Time: 12:00 AEST (Australian Eastern Daylight Time)
Day: First and third Wednesday of the month during the teaching sessions in 2025
Organisers: Dionne Ibarra (Monash), Joan Licata (ANU), Daniel Mathews (Monash), and Jonathan Spreer (Sydney)
To join the webinar please email the organisers to be added to the mailing list:
AustralianGeometricTopology@gmail.com
The webinar is hosted via ZOOM, and the access information will be sent to the mailing list approximately 1 hour before each event.
The Zoom room will open about 15 minutes before the talk for people to mingle. The talk will last for about 50 minutes. You are encouraged to ask questions during the talk, either using your microphone or via the chat function, and there is an opportunity for questions and discussion afterwards.
After the talk, participants will be assigned randomly to breakout rooms to continue small group discussions or to socialise. The organisers will make every participant a co-host of the webinar so that they can see who is in which room and move freely between breakout rooms.
Wednesday, 1 October 2025
Title: Quantized nonabelianization
Abstract: Gaiotto-Moore-Neitzke introduced the notion of spectral network in the study of supersymmetric gauge theory. It provides a way to define a nonabelianization map (UV-IR map) from gl(1) local systems on a spectral curve to gl(N) local systems on the base surface. Neitzke-Yan considered a quantization of this map between skein algebras. In this work, we consider a different quantization in the sense of braid skein algebras using both Floer and Morse approach. This is joint work with Ko Honda and Yin Tian.
Wednesday, 1 October 2025
Title: Quantized nonabelianization
Abstract: Gaiotto-Moore-Neitzke introduced the notion of spectral network in the study of supersymmetric gauge theory. It provides a way to define a nonabelianization map (UV-IR map) from gl(1) local systems on a spectral curve to gl(N) local systems on the base surface. Neitzke-Yan considered a quantization of this map between skein algebras. In this work, we consider a different quantization in the sense of braid skein algebras using both Floer and Morse approach. This is joint work with Ko Honda and Yin Tian.
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Wednesday, 15 October 2025
Title: TBA
Abstract: TBA
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Wednesday, 5 November 2025
Title: Multivariable knot polynomials, the V_n-polynomials, and their patterns.
Abstract: Quantum knot invariants are known to come from R-matrices along with some extra structures, a process called the Reshetikhin--Turaev functor. In 2019, Rinat Kashaev proved that R-matrices alone are sufficient to define knot invariants, as long as they satisfy some nondegeneracy conditions called rigidity. More recently, Stavros Garoufalidis and Rinat Kashaev developed a new method of constructing rigid R-matrices, which recovers several known knot polynomials such as the colored Jones polynomials, and gives a new family of multivariable knot polynomials, the V_n-polynomials. In this talk, I will talk about the Reshetikhin--Turaev functor in this context, the computation of V_n-polynomials and the patterns of the V_2-polynomial based on 1,701,935 knots computed. Joint work with Stavros Garoufalidis.
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Wednesday, 19 November 2025
Title: TBA
Abstract: TBA
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Wednesday, 3 December 2025
Title: TBA
Abstract: TBA
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Wednesday, 17 September 2025
Title: Hecke algebras via Morse theory of loop spaces
Abstract: Higher-dimensional Heegaard Floer homology is defined by extending Lipshitz's cylindrical reformulation to arbitrary Liouville domains. In this talk, I will present a Morse-theoretic model allowing for computations of the HDHF A∞ -algebra of cotangent bundles. We apply this model to get an explicit computation of this A∞-algebra for the cotangent bundle of the 2-dimensional sphere. The result of this computation produces a differential graded algebra which may be regarded as the derived HOMFLYPT skein algebra of the sphere. This talk is based on joint work with Ko Honda, Yin Tian, and Tianyu Yuan.
Wednesday, 20 August 2025
Title: “Did you give me a link?” - An Algorithm for Recognising Dehn Ancestryd you give me a link?” - An Algorithm for Recognising Dehn Ancestry
Abstract: Given a triangulation of a 3-manifold M, can I algorithmically decide whether M is the complement of a link? We can find an algorithm to answer this seemingly innocuous question, but to do so will require heavy machinery from topology, geometry, and number theory. In this talk, we will give an introduction to algorithms on 3-manifolds, illustrate how different tools from geometry and number theory appear, and finally give several applications of our algorithm. No prior knowledge of 3-manifolds or algorithms will be expected.
