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12:30-12:35 |
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Welcome |
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12:35-12:55 |
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Tea |
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12:55-1:00 |
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Welcome, Part II |
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1:00-1:50 |
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Colloquium: Yair Minsky |
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Skinning maps and a lost theorem of Thurston
Slides |
1:50-2:00 |
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Break from computer |
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2:00-3:00 |
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Lightning talk session 1 |
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Breakout Rooms for 8 talks in Session 1 below |
3:05-4:00 |
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Social Hour |
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On GatherTown: Come see old friends and meet new ones! |
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10:30-10:40 |
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Tea |
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10:40-10:45 |
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Conference Photo |
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10:45-11:15 |
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Kristen Hendricks |
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Homology cobordism and involutive Heegaard Floer homology
Slides |
11:20-11:50 |
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Andras Juhasz |
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Knot theory and machine learning |
11:50-12:00 |
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Break from computer |
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12:00-1:15 |
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Lunch and Lightning talk 2 |
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Breakout Rooms for 9 talks in Session 2 below |
1:15-1:45 |
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Beibei Liu
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Four manifold with no smooth spines |
1:50-2:50 |
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Social Hour |
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On GatherTown: Come see old friends and meet new ones! Discuss the talks or anything else. |
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10:30-10:40 |
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Tea |
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10:40-11:10 |
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Vera Vertesi
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Cutting and pasting open books
Slides
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11:15-11:45 |
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Marissa Loving |
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End-periodic homeomorphisms and volumes of mapping tori
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11:45-12:00 |
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Break from computer |
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12:00-1:15 |
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Lunch and Lightning talk 3 |
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Breakout Rooms for 8 talks in Session 3 below |
1:15-1:45 |
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Ursula Hamenstädt
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A boundary for the mapping class group |
1:45-2:15 |
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Last conference tea |
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Breakout Rooms to chat with Speakers, Organizers, and say good bye. |
| | Rhea Palak Bakshi | On the Kauffman bracket skein module of the connected sums of 3-manifolds |
| | Assaf Bar-Natan | The Grand Arc Graph on Infinite-Type Surfaces |
| | Fraser Binns | Braids and knot Floer homology |
| | Orsola Capovilla-Searle | Legendrian links with infinitely many genus $0$ exact Lagrangian fillings that are not Hamiltonian isotopic. |
| | Anindya Chanda | A New Class of Examples of Quasigeodesic (Anosov) Flow in Dimension 3 |
| | Sally Collins | Smooth concordance, homology cobordism, and the Mazur pattern |
| | Robert DeYeso III | Thin knots and the Cabling Conjecture |
| | Ethan Farber | A Farey tree structure on a family of pseudo-Anosov braids |
| | Rima Chatterjee | Cabling knots in overtwisted contact manifolds |
| | Tyler Gaona | Minimal volume hyperbolic 3-orbifolds with rigid and nonrigid cusps |
| | Sudipta Ghosh | Connected sums and directed systems in knot Floer homologies |
| | James Hughes | A Brief Introduction to Legendrian Weaves |
| | Khanh Le | Totally geodesic surfaces in hyperbolic knot complements |
| | Lily Li and Caleb Partin | Generalizing the Twisted Rabbit Problem |
| | Robert Quarles | A new perspective on a polynomial time knot polynomial |
| | Braeden Reinoso | Fixed points of pseudo-Anosovs, and knot Floer homology |
| | Mark Ronnenberg | Prequantum Bundle on the Traceless Character Variety of a Surface with Boundary |
| | Gage Martin | Annular Khovanov homology and meridional disks |
| | Agniva Roy | Constructions and Invariants of higher dimensional Legendrian spheres |
| | Lorenzo Ruffoni | Strict hyperbolization and special cubulation |
| | Abdoul Karim Sane | Intersection norms, Thurston norms and their dual unit balls. |
| | Roberta Shapiro | An Alexander method for infinite-type surfaces |
| | Kai Smith | Pillowcase Homology and Character Varieties of Tangles |
| | C.-M. Michael Wong | Tau invariants in monopole and instanton Floer theories |
| | Jiajun Yan | A New Construction of ALE Spaces via Gauge Theory |
Titles and Abstracts
A boundary for the mapping class group
We explain what a (topological) boundary of an infinite group
is and why such a boundary is interesting. We then
describe an explicit such boundary for the mapping class group,
the group of isotopy classes of orientation preserving diffeomorphism of a
closed surface S.
Homology cobordism and involutive Heegaard Floer homology
The homology cobordism group consists of integer homology spheres under connected sum, modulo an equivalence relation called homology cobordism. We review some history of this group and discuss applications of Heegaard Floer homology to its structure. In particular, we show that the homology cobordism group is not generated by Seifert fibered spaces. This is joint work with J. Hom, M. Stoffregen, and I. Zemke.
