Wednesday, 6 September 2023

Pedram Hekmati (Auckland)

Title: Equivariant Seiberg-Witten Floer cohomology and its applications

Abstract: Floer theory comes in various flavours and has developed into a primary tool in low-dimensional topology. In this talk, I will discuss equivariant Seiberg–Witten–Floer cohomology for finite group actions on rational homology 3-spheres. This gives rise to a series of numerical invariants, generalising the Ozsvath–Szabo d-invariant. I will survey their basic properties and applications in knot theory and as obstructions to extending group actions to bounding 4-manifolds. This is joint work with David Baraglia.

Wednesday, 16 August 2023

Roland van der Veen 

Title: Quantum invariants from Fox calculus

Abstract: Fox calculus is a simple formalism to compute a presentation

of the first homology of the cyclic cover of a knot complement

directly from a knot diagram.

The aim of this talk is to show how to use the Fox matrix to construct

a powerful quantum invariant

related to the Jones polynomial. This approach yields very efficient

computations and suggests there might

be a simple topological interpretation of such invariants in terms of

the cyclic cover.

This is joint work with Dror Bar-Natan, Arxiv:2206.12298

Handout:  Recipe for a quick, strong, knot invariant 

Wednesday, 17 May 2023

Colin Adams (Williams College)

Title: Hyperbolic Volumes of Virtual Knots and their Compositions

Abstract: Virtual knots correspond to knots in thickened surfaces and as such can possess hyperbolic metrics. For low crossing number, they are almost all hyperbolic. We will discuss hyperbolicity of virtual knots and their corresponding volumes, paying particular attention to  their behavior under composition. 

Wednesday, April 19 2023

Anna Parlak (UC Davis)

Title: Faces of the Thurston norm ball dynamically represented by multiple distinct flows

Abstract:

A pseudo-Anosov flow on a closed 3-manifold dynamically represents a face F of the Thurston norm ball if the cone on F is dual to the cone spanned by homology classes of closed orbits of the flow. Fried showed that for every fibered face of the Thurston norm ball there is a unique, up to isotopy and reparameterization, flow which dynamically represents the face.  Mosher found sufficient conditions on a non-circular flow to dynamically represent a non-fibered face, but the problem of the existence and uniqueness of the flow for every non-fibered face was unresolved.

I will explain how one can show that a non-fibered face can be in fact dynamically represented by multiple topologically inequivalent flows and discuss how two distinct flows representing the same face may be related. 

Wednesday, April 19 2023

Ethan Dlugie (UC Berleley)

Title: The Burau Representation and Shapes of Polyhedra

Abstract:  The Burau representation of braid groups has been around for almost a century. Still we don't know the full answer to whether this representation is faithful. The only remaining case is for the $n=4$ braid group, and faithfulness here has intimate connections to the question of whether the Jones polynomial detects the unknot. By essentially specializing the $t$ parameter in this representation to certain roots of unity, an interesting connection appears with the moduli space of flat cone metrics on spheres explored by Thurston. Leveraging this connection, I will explain how one can place strong restrictions on the kernel of the $n=4$ Burau representation.

Wednesday, March 15 2023

Kate Petersen (U Minnesota Duluth)

Title: Recognition Problems for 3-manifolds

Abstract:  A natural problem in low-dimensional topology is to determine the geometry of a 3-manifold when the manifold is given by a triangulation. I will discuss this problem in the context of decision problems and the complexity of answering them.  I will focus on the problem of showing that a 3-manifold is not a lens space.  This is joint work with Neil Hoffman and Eric Samperton. 

Wednesday, March 1 2023

Michael Landry (Washington University in St.  Louis)

Title:    Endperiodic maps via pseudo-Anosov flows 

Abstract:    We show that every atoroidal endperiodic map of an infinite-type surface can be obtained from a depth one foliation in a fibered hyperbolic 3-manifold, reversing a well-known construction of Thurston. This can be done almost-transversely to the canonical suspension flow, and as a consequence we recover the Handel-Miller laminations of such a map directly from the fibered structure, and generalize from the finite-genus case the relation between topological entropy, growth rates of periodic points, and growth rates of intersection numbers of curves. Fixing the manifold and varying the depth one foliations, we obtain a description of the Cantwell-Conlon foliation cones and a proof that the entropy function on these cones is continuous and convex. This is joint work with Yair Minsky and Sam Taylor.