Wednesday, 6 August 2025
Title: Algebraic invariants in knot theory
Abstract: The axioms of a quandle are algebraic translations of the three Reidemeister moves in knot theory. Quandles are commonly used to construct algebraic invariants in knot theory. In the first half of the talk, I will discuss basic quandle theory followed by homology theories related to quandles and their generalizations in relation to classical knot theory. The second half of the talk will focus on algebraic invariants in the recently-introduced field of multi-virtual knot theory, a generalization of virtual knot theory. After discussing preliminary notions related to the subject, I will generalize the usual coloring invariants in classical knot theory to the multi-virtual setting using operator quandles. The talk will showcase results obtained with several co-authors.
Wednesday, 21 May 2025
Title: The intersection polynomials of a long virtual knot
Abstract: In classical knot theory it is well known that there is a one-to-one correspondence between the set of classical knots and that of long classical knots. However, the situation is quite different in virtual knot theory. In fact, Higa, Nakamura, Nakanishi and Satoh recently proved that for any virtual knot, there are infinitely many long virtual knots whose closures are the virtual knot. In this talk we study long virtual knots by introducing twelve kinds of polynomial invariants called the intersection polynomials. We give some fundamental properties of these invariants such as the product formula and a relationship with two kinds of supporting genera. We also provide a characterization of each of the intersection polynomials.
This is joint work with Takuji Nakamura, Yasutaka Nakanishi and Shin Satoh.
Wednesday, 7 May 2025
Title: Two conjectures of Poincare
Abstract: At the beginning of the 20th century, Poincare made two conjectures, one in topolog the other in number theory. While the former is well-known, the knowledge of the latter, despite its impact on number theory, such as the formulation of the Birch & Swinnerton-Dyer conjecture, is not widespread even among number theorists. The talk will focus on the impact of this conjecture of Poincare, including my own drop in the bucket.
Wednesday, 16 April 2025
Title: Skein modules of the connected sums of 3-manifolds
Abstract: Skein modules were introduced by Józef H. Przytycki and independently by Vladimir Turaev as generalisations of the 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 give the exact structure of the KBSM of the connected sum of two solid tori and show that it is isomorphic to the KBSM of a genus two handlebody modulo some specific handle sliding relations. Moreover, in recent work joint with Lê and Przytycki, we have shown that these handle sliding relations have a nice presentation in terms of Chebyshev polynomials. Finally, I will talk about recent joint work with Kim, Shi, and Wang about the KBSM of the connected sum of two copies of S^1 \times S^2.
Wednesday, 2 April 2025
Title: Volume Conjecture for double twist knots
Abstract: In this talk, the proof of the volume conjecture for double twist knots are explained. The volume conjecture predicts that the volume of the complement of a hyperbolic knot coincides with a certain limit of the colored Jones polynomial the knot. To prove this conjecture, we introduce the complexified tetrahedron, which is a deformation of the regular ideal octahedron, and then reformulate the colored Jones polynomial by the ADO invariant. At the end, to compare the limit with the volume, we use the Neumann-Zagier potential function. This method may work for two-bridge knots.
Wednesday, 19 March 2025
Title: New invariants of Linkoids and Measures of Entanglement of Open Curves in 3-space
Abstract: Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems. In this talk, we describe a novel definition of the Jones polynomial that generalises the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, it coincides with that of the corresponding link. We will also talk about new invariants of linkoids which are introduced via a surjective map between linkoids and virtual knots. This leads to a new collection of strong invariants of linkoids that are independent of any given virtual closure. We will show that invariants of linkoids give rise to a collection of novel measures of entanglement of open curves in 3-space, which are continuous functions of the curve coordinates and tend to their corresponding classical invariants when the endpoints of the curves tend to coincide.
Wednesday, 5 March 2025
Title: Numerically investigating the space writhe and rope length of knots.