Knot theory and machine learning
We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times.
This is joint work with Alex Davies, Marc Lackenby, and Nenad Tomasev.
Four manifold with no smooth spines
It is an interesting question to ask whether a compact smooth 4-manifold which deformation retracts to a PL embedded closed surface contains a smooth spine, i.e. deformation retracts onto a smoothly embedded surface. In this talk, we will use Heegaard Floer homology and high-dimensional surgery theory to give some obstructions. We will also discuss some examples where the interior of the 4-manifold is negatively curved. This is joint work with Igor Belegradek.
End-periodic homeomorphisms and volumes of mapping tori
I will discuss volumes of mapping tori associated to irreducible end-periodic homeomorphisms of certain infinite-type surfaces, inspired by a theorem of Brock (in the finite-type setting) relating the volume of a mapping torus to the translation distance of its monodromy on the pants graph. This talk represents joint work with Elizabeth Field, Heejoung Kim, and Chris Leininger.
Skinning maps and a lost theorem of Thurston
Thurston's proof of the hyperbolization theorem for Haken manifolds involved a gluing step, in which the matching conditions for the boundary components being glued are phrased in terms of a fixed-point problem for a certain self-map of Teichmuller space. A better quantitative understanding of this process would improve our control of the relation of topology to geometry of these manifolds. Thurston stated an appealing theorem: that a finite power of the self-map has bounded image, thus controlling the process of finding the fixed point. Nobody seems to know what Thurston's proof was. With Ken Bromberg and Dick Canary, we provide a proof that involves building uniform models for the internal geometry of hyperbolic manifolds of a given topological type. I will try to explain the background and the ingredients of this theorem.
Cutting and pasting open books
Just like Heegaard decompositions, open book decompositions give an efficient way of describing (closed, oriented) 3-manifolds. An open book decomposition is defined as a fibration of the complement of a link. Open book decompositions naturally give Heegaard decompositions, but the extra structure given by the open book gives rise to an extra structure on the 3-manifold: a contact structure.
In this talk I will describe a generalisation of open books to 3-manifolds with boundary that behaves well with gluing. This structure is the essential ingredient to define an invariant of contact structures on 3-manifolds with boundary in bordered Floer homology.
I will concentrate mostly on the geometric/combinatorial part of the result, spending some time on a recent result that recognises generic intersections of open books with surfaces.
Part of this work is joint with Akram Alishahi, Viktòria Földvári, Kristen Hendricks, Joan Licata and Ina Petkova.
Lightning talk
On the Kauffman bracket skein module of the connected sums of 3-manifolds
Skein modules were introduced by Józef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of the connected sum of two solid tori.
The Grand Arc Graph on Infinite-Type Surfaces
In this talk, I will be defining the grand arc graph for infinite-type surfaces. This simplicial graph is often infinite-diameter and delta hyperbolic, and the mapping class group acts continuously on its boundary. I will also describe some ideas about describing the boundary of this graph as a space of laminations on an infinite-type surface. This is based on joint work with Yvon Verberne.
Braids and knot Floer homology
I will discuss generalised braids, the BRAID invariant, lower bounds on the rank of knot Floer homology and why they are useful. This talk is based on joint work in progress with Subhankar Dey.
Legendrian links with infinitely many genus $0$ exact Lagrangian fillings that are not Hamiltonian isotopic.
One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings. We show that a family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. One can apply such results to find closed exact Lagrangians in Weinstein manifolds. In particular, we find infinitely many exact Lagrangian spheres and tori 4-dimensional Milnor fibers of T_{p,q,r} singularities.
A New Class of Examples of Quasigeodesic (Anosov) Flow in Dimension 3
A flow on a Riemannian manifold is called ‘quasigeodesic’ if, when lifted to the universal cover, its flowlines are length minimizing up to a multiplicative and/or an additive constant. Quasigeodisity of a flow is a metric invariant property and quasigeodesic flows play a very important role in the study of the underlying manifolds. One of the most important class of examples is suspension flows induced by Psudo-Anosov maps on surfaces (Cannon and Thurston). In this talk we will describe a new class of quasigeodesic Anosov flows in a non-hyperbolic non-Seifert fiber setting.
Cabling knots in overtwisted contact manifolds
Knots in overtwisted manifolds are not well explored. There are two types of knots in an overtwisted manifold -loose (having overtwisted complement) and non-loose (with tight complement). While we understand loose knots, non-loose knots remain a mystery. In this talk, we focus on the cabling operation of non-loose knots and discuss how it is different from cabling knots in tight manifolds. In particular, I will show which conditions preserve non-looseness. This is a joint work in progress with Etnyre, Min and Mukherjee.