Wednesday, 30 November 2022

Robert DeYeso (University of Iowa)

Title:    Thin knots satisfy the Cabling Conjecture  

Abstract:    The Cabling Conjecture of González-Acuña and Short holds that only cabled knots admit Dehn surgery to a manifold containing an essential sphere. We use Heegaard Floer homology via immersed curves techniques to show that (Knot Floer) thin knots satisfy the conjecture.  Part of this work is joint with Holt Bodish.

Wednesday, 9 November 2022

Lizzie Buchanan (Dartmouth College)

Title:  Diagrams of fibered positive links, and their consequences 

Abstract:  The maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree, and we have a similar result for links. This allows us to complete the positivity classification of all knots up to 12 crossings. The construction of this bound comes from delving into the nature of reduced positive diagrams of fibered links. We explore the properties of such diagrams, and see what they can tell us about a given link.

Wednesday, 19 October 2022

Thomas Le Fils (The University of Sydney)

Title: Holonomy of complex projective structures and periods of translation surfaces.

Abstract: A complex projective structure on a surface is the datum of an atlas of charts in the Riemann sphere with Mobius transformations as transition maps.

Such a structure gives rise to a homomorphism of the fundamental group of the surface into $\mathrm{PSL}_2(\mathbb C)$: its holonomy. I will address the question of characterizing the homomorphisms that arise as the holonomy of a projective structure with branch points on a surface.

I will give particular attention to the case of translation surfaces i.e. projective structures obtained by gluing sides of polygons with translations.

Wednesday, 28 September 2022

Seppo Niemi-Colvin (Indiana University Bloomington)

Title:   Invariance of Knot Lattice Homology and Homotopy

Abstract:  Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic complex surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.

Wednesday, 7 September 2022

Daren Chen (Stanford University)

Title:  Khovanov-type homology of null homologous links in RP^3

Abstract:  Khovanov homology is originally defined for links in S^3, and it was extended for links in I-bundles over surface by Asaeda, Przytycki and Sikora. In this talk, we will exhibit some generalization of their construction for null homologous links in RP^3. On the other side of the story, Ozsvath and Szabo defined a spectral sequence relating the Heegaard Floer homology of the branched double cover of S^3 over L to the Khovanov homology of L. We will extend this construction for null homologous links in RP^3 as well, relating the Heegaard Floer homology of the branched double cover of RP^3 over L to our Khovanov-type homology of L.

Wednesday, 17 August 2022

Dionne Ibarra (Monash)

Title: 

On framing of links in 3-manifolds 

Abstract:  

In this talk we will present work on the change of framing of knots and links via ambient isotopy in 3-manifolds by using McCullough's results on a generalized definition of Dehn twist homeomorphisms. In particular, we will discuss work by Cahn, Chernov, and Sadykov for framed knots and then the generalization of their work (joint work with Bakshi, Przytycki, Montoya-Vega, and Weeks) to links by showing 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 2-sphere. We will then use spin structures to show that the ambient isotopy is a composition of even powers of Dehn homeomorphisms along the disjoint union of non-separating 2-spheres.

Wednesday, 3 August 2022, 12:00 AEST

Andrew Kricker (Nanyang Technological University)

Title: On the Garoufalidis-Kashaev meromorphic 3D-index and some of its asymptotics.

Abstract:

This talk will introduce arXiv:2109.05355, which is joint work with Craig Hodgson and Rafael Siejakowski.