Abstract: The space writhe of a knot measures how 'coiled' a knot is in 3-space, and is taken as the average writhe of a configuration of a knot from all viewing points. By performing dynamical simulations, we can model a knot as a polymer to explore the relation between the space writhe and 'standard' writhe for various configurations.
Additionally, we will discuss a method for determining upper bounds on the rope length of (nQ,Q) torus links by optimising a superhelical 'building block' of Q helices. This is joint work with Alexander Klotz.
Wednesday, 16 October 2024
Title: A simpler braid description for all links in the 3-sphere
Abstract: By Alexander's theorem, every link in the 3-sphere can be represented as the closure of a braid. Lorenz links and twisted torus links are two families that have been extensively studied and are well-described in terms of braids. In this talk, we will present a natural generalization of Lorenz links and twisted torus links that produces all links in the 3-sphere. This provides a simpler braid description for all links in the 3-sphere.
Wednesday, 18 September 2024
Title: The Unknotting Number
Abstract: Knots theory is a branch of topology that studies knots. A knot is a simple closed curve in the 3-space. The most fundamental knot in knot theory is known as the unknot. Two knots are considered equivalent if one can be made into the other through a series of deformations known as ambient isotopies. Knot Theory mainly revolves around determining whether two knots are isotopic. Many knot invariants have been introduced to tell knots apart and the ‘Unknotting Number’ is one such invariant.
The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot and the minimum is taken over all the regular projections of the knot. In this talk I will be giving a brief introduction to knot theory, knot invariants and will be discussing the unknotting number, its properties and application to the topology of biomolecules.
Wednesday, 4 September 2024
Title: On exact Lagrangian fillings of Legendrian links
Abstract: An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. One way to distinguish exact Lagrangian polytopes is by considering Newton polytopes of augmentations. We will discuss properties of Newton polytopes arising from Lagrangian fillings.
Wednesday, 21 August 2024
José Andrés Rodríguez Migueles
Title: Hyperbolicity of link complements in Seifert fibered spaces
Abstract: Let L be a link in a Seifert fibered space M over a hyperbolic 2-orbifold O that projects injectively to filling multicurve of closed geodesics G_L in O. In this talk I will show that the complement of L in M admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.
Wednesday, 15 May 2024
Justin Lanier (University of Sydney)
Title: Mapping class groups and dense conjugacy classes
Abstract: I’ll start by discussing some examples of topological groups that have dense conjugacy classes, so that there exists an element that well approximates every element in the group, up to conjugacy. I’ll also give an overview of infinite-type surfaces (those with infinite genus or infinitely many punctures) and the emerging study of their mapping class groups. One difference from the finite-type setting is that these mapping class groups come with natural non-discrete topologies. I’ll discuss joint work with Nick Vlamis where we characterize which surfaces have mapping class groups with dense conjugacy classes. I’ll also discuss our follow-up work where we do the same for finite index subgroups of these mapping class groups, as well as for homeomorphism groups of surfaces.
Wednesday, 1 May 2024
Title: Kirby Belts and the Skein Lasagna Module of S2xS2
Abstract: The skein lasagna module of a chosen link homology theory is a new kind of smooth 4-manifold invariant capable of detecting exotic phenomena. In this talk, we define and investigate a certain homotopy colimit of a directed system of tangle complexes whose homology is the skein lasagna module for $\mathfrak{gl}_{2}$ Khovanov-Rozansky homology of S^2\times B^2 with geometrically essential links in the boundary. We then use this new computational technique to compute the skein lasagna module of S^2\times S^2, and to show that the skein lasagna module of S^2\times B^2 recovers Khovanov homology for links in S^1\times S^2.
Wednesday, 17 April 2024
Title: Deformation inequivalent symplectic structures and Donaldson's four-six question
Abstract: Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.