Smooth concordance, homology cobordism, and the Mazur pattern
The 0-surgery manifolds of two knots K_1 and K_2 are homology cobordant rel meridians if there exists a Z-homology cobordism X between them such that the two positively oriented knot meridians are in the same homology class of H_1(X; Z). One can consider how knot type of the two knots influences the existence of such a cobordism. In this talk, we give a pair of rationally slice knots which are not smoothly concordant but whose 0-surgery manifolds are homology cobordant rel meridians. One knot in the pair is the figure eight knot, which is order 2 in the smooth concordance group. All previous examples of such pairs of knots were all infinite order.
Thin knots and the Cabling Conjecture
The Cabling Conjecture of González-Acuña and Short holds that only cable knots admit Dehn surgery to a manifold containing an essential sphere. We approach this conjecture for thin knots using Heegaard Floer homology, primarily via immersed curves techniques inspired by Hanselman's work on the Cosmetic Surgery Conjecture. We show that almost all thin knots satisfy the Cabling Conjecture, with possible exception coming from a (conjecturally non-existent) collection of thin, hyperbolic, L-space knots. This result serves as a reproof that the Cabling Conjecture is satisfied by alternating knots, and also a new proof that thin, slice knots satisfy the Cabling Conjecture.
A Farey tree structure on a family of pseudo-Anosov braids
Braids are ubiquitous in low-dimensional topology. In this talk, we think of braids as transformations on a punctured disc. We describe a family of braids possessing many intriguing properties: they are (1) positive, (2) pseudo-Anosov, and (3) represent every positive non-integral fractional Dehn twist coefficient. This family exhibits the combinatorial structure of the Farey tree, which dictates the dynamics of these braids.
Minimal volume hyperbolic 3-orbifolds with rigid and nonrigid cusps
A complete hyperbolic 3 orbifold is a quotient of hyperbolic space by a discrete subgroup of isometries. For cusped orbifolds, we can obtain lower bounds on the volume of the orbifold in terms of the volumes of its cusps by a sphere packing result of Boroczsky. A cusp in the orbifold lifts to a packing of horoballs in hyperbolic space. In this talk I'll show how an analysis of the horoball packing can reveal enough volume in either the cusp or the compact part of the orbifold to classify the minimal orbifold with one rigid and one nonrigid cusp.
Connected sums and directed systems in knot Floer homologies
Knot Floer homology is an invariant of knot which was first introduced in the context of Heegaard Floer homology and later extended to other Floer theories. In this talk, we discuss a new approach to the connected sum formula using direct limits. Our methods apply to versions of knot Floer homology arising in the context of Heegaard, instanton and monopole Floer homology. Using the same argument, we also deduce the oriented skein relation for the minus version of instanton knot homology. This is joint work with Ian Zemke.
A Brief Introduction to Legendrian Weaves
Given a Legendrian link L in the contact 3-sphere, one can hope to classify the set of Lagrangian fillings of L, i.e. exact Lagrangian surfaces in the symplectic 4-ball with boundary equal to L. The method of Legendrian weaves, developed recently by Casals and Zaslow, is a diagrammatic calculus that combinatorializes the construction of Lagrangian fillings and the computation of certain invariants. In this talk I will introduce the basics of this construction and briefly mention an application to proving the existence of infinitely many distinct Lagrangian fillings.
Totally geodesic surfaces in hyperbolic knot complements
The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them. There has been considerable work in understanding the existence of totally geodesic surfaces in hyperbolic 3-manifolds. Most recently, Bader, Fisher, Miller, and Stover showed that having infinitely many maximal totally geodesic surfaces implies that the 3-manifold is arithmetic. In this talk, we will discuss various new techniques to count totally geodesic surfaces in hyperbolic knot complements and obstructions to the existence to totally geodesic surfaces in hyperbolic 3-manifolds. As an application, we will also present examples of infinitely many non-commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly k totally geodesic surfaces for every positive integer k. We will also report on current project on counting totally geodesic surfaces in the complement of knot with small crossing number. This is joint work with Rebekah Palmer.
Generalizing the Twisted Rabbit Problem
Hubbard’s twisted rabbit problem seeks to characterize the image of the rabbit polynomial under powers of Dehn twists around its ears. Bartholdi-Nekrashevych first solved this problem, and recent work of Belk-Lanier-Margalit-Winarski supplies a topological viewpoint on such polynomial recognition problems. Using the topological perspective, this talk discusses two directions of generalization of the twisted rabbit problem. We first allow for more general classes of Dehn twists and then describe an inductive structure relating topological polynomials of different portraits.