Garoufalidis and Kashaev have defined a fascinating topological invariant which associates to a 3-manifold with toroidal boundary a meromorphic function of two complex variables. It is defined from an ideal triangulation by a "state-integral", where a state is an element of S^1 assigned to every edge of the triangulation and the integrand is a product of quantum dilogarithms obtained from the combinatorics of the triangulation. This invariant is a sort of generating function for the q-series 3D-index of Dimofte, Gaiotto and Gukov, and in fact appears to prove topological invariance of it. To understand the significance of this invariant better, we have studied a certain asymptotic limit of this function at the origin as the quantum parameter q approaches 1. Based on numerical investigations and integral heuristics we propose a conjecture for these asymptotics involving surprisingly rich structure from the geometry of the manifold and its collection of boundary-parabolic PSL(2,C)-representations. 

Wednesday, 22 June 2022, 12:00 AEST

Daniele Celoria (University of Melbourne)

Title: A discrete Morse perspective on knot projections

Abstract: We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse matching (dMm) on the 2-sphere, extending a construction due to Cohen. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMms, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of simple moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations. This is joint work with Naya Yerolemou.

Wednesday, 8 June 2022, 12:00 AEST

Agnese Barbensi (University of Melbourne)

Title: Homology of homologous knotted proteins

Abstract: The classification and characterisation of knotted protein structures often requires noise-free and complete data. Here we develop a persistent homology (PH) based pipeline to analyse geometric features of protein entanglement. We show that PH successfully cluster trefoil proteins by structure similarity and knot-depth type. Moreover, we show that persistence landscapes can be used to quantify the topological difference between a family of knotted and unknotted proteins in the same structural homology class, and we show how to localise and interpret this difference geometrically. Crucially, the topological and geometric quantification we find is robust to noisy input data.

Wednesday, 25 May 2022, 12:00 AEST

Beatrice Bleile (University of New England)

Title: Realisation and Splitting for Poincar\'e Duality Pairs in Dimension Three

Abstract: The homotopy type of a $PD^3$--pair with aspherical boundary components is determined algebraically by the isomorphism class of its fundamental triple. Turaev's realisation condition for $PD^3$--complexes extends to $PD^3$--pairs and provides necessary and sufficient conditions under the additional condition of $\pi_1$--injectivity. We remove this additional condition and prove a Realisation Theorem and Splitting Theorems for all $PD^3$--pairs with aspherical boundary components.

Wednesday, 13 April 2022, 12:00 AEST -hosted by Tillus from Sydney

Gye-Seon Lee (Seoul National University)

Title: Exotic quasi-Fuchsian groups

Abstract: Let G be the isometry group of (d+1)-dimensional hyperbolic space. A subgroup H of G is quasi-Fuchsian if H is a convex cocompact discrete subgroup of G and the limit set of H is homeomorphic to the (d-1)-dimensional sphere. In this talk, I will explain how to construct examples of quasi-Fuchsian groups of G which are not quasi-isometric to the hyperbolic d-space using the Tits-Vinberg representation of Coxeter groups. 

Joint work with Ludovic Marquis.

Wednesday, 30 March 2022, 12:00 AEST -hosted by Tillus from Sydney

Franco Vargas Pallete (Yale)

Title: Peripheral birationality for 3-dimensional convex co-compact PSL(2,C) varieties

Abstract: It is a consequence of a well-known result of Ahlfors and Bers that the PSL(2, C) character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding PSL(2, C) character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.

#29 - Wednesday 2 December 2020, 12:00 AEDT (Hosted by Tillus)

Sara Maloni (Virginia)

Title: Convex hulls of quasicircles in hyperbolic and anti-de Sitter space

Abstract: Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a universal version of these conjectures (and prove the existence part) by considering convex sets spanning quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. This work generalizes Alexandrov and Pogorelov's results about the characterization metrics induced on the boundary of a compact convex subset of hyperbolic space. Time permitting, we will discuss why in hyperbolic space quasicircles can't be characterized by the width of their convex hulls, except when the convex hulls have small width. This is different than the anti-de Sitter setting. (This is joint work with Bonsante, Danciger and Schlenker.)

#28 - Wednesday 25 November 2020, 12:00 AEDT (Hosted by Tillus)

Murray Elder (UTS)

Title: Rewriting systems, plain groups, and geodetic graphs 

Abstract: I will describe a new proof, joint with Adam Piggott (UQ), that groups presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 are exactly the plain groups (free products of finite and infinite cyclic groups). Our proof relies on a new result about properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.