Wednesday, 3 April 2024
Title: High-dimensional anti-surgery for Weinstein manifolds
Abstract: A Legendrian knot in the boundary of a Weinstein domain of dimension at least 6 which bounds a Lagrangian disk can be considered the boundary of the co-core of a handle. A Weinstein anti-surgery amounts to carving out this handle from the Weinstein domain. In this talk, I’ll explain an algorithm which constructs explicit handle decompositions of many of these high-dimensional Weinstein anti-surgery manifolds using a new high-dimensional Legendrian isotopy. I’ll give a specific application of this algorithm to Lazarev and Sylvan’s class of Weinstein manifolds which they called P-flexible, formed from handle attachment along P-loose Legendrians. This talk is based on work in progress with Ipsita Datta, Oleg Lazarev, and Chindu Mohanakumar.
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Wednesday, 20 March 2024
Title: Classification of Tight Contact Structures on Hyperbolic Manifolds
Abstract: The classification of contact structures is one of the important problems in contact geometry. In this talk, we will give an overview of the classification of tight contact structures. We will first discuss the general strategy and the classification of tight contact structures on Seifert fibered spaces. After that, we will talk about the classification of tight contact structures on hyperbolic 3-manifolds, particularly hyperbolic L-spaces. This is a joint work with Isacco Nonino.
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Wednesday, 6 March 2024
Title: From Generalized Kauffman-Harary Conjecture to Incompressible surface
Abstract: For a reduced alternating diagram of a knot with a prime determinant p, the Kauffman-Harary conjecture states that every non-trivial Fox p-coloring of the knot assigns different colors to its arcs. In 2022, we prove a generalization of the conjecture stated nineteen years ago by Asaeda, Przytycki, and Sikora: for every pair of distinct arcs in the reduced alternating diagram of a prime link with determinant $\delta,$ there exists a Fox $\delta$-coloring that distinguishes them. To explore the geometric approach of GKH, we study the properties of incompressible surfaces in double branched cover along the branching set L in $S^3$. We attempt to extend Mensaco's meridian theorem to double branched cover of alternating prime non-split links.
Wednesday, 1 November 2023
Title: Using nets in topology
Abstract: In analysis, sequences are an important tool, appearing in many proofs. In proofs involving general topological spaces, however, sequences rarely occur. The reason is that sequences are functions defined on a countable set, the natural numbers, and countable sets are not big enough to capture the topology of a general space.
A net is a generalization of a sequence where now the domain can be a set of any cardinality. Many proofs for metric spaces have natural generalizations to general topological spaces, where sequences are replaced by nets.
Nets were developed by Moore and Smith one century ago, but many people working in topology today are not closely familiar with them. In this expository talk, I will introduce nets from scratch and show how they can be used to prove standard results in topology, such as Tychonoff's Theorem.
Wednesday, 18 October 2023
Title: Character varieties for 2-bridge links
Abstract: I will describe an efficient technique for computing the PSL(2,C) character variety for a 2-bridge link.
Wednesday, 20 September 2023
Title: A topological characterisation of the Kashiwara-Vergne groups
Abstract: Solutions to the Kashiwara--Vergne equations are a "higher dimensional"' version of Drinfeld associators. In this talk we will describe the symmetry groups of Kashiwara-Vergne solutions as automorphisms of the (completed) wheeled props of "welded tangled foams"' - a class of knotted surfaces in R4. In addition, we will describe the graded Grothendieck-Teichmüller group as a sub-group of one of these symmetry groups.
Wednesday, 6 September 2023
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
Wednesday 24 November 2021, 12:00pm AEDT
Title: Big Mapping Class Groups
Abstract: In this talk, we will introduce infinite-type surfaces and study their associated big mapping class groups. Although infinite-type surfaces are a natural next step in the study of surface topology, they also can be found quite frequently in the wilds of mathematics: from dynamics, to descriptive set theory, to the topology of 3-manifolds. We will discuss some of these connections as well as some of my own contributions to this vibrant area of study.
Wednesday 3 November 2021, 12:00 pm AEDT
Title: Topological descriptions of protein folding
Abstract: Knotting in proteins was once considered exceedingly rare. However, systematic analyses of solved protein structures over the last two decades have demonstrated the existence of many deeply knotted proteins, and researchers now hypothesize that the knotting presents some functional or evolutionary advantage for those proteins. Unfortunately, little is known about how proteins fold into knotted configurations. In this talk, we approach this problem from a theoretical point of view, using topological techniques. In particular, based on computational and experimental evidence, we propose a new theoretical pathway for proteins to form knots. We then use topological techniques to compare the configurations obtained from the theoretical pathways with known configurations of actual proteins. This is joint work with Erica Flapan and Adam He.