Annular Khovanov homology and meridional disks
Both Khovanov homology and Floer theoretic knot invariants have extensions to annular links. In this talk we will highlight a key difference between these annular theories. Specifically, there are infinite families of annular links for which the maximum non-zero annular Khovanov grading grows infinitely large but the maximum non-zero annular Floer-theoretic gradings are bounded. These examples provide further evidence for the wrapping conjecture of Hoste-Przytycki.
A new perspective on a polynomial time knot polynomial
Bar-Natan and van der Veen presented a simple, strong knot invariant that is closely related to the Alexander polynomial. In this talk we will explore some symmetry properties of this invariant and discuss a new perspective on the invariant that allows for a different computational method.
Fixed points of pseudo-Anosovs, and knot Floer homology
It's a long-standing open question whether knot Floer homology detects the torus knot T(2,5). In ongoing work-in-progress with Ethan Farber and Luya Wang, we answer this question in the affirmative. To do this, we complete a strategy recently outlined by Baldwin, Hu, and Sivek which relates the detection question to another question about fixed points of pseudo-Anosov maps: if there are no genus-two pseudo-Anosovs with a single fixed point, then knot Floer homology detects T(2,5). We prove, somewhat surprisingly, that there are in fact genus-two pseudo-Anosovs with a single fixed point. But, we classify all such maps in a special stratum, and use this classification to finish the detection result mentioned above.
Prequantum Bundle on the Traceless Character Variety of a Surface with Boundary
A prequantum bundle is a principal U(1)-bundle over a symplectic manifold, equipped with a connection whose curvature induces the symplectic form on the base space. The traceless character variety is a moduli space associated to a manifold, which is well-known to be symplectic, and is of interest in studying singular instanton homology. In this talk, I will briefly describe how to construct a prequantum bundle over the traceless character variety of a surface with boundary, in a way that is tailored to 3-manifolds containing a tangle.
Constructions and Invariants of higher dimensional Legendrian spheres
In study of low dimensional contact topology, contact structures obtained by surgery along Legendrians has been a rich source of constructions. In higher dimensions, a first step to doing that is coming up with constructions of families of Legendrian spheres. In this talk we describe a natural way to construct Legendrian spheres from stabilising spheres in open book decompositions, and how to obtain their front projections in nice scenarios.
Strict hyperbolization and special cubulation
Gromov introduced some “hyperbolization” procedures to turn a given polyhedron into a space of non-positive curvature, in a way that preserves some of the topological features of the original polyhedron. For instance, a manifold is turned into a manifold. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. This has been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. In joint work with J. Lafont, we construct actions of the resulting groups on CAT(0) cube complexes. As an application, we obtain that they are virtually linear over the integers.
Intersection norms, Thurston norms and their dual unit balls.
Dual unit balls of Thurston norms are invariants of 3-manifolds that encode important properties. The characterization of polyhedra that appears like dual unit balls of Thurston norms is an open question. We will show, using Intersection norms on surfaces, how to construct a large family of polyhedra which are dual unit balls of Thurston norms.
An Alexander method for infinite-type surfaces
The Alexander method is a combinatorial tool that helps distinguish elements of the mapping class group of a surface, MCG(S). The most general version for finite-type surfaces was codified by Farb-Margalit. We build on the work of Hernández-Morales-Valdez to extend the Alexander method of Farb-Margalit to all infinite-type surfaces. We explain our theorem via several examples. Then, we discuss several applications of the Alexander method: verifying relations in the MCG(S), proving that centralizers of certain subgroups of the MCG(S) are trivial, and providing a simpler basis for the topology of MCG(S).
Pillowcase Homology and Character Varieties of Tangles
Pillowcase homology is a knot homology theory constructed by decomposing a knot into two tangles and counting the intersections of their traceless SU(2) character varieties inside a space called the pillowcase. In this talk I will give an overview of pillowcase homology and state some results about the structure of the character varieties of tangles and their perturbations.
Tau invariants in monopole and instanton Floer theories
The tau invariant in Heegaard Floer theory is a powerful concordance invariant that, for example, bounds the smooth four-genus from below. Its counterparts in monopole and instanton Floer theories each admit two rather different definitions that mirror properties of the Heegaard Floer tau. In this talk, I will outline a result that proves that those two definitions of tau agree, as a natural next step in understanding the connections between the three Floer theories for knots. This is joint work with Sudipta Ghosh and Zhenkun Li.
A New Construction of ALE Spaces via Gauge Theory
In this talk, we will briefly introduce the background and history of ALE spaces first constructed by Kronheimer in his PhD thesis. Then we give a new construction of the same spaces via gauge theory. In particular, we realize each ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a hyperkähler gauge group action.