#27 - Wednesday 18 November 2020, 12:00 AEDT (Hosted by Tillus)

Michelle Chu (UIC)

Title: Prescribed virtual torsion in the homology of 3-manifolds 

Abstract: Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.

#26 - Wednesday 11 November 2020, 12:00 AEDT (Hosted by Joan)

Eli Grigsby (Boston College)

Title: On the topological expressiveness of neural networks 

Abstract: One can regard a neural network as a particular type of function  F:R^n to R^m, where  R^n  is a (typically high-dimensional) Euclidean space parameterizing some data set, and the value, F(x), of the function on a data point x is used to predict the answer to a question of interest. For example, when the question of interest is a binary classification task (e.g., "Is this e-mail spam?"), the neural network output is 1-dimensional, and the neural network partitions the domain into decision regions labeled "yes" or "no" depending on whether they are in the super-level or sub-level set of a chosen threshold, t.

It is a classical result in the subject that a sufficiently complex neural network can approximate any function on a compact set. In 2017, J. Johnson and B. Hanin-M. Sellke independently proved that universality results of this kind depend on the architecture of the neural network (the number and dimensions of its hidden layers). Their argument(s) were novel in that they provided explicit topological obstructions to representability of a function by a neural network, subject to certain simple constraints on its architecture. I will begin by telling you just enough about neural networks to understand and appreciate their result. Then I will describe a joint on-going project with K. Lindsey aimed at developing a general framework for understanding how the architecture of a neural network constrains the topological features of its decision regions.

#25 - Wednesday 28 October 2020, 12:00 AEDT (Hosted by Jessica)

Christine Lee (South Alabama)

Title: Plamenevskaya's invariant and the stable Khovanov homology of twisted torus knots.

Abstract: A transverse link is a link in the 3-sphere that is everywhere transverse to the standard contact structure. Transverse links are considered up to transverse isotopy, with classical invariants such as the self-linking number and regular isotopy class. One of the first connections between transverse links and quantum invariants was made by Plamenevskaya in 2006, when she defined a invariant of transverse links from Khovanov homology. Since then, the relationship between this fascinating invariant and contact geometry remains of intense interest. In this talk, I will discuss open questions relating Plamenevskaya's invariant to the contact-geometric properties of braids such as the Fractional Dehn Twist Coefficient, and how my recent works with Hubbard, and with Caprau, Gonzalez, Sazdanovic, and Zhang apply to study those questions.

#24 - Wednesday 21 October 2020, 12:00 AEDT (Hosted by Jessica)

Ben Burton (University of Queensland)

Title:  Sub-exponential time knot polynomials: from theory to practice  

Abstract: Many polynomial invariants of knots and links, including the Jones and HOMFLY-PT polynomials, are widely used in practice but #P-hard to compute. Several of these are now known to have sub-exponential time algorithms, using techniques from parameterised complexity. Here we outline how these algorithms work, with a particular focus on practitioners. In particular, we talk through how to manage the often-perilous transition from “fast in theory” to “fast in practice”, using Regina’s implementation of the HOMFLY-PT polynomial as a companion on our tour.  

#23 - Wednesday 14 October 2020, 12:00 AEDT (Hosted by Jessica)

Anastasiia Tsvietkova (IAS Princeton/ Rutgers Newark)

Title:  Polynomially many genus g surfaces in a hyperbolic 3-manifold 

Abstract: We will discuss a universal upper bound for the number of non-isotopic genus g surfaces embedded in a hyperbolic 3-manifold, polynomial in hyperbolic volume. The surfaces are all closed essential surfaces, oriented and connected. This is joint work with Marc Lackenby. 