Wednesday 20 October 2021, 12:00 pm AEDT
Title: Ellipsoid embedding functions of Hirzebruch surfaces
Abstract: Symplectic embeddings encode coordinate changes of physical systems, and their properties embody the dichotomy between symplectic rigidity (the features of symplectic geometry which are similar to complex geometry) and flexibility (the features of symplectic geometry which are governed solely by algebraic invariants). In 2012, McDuff and Schlenk proved that the "ellipsoid embedding function" of the ball -- a graphical illustration of the ways varying ellipsoids symplectically embed into the standard symplectic ball -- exhibits an intricate infinite staircase pattern governed by the Fibonacci numbers. We will discuss the tools used to compute the ellipsoid embedding functions of Hirzebruch surfaces, including embedded contact homology capacities and Diophantine equations, and present many new infinite staircases governed by the combinatorial structure of the Cantor set.
Wednesday 6 October 2021, 12:00 pm AEDT (Note there has been a time change in parts of Australia!)
Title: Systoles and cosmetic surgeries
Abstract: The cosmetic surgery conjecture, posed by Cameron Gordon in 1990, is a uniqueness statement that (essentially) says a knot in an arbitrary 3-manifold is determined by its complement N. In the past three decades, this conjecture has been extensively studied, especially in the setting where the knot complement N embeds into the 3-sphere. Many different invariants of knots and 3-manifolds have been applied to this problem.
After surveying some of this recent work, I will describe a recent result that uses hyperbolic methods, particularly short geodesics, to reduce the cosmetic surgery conjecture for any particular N to a finite computer search. This is joint work with Jessica Purcell and Saul Schleimer.
Wednesday 15 September 2021, 12:00 pm AEST
Title: Global homological mirror symmetry for genus two curves
Abstract: Mirror symmetry is a nontrivial duality between complex and symplectic geometries. As a complex manifold, a genus-2 curve is a hypersurface in an abelian surface. Cannizzo's thesis proved a homological mirror symmetry (HMS) result for genus two curves, with the mirror being a Landau-Ginzburg model (Y, W), Y is a locally toric Calabi-Yau 3-fold and W: Y--> \mathbb C is a symplectic fibration with a singular fiber above 0. The critical locus is a “banana” configuration of three 2-spheres with the same symplectic area; this gives rise to a 1-parameter family of symplectic structures which is mirror to a 1-parameter family of complex structures on the genus-2 curve. We extend this construction to prove a global HMS result to cover the 3-dimensional moduli space of complex structures on the genus-2 curve. Some ingredients involved include the construction of more general symplectic structures where the areas of the three 2-spheres may vary independently, as well as the computation of the monodromy of a fiber around the singularity of the fibration. This is a joint work with Haniya Azam, Catherine Cannizzo, and Chiu-Chu Melissa Liu. (I will begin the talk with a gentle introduction to HMS for a genus 1 curve, i.e. a torus, before moving on to genus 2. I will not assume the audience knows anything about toric manifolds, Calabi-Yau manifolds, mirror symmetry, derived categories, etc.) Wednesday 1 September 2021, 12:00 pm AEST
Title: Making the Jones polynomial more geometric
Abstract: The colored Jones polynomials are conjectured to detect geometric information about knot complements, such as hyperbolic volume. These relationships ("volume conjectures") are known in a number of special cases but are in general quite mysterious. In this talk I will discuss a program to better understand them by constructing holonomy invariants, which depend on both a knot K and a representation of its knot group into SL_2(C). By defining a version of the Jones polynomial that knows about geometric data, we hope to better understand why the ordinary Jones polynomial does too. Along the way we can obtain more powerful quantum invariants of knots and other topological objects.
