# NCNGT 2020

### Floer theory and low-dimensional topology: Matthew Stoffregen (MIT) and Linh Truong (IAS).

Connections to contact and symplectic geometry:

Ascending surfaces, complex curves, and knot Floer homology- Kyle Hayden (Columbia University)

Abstract: Ascending surfaces are a natural class of surfaces in certain symplectic 4-manifolds that behave like complex curves, but they are much more flexible. I'll present a key structural result about these surfaces and highlight two applications: one to the study of complex curves, the other to recent results of Juhasz-Miller-Zemke regarding the transverse invariant in knot Floer homology.

Right-veering open books and the Upsilon invariant- Diana Hubbard (Brooklyn College, CUNY)

Abstract: Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss an application of this work to the Slice-Ribbon conjecture.

GRID invariants obstruct decomposable Lagrangian cobordisms- C.-M. Michael Wong (Louisiana State Unversity)

Abstract: Ozsvath, Szabo, and Thurston defined several combinatorial invariants of Legendrian links in the 3-sphere using grid homology, which is a combinatorial version of link Floer homology. These, collectively called the GRID invariants, are known to be effective in distinguishing some Legendrian knots that have the same classical invariants. In this talk, we show that the GRID invariants provide an obstruction to the existence of decomposable Lagrangian cobordisms between Legendrian links. This obstruction is stronger than the obstructions from the Thurston-Bennequin and rotation numbers, and is closely related to a recent result by Golla and Juhasz. This is joint work with John Baldwin and Tye Lidman.

Gauge theory:

Equivariant singular instanton homology, I: Applications to 4D clasp numbers-Aliakbar Daemi (Washington University in St Louis)

Abstract: Any knot K is the boundary of a normally immersed disc in the 4-ball, and the 4D clasp number of K is the smallest number of double points of any such immersed disc. The 4D clasp number of K is bounded below by the slice genus of K. Motivated by the Caporaso-Harris-Mazur conjecture about algebraic curves in a quintic surface, Kronheimer and Mrowka asked whether the difference between the 4D clasp number and the slice genus can be arbitrarily large. In this talk I will introduce a knot invariant, called Gamma, and review some of its properties. Then I explain how this invariant can be used to answer Kronheimer and Mrowka's question. This is joint work with Chris Scaduto.

Equivariant singular instanton homology, II: Introduction to the constructions- Chris Scaduto (University of Miami)

Abstract: The Gamma-invariant introduced in part I of this series is one of several outputs from equivariant singular instanton theory. This framework associates to a knot a suite of invariants which are morally derived from Morse theory of the Chern-Simons functional on an infinite-dimensional space with a circle action. After some general background, I will focus on the algebraic structures that are forefront to the theory. This is joint work with Ali Daemi.

Equivariant singular instanton homology, III: Singular Froyshov invariants and the Gamma-invariant--Aliakbar Daemi (Washington University in St Louis)

Abstract: A homology concordance is an embedded cylinder in a homology cobordism. Equivariant singular instanton homology is functorial with respect to homology concordances. We use this to produce a family of homology concordance invariants. The simplest elements of this family are integer valued invariants which are obtained by imitating the definition of the Froyshov homomorphisms in the context of three manifold invariants. The Gamma invariant used in the first talk is a refinement of these singular Froyshov invariants. This is joint work with Chris Scaduto.

Equivariant singular instanton homology, IV: Further applications- Chris Scaduto (University of Miami)

Abstract: In this final talk I will present additional applications of the theory discussed in the previous parts of the series. One application is that certain topological assumptions imply the existence of non-trivial SU(2) representations for fundamental groups of knotted surface complements. Another application confirms a conjecture of Poudel-Saveliev and computes the irreducible mod 4 graded instanton homology of torus knots. This is joint work with Ali Daemi.

Framed Instanton Floer homology revisited via sutures- Zhenkun Li (MIT)

Abstract: Framed Instanton Floer homology was introduced by Kronheimer and Mrowka for closed oriented 3-manifolds. It is conjectured to be isomorphic to the hat version of Heegaard Floer homology, and recently many computational results were achieved by several groups of people. In this talk, I will explain how the framed Instanton Floer homology of a closed oriented 3-manifold Y can be related to the sutured instanton Floer homology of the complement of a torsion knot inside Y with some suitable sutures. Then several applications follow. This is partially jointed with Sudipta Ghosh and C.-M. Michael Wong.

SL(2,C) Floer Homology, I: The 3-manifold invariant HP(Y)- Ikshu Neithalath (UCLA)

Abstract: We will construct HP(Y), the SL(2,C) Floer homology of a 3-manifold Y, as defined by Abouzaid and Manolescu. To do so, we will give a brief overview of the necessary algebro-geometric tools such as character varieties and perverse sheaves of vanishing cycles.