#22 - Wednesday 7 October 2020, 12:00 AEDT (Hosted by Jessica)

Jeffrey Danciger (UT Austin)

Title:  Exotic real projective Dehn surgery space 

Abstract: We study properly convex real projective structures on closed 3-manifolds. A hyperbolic structure is one special example, and in some cases the hyperbolic structure may be deformed non-trivially as a convex projective structure. However, such deformations seem to be exceedingly rare. By contrast, we show that many closed hyperbolic 3-manifolds admit a second convex projective structure not obtained through deformation. We find these structures through a theory of properly convex projective Dehn filling, generalizing Thurston’s picture of hyperbolic Dehn surgery space. Joint work with Sam Ballas, Gye-Seon Lee, and Ludovic Marquis. 

#21 - Wednesday 30 September 2020, 12:00 AEST (Hosted by Jessica)

Genevieve Walsh (Tufts)

Title: Incoherence of free-by-free and surface-by-free groups  

Abstract: A group is coherent if every finitely generated subgroup is finitely presented, and incoherent otherwise. Many well-known groups are coherent: free groups, surface groups, and the fundamental groups of compact 3-manifolds (due to Scott). We consider groups of the form $F_m \by F_n$ or $S_g \by F_n$ where $S_g$ is the fundamental group of a closed surface of genus $g$. We show that these groups are incoherent whenever $g, n$ are at least 2. One possible method to show that such groups are incoherent would be to show that they virtually algebraically fiber. That is, that there is a finite index subgroup which surjects Z with finitely generated kernel. While this works in some cases, we additionally show that there are free-by-free and surface-by-free groups which do not virtually algebraically fiber. This is joint work with Robert Kropholler and Stefano Vidussi. 

#20 - Wednesday 23 September 2020, 12:00 AEST (Hosted by Dan)

Robert Tang (XJTLU )

Title: Large-scale geometry of the saddle connection graph 

Abstract: The saddle connection graph is an analogue of the well-known arc graph in the context of (half-)translation surfaces. In this talk, I will discuss the large-scale geometry of the saddle connection graph. In particular, we classify these graphs up to quasi-isometry, and characterise their Gromov boundary in terms of foliations. The talk will involve lots of Euclidean geometry and pictures. This is joint work with V. Disarlo, H. Pan, and A. Randecker. 

#19 - Wednesday 9 September 2020, 12:00 AEST (Hosted by Dan)

Tamas Kalman (Tokyo Institute of Technology) 

Title: Floer homology, the HOMFLY polynomial, and combinatorics 

Abstract: All oriented links $L$ have special diagrams. Based on such a diagram we construct a sutured handlebody $M$ which embeds in the branched double cover of the link. From the sutured Floer homology of $M$ we recover the Alexander polynomial $\Delta$ of $L$ via a simple forgetful map. More surprisingly, in cases when the diagram is also positive (so that $L$ is a special alternating link), $\mathrm{SFH}(M)$ can be used to compute those coefficients of the HOMFLY polynomial of $L$ whose sum is the leading coefficient of $\Delta$. To extract this information algebraically, we need the notion of the interior polynomial of a bipartite graph. The talk involves joint results with A. Juh\'asz, H. Murakami, A. Postnikov, J. Rasmussen, D. Thurston and, if time permits, D. Mathews. 

#18 - Wednesday 2 September 2020, 12:00 AEST (Hosted by Dan)

Ko Honda (UCLA)

Title: Convex hypersurface theory in higher-dimensional contact topology

Abstract: Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds. In particular they have been successfully applied to many classification problems. After briefly reviewing convex surface theory in dimension three, we explain how to generalize many of their properties to higher dimensions. This is joint work with Yang Huang. 

#17 - Wednesday 26 August 2020, 12:00 AEST (Hosted by Joan)

Zsuzsanna Dancso (Sydney)

Title: The algebraic structures of classical, welded and virtual tangles

Abstract: Story: Listening to each other's talks, Marcy Robertson and I had a vague suspicion that we were thinking about the same objects. We set out with Iva Halacheva to explain definitions to each other: much harder than we expected. Once we succeeded, we were so pleased we wrote a paper about it. The translation has the potential to connect several quite separate research communities.