Wednesday 18 August 2021, 12:00 pm AEST
Title: Configurations spaces of particles: homological solid, liquid, and gas
Abstract: Configuration spaces of points in the plane are well studied and the topology of such spaces is well understood. But what if you replace points by particles with some positive thickness, and put them in a container with boundaries? It seems like not much is known. To mathematicians, this is a natural generalization of the configuration space of points, perhaps interesting for its own sake. But is also important from the point of view of physics––physicists might call such a space the "phase space" or "energy landscape" for a hard-spheres system. Since hard-spheres systems are observed experimentally to undergo phase transitions (analogous to water changing into ice), it would be quite interesting to understand topological underpinnings of such transitions.
We have just started to understand the homology of these configuration spaces, and based on our results so far we suggest working definitions of "homological solid, liquid, and gas". This is joint work with a number of collaborators, including Hannah Alpert, Ulrich Bauer, Kelly Spendlove, and Robert MacPherson.
Wednesday 11 August 2021, 12:00pm AEST
Title: Invariants of 1-dimensional tangles via factorization homology
Abstract: This talk will tour through a slick method (factorization homology) of constructing, from categorical data, invariants of 1-dimensional tangles in n-manifolds. A key ingredient will be constructing an action of the orthogonal group O(n+1) on n-monoidal 1-categories with duals (note: 2-monoidal means braided-monoidal, and this orthogonal action recovers all symmetries of braided-monoidal data we are aware of). These invariants will recover the generalized Jones polynomials and Skein Modules, as well as imply the 1-dimensional tangle hypothesis. Examples along these lines will be emphasized. Interaction and input will be very welcome!
(This is a report on joint work with John Francis.)
Wednesday 4 August 2021, 12:00 pm AEST
Title: An algorithm to compute the crosscap number of a knot
Abstract: The crosscap number of a knot is the non-orientable counterpart of its genus. It is defined as the minimum of one minus the Euler characteristic of S, taken over all non-orientable surfaces S bounding the knot. Computing the crosscap number of a knot is tricky, since normal surface theory - the usual tool to prove computability of problems in 3-manifold topology, does not deliver the answer "out-of-the-box".
In this talk, I will review the strengths and weaknesses of normal surface theory, focusing on why we need to work to obtain an algorithm to compute the crosscap number. I will then explain the theorem stating that an algorithm due to Burton and Ozlen can be used to give us the answer.
This is joint work with Jaco, Rubinstein, and Tillmann.
2021 #15: Wednesday 14 July 2021, 12:00 pm AEST
Title: Generalisations of the Rips Filtration for quasi-metric spaces and asymmetric functions with corresponding stability results
Abstract: Rips filtrations over a finite metric space and their corresponding persistent homology are prominent methods in Topological Data Analysis to summarize the ``shape'' of data. For finite metric space $X$ and distance $r$ the traditional Rips complex with parameter $r$ is the flag complex whose vertices are the points in $X$ and whose edges are $\{[x,y]: d(x,y)\leq r\}$. From considering how the homology of these complexes evolves as we increase $r$ we can create persistence modules (and their associated barcodes and persistence diagrams). Crucial to their use is the stability result that says if $X$ and $Y$ are finite metric space then the bottleneck distance between persistence modules constructed by the Rips filtration is bounded by $2d_{GH}(X,Y)$ (where $d_{GH}$ is the Gromov-Hausdorff distance). Using the asymmetry we construct four different constructions analogous to the persistent homology of the Rips filtration and show they also are stable with respect to a natural generalisation of the Gromov-Hasdorff distance called the correspondence distortion distance. These different constructions involve ordered-tuple homology, symmetric functions of the distance function, strongly connected components and poset topology.
2021 #14: Wednesday 7 July 2021, 12:00 pm AEST
Title: Non-trivial smooth families of K3 surfaces
Abstract: We will show that the fundamental group of the diffeomorphism group of a K3 surface contains a free abelian group of countably infinite rank as a direct summand. Our construction relies on some deep results concerning Einstein metrics on K3, such as the global Torelli theorem. Non-triviality is detected using a families version of the Seiberg-Witten invariants.
