concordance one studies knots in the 3-sphere up to a 4-dimensional equivalence relation: two knots are concordant if they appear as the boundary of an annulus embedded in a very simple 4-manifold. This innocuous-looking definition is a wellspring of many difficult questions in low dimensional topology.
In this talk we shall consider a problem of concordance in 3-manifolds other than the 3-sphere: thickened surfaces. Given two links in (possibly different) thickened surfaces, we'll define what it means for them to be concordant. We'll see that the 4-manifold appearing in such concordances is a priori more complicated than that of the 3-sphere case. But do we ever actually need this complexity? We'll exhibit pairs of concordant links that are not concordant in simple spaces, necessitating a complex one. We'll give an idea of how an augmented version of Khovanov homology can be used to prove this, and tie it in to the Slice-Ribbon conjecture.
Oguz Savk : Brieskorn spheres and homology cobordism
The initial aim of the talk is to present the description of an effective method for computing d-invariants of infinite families of Brieskorn spheres. Associating our work with the recent results in involutive Floer homology, these spheres will be new 3-manifolds generating infinite rank summands in the integral homology cobordism group. The next intention of the talk is to bring your attention to new infinite families of Brieskorn spheres bounding rational homology balls but not integral homology balls. They will be useful in understanding the homomorphism between homology cobordism groups.
Puttipong Pongtanapaisan : Meridional rank and bridge number of knotted surfaces
An effective way to study a complicated knot in the three-sphere is to decompose it into two collections of unknotted arcs and analyze how the two simple pieces are glued back together. This decomposition, called a bridge splitting, gives rise to a measure of complexity for the knot called the bridge number, which is the smallest number of unknotted arcs in each collection. The bridge number is known to behave nicely under connected sum and is conjectured to be equal to the minimal number of meridians needed to generate the knot group.
As a four-dimensional analog of a bridge splitting, Meier and Zupan showed that any knotted surface in the four-sphere can be decomposed into three collections of unknotted disks. In this talk, I will discuss how to adapt techniques commonly used to study bridge splittings to investigate the bridge number of knotted surfaces.
Artem Kotelskiy: This talk will be an introduction to Khovanov homology and Heegaard Floer homology of 3-manifolds and knots inside them. The focus will be on examples, applications and interconnections between those theories. In particular, I will discuss the construction of the Lagrangian Floer homology of two curves on a surface, which, being a central idea in our research, will play a major role in Claudius's talk.
Claudius Zibrowius: Over the past four years, a number of immersed curve invariants have emerged that have proven to be effective in answering open problems in low-dimensional topology. Picking up the thread of Artem's talk, I will give a broad overview of these invariants and their applications.
Katherine Raoux: Minimal genus problems and Knot Floer homology
“Given a 2-dimensional homology class, what is the minimal genus surface representing this class?”
“Given a knot, what is the minimal genus surface with boundary the knot?”
These are broad questions with rich histories in the context of both 3 and 4-manifold topology. Because these questions are often difficult to answer precisely, we tend to settle for lower bounds. In recent years, tools from Heegaard Floer theory, particularly the Ozsváth-Szabó tau-invariant, have been used to produce such lower bounds. I will discuss the tau-invariant, a recent generalization due to Hedden and myself, and corresponding genus bounds.
Priyam Patel: Infinite-type surfaces
A surface is of finite type if it is homeomorphic to a compact surface with at most finitely many points removed. Otherwise, the surface is of infinite-type. There has been a recent surge of interest in infinite-type surfaces and their mapping class groups (the group of homeomorphisms of the surface up to a natural equivalence called isotopy), which arise naturally in a variety of contexts. In this talk, I will give a crash course on infinite-type surfaces and highlight some of the main avenues of research in this area. I will end by discussing some recent joint work with Tarik Aougab and Nick Vlamis regarding the isometry groups of infinite-type surfaces.
Gage Martin: Khovanov homology and link detection
Khovanov homology is a combinatorially defined link homology theory. Due to the combinatorial definition, many topological applications of Khovanov homology arise via connections to Floer theories. A specific topological application is the question of which links Khovanov homology detects. In this talk, we will give an overview of Khovanov homology and link detection, mention some of the connections to Floer theoretic data used in detection results, and give ideas of the proof that Khovanov homology detects the torus link T(2,6).
Sam Ballas: Gluing equations for projective structures on 3-manifolds
One of Thurston’s many amazing ideas are his hyperbolic gluing equations. Roughly speaking, given an ideally triangulated 3-manifold M, one can construct a set of complex polynomial equations whose solutions correspond to hyperbolic structures on M. In this talk I will describe some recent work generalizing these equations in the context of real projective structures. Specifically, we construct a system of equations whose solutions can be used to construct a projective structure on M. Furthermore, solutions to our equations detect several interesting classes of projective structures including hyperbolic structures, anti-de Sitter structures, and convex projective structures. This work is joint with A. Casella.
Anthony Conway: The talk will decompose into two parts. Firstly, we survey some questions about the knot concordance group C. Secondly, we zoom in on one of these questions, due to Rudolph, which asks for a description of the subgroup of C generated by algebraic knots.
Sami Douba: Geometric group theorists are conditioned to think of finitely generated groups as discrete objects. However, many groups of interest to geometric group theorists can be realized as dense subgroups of compact Lie groups. It is a consequence of several powerful theorems that the fundamental group of a closed 3-manifold admits a faithful finite-dimensional orthogonal representation if the manifold can be endowed with a Riemannian metric of everywhere nonpositive sectional curvature. I will describe some 3-manifolds whose fundamental groups cannot be embedded in compact Lie groups, and, time permitting, discuss deeper questions relating nonpositive curvature to the existence of such embeddings.