Formal abstract: Circuit algebras are a generalisation of Jones's planar algebras, in which one drops the planarity condition on ``connection diagrams.'' They provide a useful language for the study of welded and virtual tangles in low-dimensional topology. In this talk we present an equivalence of categories between circuit algebras and the category of linear wheeled props -- a type of strict symmetric tensor category with duals that arises in homotopy theory, deformation theory and the Batalin-Vilkovisky quantisation formalism. This parallels a well-studied classification result of planar algebras as pivotal categories with a self-dual generator (originally by Morrison-Peters-Snyder). Joint work with Iva Halacheva and Marcy Robertson. 

#16 - Wednesday 19 August 2020, 12:00 AEST (Hosted by Joan)

Kyle Hayden (Columbia)

Title: A softer side of complex curves 

Abstract: There is a rich, symbiotic relationship between knot theory and the study of complex curves. 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 smooth (in fact, 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. 

#15 - Wednesday 12 August 2020, 12:00 AEST (Hosted by Joan)

Jeremy van Horn-Morris (Arkansas) 

Title: Towards a braid theory of codimension 2 contact submanifolds 

Abstract: In dimension 3, the theory of codimension 2 contact submanifolds is better known as the transverse knot theory of a contact manifold, a theory which has a complete description in terms of braid theory. In higher dimensions, very little is known but there is a small but growing list of results. I will explain a method developed with A. Kaloti to use open books and lefschetz fibrations to study codimension 2 contact embeddings. I will give a lot of background and present some initial applications and will highlight the similarities to and differences from the analogous story in dimension 3. (Hosted by Joan from Canberra) 

#14 - Wednesday 5 August 2020, 12:00 AEST (Hosted by Joan)

Josh Howie (UC Davis)

Title: Alternating genera of torus knots

Abstract: The alternating genus of a knot is the minimum genus of a surface onto which the knot has an alternating diagram satisfying certain conditions. Very little is currently known about this knot invariant. We study spanning surfaces for knots, and define an alternating distance from the extremal spanning surfaces. This gives a lower bound on the alternating genus and can be calculated exactly for torus knots. We prove that the alternating genus can be arbitrarily large, find the first examples of knots where the alternating genus is exactly 3, and classify all toroidally alternating torus knots. 

#13 - Wednesday 29 July 2020, 12:00 AEST (Hosted by Joan)

Alexander Zupan (University of Nebraska-Lincoln)

Title: Cubic graphs and bridge trisections

Abstract: An elementary but very useful exercise in knot theory is to show that for any generic immersed curve in the plane, there is a choice of crossing information at each double point so that the resulting knot diagram determines the unknot in 3-space. Here's a related problem: A graph G is cubic if every vertex has valence three and Tait-colored if each edge is colored red, blue, or green, where each vertex is incident to edges of all three colors. For any immersion of G into the plane such that the red and blue edges are embedded and the green edges are allowed to meet in double points, is there a choice of crossing information so that every bi-colored cycle is unknotted? Although an ad hoc approach works in simple cases, an algorithm to produce crossing choices (like the solution to the elementary problem) seems intractable. We will show that the answer is yes, but the proof strategy takes a route through bridge trisections of knotted surfaces in 4-space -- in fact, in this setting we can prove something significantly stronger. This talk is based on joint work with Jeffrey Meier and Abigail Thompson.

#12 - Wednesday 22 July 2020, 12:00 AEST (Hosted by Joan)

Maggie Miller (Princeton)

Title: Characterizing handle-ribbon knots

Abstract: Kauffman conjectured that a knot K is slice if and only if it bounds a genus-g Seifert surface containing a g-component slice link as a cut system. It’s very easy to show that a knot is ribbon if and only if it bounds a genus-g Seifert surface containing a g-component unlink as a cut system. Alex Zupan and I proved something in the middle of these statements: a knot is handle-ribbon (aka strongly homotopy-ribbon, aka something I will define in the talk) if and only if it bounds a genus-g Seifert surface containing a g-component R link L as a cut system – meaning that zero-surgery on L yields #_g S^1 x S^2. This gives a 3-dimensional definition of a 4-dimensional property. I’ll talk about these 3.5D knot properties and maybe how we use these techniques to extend a statement of Casson and Gordon. (The work in this talk is joint with Alexander Zupan from the University of Nebraska--Lincoln.)