2021 #13: Wednesday 30 June 2021, 12:00pm AEST
Title: Knot homologies and symmetry
Abstract: We will recall some classical results about knot theory and 3-manifold topology in the presence of finite-order symmetries. We’ll then talk about how some of these properties lift to, or are reflected by, more recent invariants of knots and manifolds, particularly Heegaard Floer homology, and give some intuition for where these results come from.
2021 #12: Wednesday 23 June 2021, 12:00pm AEST
Title: Constructing new partially hyperbolic diffeomorphisms
Abstract: In the study of chaotic dynamical systems, a certain class of systems known as partially hyperbolic systems has risen to prominence. These systems share many of the properties of uniformly hyperbolic systems, but whereas the known examples of uniformly hyperbolic systems are few, the weaker definition of partial hyperbolicity allows a much greater wealth of examples. In this talk, I will explain the notion of a partially hyperbolic system and introduce a new topological technique for "gluing together" already known examples of partially hyperbolic systems to produce new ones. This is joint work with Christian Bonatti, Andrey Gogolev, and Rafael Potrie.
2021 #11: Wednesday 9 June 2021, 12:00pm AEST
Title: Planar bipartite biregular degree sequences.
Abstract: A finite sequence of natural numbers is called graphical if it is the degree sequence of a simple graph (no loops, no multiple edges); similarly, a pair of sequences is called bipartite graphical if they are the degree sequences of the parts of a simple bipartite graph. While the necessary and sufficient conditions for graphicality are well known (Erdős–Gallai theorem, and respectively Gale–Ryser theorem in the bipartite case), the characterisations of planar graphical sequences and bipartite planar graphical pairs are pretty much wide open. In the talk, I will overview known results and give the characterisation of constant bipartite planar graphical pairs – it turns out that all but two pairs from among those satisfying obvious restrictions are bipartite planar.
2021 #10: Wednesday 2 June 2021, 12:00pm AEST
Title: The Thurston norm via spun normal surfaces
Abstract: The Thurston norm is a norm on the second homology of a hyperbolic 3-manifold which, given a homology class, returns the minimal complexity over all surfaces representing the class. The unit ball for this norm is a polyhedron symmetric about the origin, and by understanding the surfaces representing the vertices of this polyhedron one gets a wealth of information about the embedded surfaces in the manifold. In 2008, Cooper and Tillmann gave an algorithm for computing the Thurston norm ball of a closed manifold, using normal surfaces. I'll discuss work with Cooper and Tillmann in which we give a similar algorithm for cusped hyperbolic 3-manifolds, using spun normal surfaces. I’ll also give a demonstration of a computer program, available at https://pypi.org/project/tnorm/, that implements this algorithm and the algorithm of Cooper and Tillman, and thus computes the Thurston norm unit ball for any finite volume hyperbolic 3-manifold.
2021 #9: Wednesday 26 May 2021, 12:00pm AEST
Title: Recovering knot diagrams from triangulations
Abstract: While the study of knots originated from manipulating knot diagrams, one can also study knots by analyzing their complements. In fact, Gordon and Luecke showed two knots are equivalent if and only if their complements are homeomorphic. There are well-known procedures for constructing a knot complement from a knot diagram. We will analyze the problem from the other perspective: constructing a knot diagram from a triangulated knot complement. Specifically, we will give upper bounds for crossing number and bridge number associated to such a construction and discuss some of the challenges for a future implementation. This is joint work with Robert Haraway, Saul Schleimer and Eric Sedgwick.
2021 #8: Wednesday 12 May 2021, 12:00pm AEST
Title: Taut foliations, Dehn surgery, and Braid Positivity
Abstract: The L-space conjecture predicts a surprising relationship between the algebraic, geometric, and Floer-homological properties of a 3-manifold Y. In particular, it predicts exactly which 3-manifolds admit a "taut foliation". In this talk, I'll discuss some of my past and forthcoming work investigating these connections, with a focus towards "braid positive knots" (i.e. the knots realized as the closure of positive braids). I'll also present some applications, including obstructions to braid positivity, and a new unknot detector. Finally I'll briefly sketch a strategy for building taut foliations in manifolds obtained by Dehn surgery along knots in the three sphere. No background in foliations or Floer homology theories will be assumed. All are welcome!