#11 - Wednesday 15 July 2020, 12:00 AEST (Hosted by Joan)

Norman Do (Monash)

Title: Hurwitz numbers and topological recursion

Abstract: Hurwitz numbers count branched covers of the sphere and have been studied for well over a century. On the other hand, topological recursion emerged from the mathematical physics literature in 2007 and is now known or conjectured to govern a vast array of "enumerative'' problems including: tilings of surfaces; Mirzakhani's volumes of moduli spaces of hyperbolic surfaces; quantum knot invariants; Gromov-Witten invariants of various target manifolds; and much more. In this talk, we will tell the story of how and why Hurwitz numbers are related to topological recursion. We will conclude with a discussion of recent work with Gaëtan Borot, Maksim Karev, Danilo Lewański and Ellena Moskovsky that relates so-called double Hurwitz numbers to topological recursion.

#10 - Wednesday 8 July 2020, 12:00 AEST (Hosted by Joan)

Kasra Rafi (Toronto)

Title: Large scale geometry of big mapping class groups

Abstract: We apply the framework of Rosendal for the study of the coarse geometry of non locally compact groups in the setting of Mapping Class Groups of surfaces of infinite type. Under mild conditions, we give a classification of which Mapping Class Groups have a well defined quasi-isometry type and, amount those, which Mapping Class Groups are quasi-isometric to a point (Joint with Kathryn Mann). We also give a classification of Mapping Class Groups that act non-trivially on a hyperbolic space (Joint with Horbez and Qing). 

#9 - Wednesday 1 July 2020, 12:00 AEST (Hosted by Joan)

Josh Greene (Boston)

Title: The rectangular peg problem

Abstract: 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.

#8 - Wednesday 24 June 2020, 12:00 AEST (Hosted by Tillus)

Tengren Zhang (NUS)

Title: Affine actions with Hitchin linear part

Abstract: We prove that if a surface group acts properly on R^d via affine transformations, then its linear part is not the lift of a PSL(d,R)-Hitchin representation. To do this, we proved two theorems that are of independent interest. First, we showed that PSO(n,n)-Hitchin representations, when viewed as representations into PSL(2n,R), are never Anosov with respect to the stabilizer of the n-plane. Following Danciger-Gueritaud-Kassel, we also view affine actions on R^{n,n-1} as a geometric limit of isometric actions on H^{n,n-1}. The second theorem we prove is a criterion for when an affine action on R^{n,n-1} is proper, in terms of the isometric actions on H^{n,n-1} that converge to it. This is joint work with Jeff Danciger.

#7 - Wednesday 17 June 2020, 12:00 AEST (Hosted by Tillus)

Lisa Piccirillo (Brandeis/MIT)

Title: The trace embedding lemma and PL surfaces

Abstract: 4-manifold topologists have long been interested in understanding smooth (resp. topological) embedded surfaces in smooth (resp. topological) 4-manifolds, and as such have developed rich suites of tools for obstructing the existence of smooth (resp. topological) surfaces. Understanding PL surfaces in smooth 4-manifolds has historically garnered less interest, but several problems about PL surfaces have recently arisen in modern lines of questioning. Presently there are far fewer tools available to obstruct PL surfaces. In this talk, I’ll discuss how to use a classical observation, called the trace embedding lemma, to repurpose smooth surface obstructions as PL surface obstructions. I’ll discuss applications of these retooled obstructions  to problems about spinelessness, exotica, and geometrically simply connectedness. This is joint work with Kyle Hayden.

#6 - Wednesday 10 June 2020, 12:00 AEST 

No Talk in support of #Strike4BlackLives

To get started, see:

https://blogs.ams.org/inclusionexclusion/2020/06/06/shutdownmath/

https://www.shutdownstem.com/about

#5 - Wednesday 3 June 2020, 12:00 AEST (Hosted by Tillus)

Dan Margalit (GaTech)

Title: Homomorphisms of braid groups

Abstract: In the early 1980s Dyer-Grossman proved that every automorphism of the braid group is geometric, meaning that it is induced by a homeomorphism of the corresponding punctured disk.  I'll discuss two recent generalizations.  With Lei Chen and Kevin Kordek, we prove that every homomorphism from the braid group on n strands to the braid group on (up to) 2n strands is geometric.  With Kordek, we prove that every homomorphism from the commutator subgroup of the braid group to the braid group is geometric. Both results can be interpreted in terms of maps between spaces of polynomials.  We will begin with some background, explain the statements of both theorems, and discuss the basic ideas behind both of the proofs.

#4 - Wednesday 27 May 2020, 12:00 AEST (Hosted by Tillus)

Priyam Patel (Utah)

Title: Isometry groups of infinite-genus hyperbolic surfaces

Abstract: Allcock, building on the work of Greenburg, proved that for any countable group G, there is a a complete hyperbolic surface whose isometry group is exactly G. When the group is finite, Allcock’s construction yields a closed surface, but when it is not finite, the construction gives an infinite-genus surface. 

In this talk, we discuss a related question. We fix any infinite-genus surface S and characterize all groups that can arise as the isometry group for a complete hyperbolic structure on S. In the process, we give a classification type theorem for infinite-genus surfaces and, if time allows, two applications of the main result. This talk is based on joint work with T. Aougab and N. Vlamis.

#3 - Wednesday 20 May 2020, 12:00 AEST (Hosted by Tillus)

Jessica Purcell (Monash)

Title: Geometric triangulations and highly twisted links

Abstract: Every 3-manifold can be triangulated, i.e. decomposed into tetrahedra. If the 3-manifold has geometry, we would like the corresponding tetrahedra to have geometry. For example, if the 3-manifold is hyperbolic, we would like the tetrahedra to be convex hyperbolic tetrahedra, with positive volume; this is called a geometric triangulation. However, it is still unknown whether every cusped hyperbolic 3-manifold admits such a triangulation. In this talk, I will show that the complement of every knot in the 3-sphere admits a geometric triangulation, provided it is sufficiently highly twisted. This is joint work with Sophie Ham.

#2 - Wednesday 13 May 2020, 12:00 AEST (Hosted by Tillus)

Diarmuid Crowley (Melbourne)

Title: Stably diffeomorphic manifolds

Abstract: The classification of 4-manifolds up to the addition of copies of S^2 x S^2 was pioneered by Wall and then Cappell and Shaneson in order to circumvent some of the complexities of 4-dimensional topology.

Later Kreck showed that stable classification, i.e. classification of 2q-manifolds up connected sum with copies of S^q x S^q, was a highly effective technique in all dimensions.  However, most work following Kreck looked at cases where stable-diffeomorphism implies diffeomorphism.

In this talk I will report on recent work aimed at distinguishing different diffeomorphism classes of manifolds within the same stable diffeomorphism class.  This work includes a current joint project with Anthony Conway, Mark Powell and Joerg Sixt and also parts of the PhD thesis of my student Csaba Nagy.

#1 - Wednesday 6 May 2020, 12:00 AEST  (Hosted by Tillus)

Henry Segerman (Oklahoma State University)

Title: From veering triangulations to link spaces and back again

Abstract: Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. In unpublished work, Guéritaud and Agol generalise an alternative construction to any closed manifold equipped with a pseudo-Anosov flow without perfect fits.

Schleimer and I build the reverse map. As a first step, we construct the link space for a given veering triangulation. This is a copy of R^2, equipped with transverse stable and unstable foliations, from which the Agol-Guéritaud construction recovers the veering triangulation. The link space is analogous to Fenley's orbit space for a pseudo-Anosov flow.

Along the way, we construct a canonical circular ordering of the cusps of the universal cover of a veering triangulation. I will also talk about work with Giannopolous and Schleimer building a census of transverse veering triangulations. The current census lists all transverse veering triangulations with up to 16 tetrahedra, of which there are 87,047