Title: The adjoint Reidemeister torsion of hyperbolic 3-manifold and the vanishing identity.
Abstract: In this talk, I would like to introduce a recent conjecture on the adjoint Reidemeister torsion of hyperbolic 3-manifold. It was derived from the interaction with quantum field theory and thus its geometric or topological meaning is quite unclear so far. I would like to present some partial results and discuss some connections to the global residue theorem. Some parts of the talk are based on joint works with Dongmin Gang, Seonhwa Kim, and Joan Porti.
Title: Tropical pictures of contact manifolds and their pseudo-holomorphic fauna
Title: Infinite-type surfaces and the omnipresent arcs
Abstract: In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and $\delta$-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost! In this talk I will introduce the omnipresent arc graph and we will see that for a large collection of infinite-type surfaces, the graph is infinite-diameter and $\delta$-hyperbolic. The talk will feature a new characterization of infinite-type surfaces, which provided the impetus for this project.
This is joint work with Federica Fanoni and Alan McLeay.
Title: Fillings of contact manifolds and J-holomorphic curves
Abstract: A filling of a contact manifold is a symplectic manifold whose boundary is the contact manifold in question. The classical examples of contact manifolds, the 2n-1 dimensional sphere and the unit cotangent bundle of a closed manifold, are, by construction, filled by the 2n dimensional ball and by the cotangent disk bundle, respectively. On the other hand, overtwisted contact 3-manifolds admit no fillings by Gromov and Eliashberg. The question is then: given a contact manifold, does it admit a filling? if so, can we classify the fillings? (This problem comes in a few different flavours, since different compatibility conditions between the symplectic manifold and the contact structure on the boundary give rise to a hierarchy of notions of fillings.) J-holomorphic curves provide an important tool for approaching this problem. In particular, genus 0 curves play a particularly important role. We will discuss why these curves are so important and take an idiosyncratic tour through the literature, from work of McDuff to Etnyre and Eliashberg and finally to my work with Van Horn-Morris and Wendl on Spinal Open Book decompositions. This talk will focus on the 3 dimensional case, though I will mention a couple higher dimensional phenomena and results.
Title: A generalization of the Tits Conjecture for Artin groups
Abstract: Artin groups are a family of groups generalizing braid groups. Tits conjectured that the squares of the standard generators of an Artin group generate the "obvious" right-angled Artin group. The conjecture was proven in 2001 by Crisp and Paris. I will introduce a generalization of this conjecture, where we ask whether a larger collection of elements generates another "obvious" right-angled Artin subgroup. This alleged right-angled Artin group is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. I will discuss some classes of Artin groups that we can prove it for, and give some applications. This is joint work with Kevin Schreve.
Title: Infinitely many Lagrangian fillings
Abstract: An interesting problem in contact topology is to understand the Lagrangian surfaces that "fill" a given Legendrian knot or link in a contact 3-manifold. A key breakthrough in the past year or so has been the discovery that some families of Legendrian links have infinitely many different fillings. There are now a variety of approaches to proving results along these lines, using techniques from microlocal sheaf theory, cluster algebras, and Floer theory. I'll focus on this last approach, and describe a concrete way to construct Legendrian links with infinitely many fillings that can be distinguished using Legendrian contact homology. This is joint work with Roger Casals
Title: Butterfly Surfaces and Strongly Invertible Knots
Abstract: A standard measure of the complexity of a knot K in S^3 is its 3-genus (or 4-genus) - the minimal genus of a surface in S^3 (or B^4) with boundary K. Since a larger genus surface may be found by attaching handles to a minimal genus surface, the genus completely characterizes which surfaces (up to homeomorphism) can have boundary K. It is more complicated to classify equivariant surfaces (up to equivariant homeomorphism) bounding a knot with an involution since surfaces with involutions are not classified simply by their genus. In this talk I will give obstructions to a strongly invertible knot bounding an equivariant surface with a separating fixed arc, even though every strongly invertible knot bounds an equivariant surface. This is joint work with Ahmad Issa.
#29 - Wednesday 2 December 2020, 12:00 AEDT (Hosted by Tillus)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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
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)
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)
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)
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)
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)
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