SL(2,C) Floer Homology, II: Invariants for Knots- Ikshu Neithalath (UCLA)

Abstract: We will sketch the construction of SL(2,C) Floer homology for knots as defined by Cote and Manolescu. We will then discuss some joint work with Cote on the properties of the knot invariant as well as some independent work computing the 3-manifold invariant for surgeries on knots.

Heegaard Floer homology

Corks, Involutions, and Heegaard Floer Homology- Irving Dai (MIT)

Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. As an application, we define a modification of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits an infinitely-generated subgroup of strongly non-extendable corks. We establish several new families of corks and prove that various known examples are strongly non-extendable. This is joint work with Matthew Hedden and Abhishek Mallick. This talk will be complementary to the talk given by Abhishek Mallick (in the "Knots, Surfaces, and 4-Manifolds" topic group), and will discuss the Heegaard Floer theory underlying various detection results.

Knot Floer homology and relative adjunction inequalities- Katherine Raoux (Michigan State University)

Abstract: In this talk, we present a relative adjunction inequality for 4-manifolds with boundary. We begin by constructing generalized Heegaard Floer tau-invariants associated to a knot in a 3-manifold and a nontrivial Floer class. Given a 4-manifold with boundary, the tau-invariant associated to a Floer class provides a lower bound for the genus of a properly embedded surface, provided that the Floer class is in the image of the cobordism map induced by the 4-manifold. We will also discuss several applications to links and contact manifolds. This is joint work with Matt Hedden.

Involutive Heegaard Floer homology and surgeries- Ian Zemke (Princeton University)

Abstract: In this talk, we investigate the behavior of involutive Heegaard Floer homology with respect to surgeries. We prove several surgery exact sequences, and also a mapping cone formula, for involutive Heegaard Floer homology. In this talk, we will describe a few elements of the proof, and also some applications to the homology cobordism group. This project is joint work in progress with Kristen Hendricks, Jen Hom and Matt Stoffregen.

Non-positively curved groups:

Actions of big mapping class groups on the arc graph- Carolyn Abbott (Columbia)

Abstract: Given a finite-type surface (i.e. one with finitely generated fundamental group), there are two important objects naturally associated to it: a group, called the mapping class group, and an infinite-diameter hyperbolic graph, called the curve graph. The mapping class group acts by isometries on the curve graph, and this action has been extremely useful in understanding the algebraic and geometric properties of mapping class groups. One particularly important class of elements of the mapping class group are those which act as loxodromic isometries of the curve graph; these are called “pseudo Anosov” elements. Given an infinite-type surface with an isolated puncture, one can associate two analogous objects: the big mapping class group and the (relative) arc graph. In this talk, we will consider the action of big mapping class groups on the arc graph, and, in particular, we will construct an infinite family of “infinite-type” elements that act as loxodromic isometries of the arc graph, where an infinite-type element is (roughly) one which is not supported on any finite-type subsurface. This is joint work with Nick Miller and Priyam Patel.

Morse Quasiflats- Jingyin Huang (Ohio State)

Abstract: We introduce the notion of Morse quasiflats, which is a common generalization of Morse quasigeodesics and quasiflats of top-rank. In the talk, we will provide motivations and examples for Morse quasiflats, as well as a number of equivalent definitions and quasi-isometric invariance (under mild assumptions). We will also show that Morse quasiflats are asymptotically conical, and provide several first applications. Based on joint work with B. Kleiner and S. Stadler.

Effective ping-pong in CAT(0) cube complexes- Kasia Jankiewicz (University of Chicago)

Abstract: Ping-pong lemma is a useful tool for finding elements in a given group that generate free semigroups. We use it to prove uniform exponential growth of certain groups acting on CAT(0) cube complexes. We also construct examples of groups with arbitrarily large gap between their cohomological and cubical dimensions. This is partially joint work with Radhika Gupta and Thomas Ng.

Mod p and torsion homology growth in nonpositive curvature- Kevin Schreve (University of Chicago)

Abstract: We compute the growth of mod p homology in finite index normal subgroups of right-angled Artin groups. We give examples where it differs from the rational homology growth, find such a group with exponential torsion growth, and for odd primes p construct closed locally CAT(0) manifolds with nontrivial mod p homology growth outside the middle dimension.

Bounded cohomology and norms on groups:

Addition of geometric volume classes- James Farre (Yale)

Abstract: We study the algebraic structure of three dimensional bounded cohomology generated by volume classes for infinite co-volume, finitely generated Kleinian groups. While bounded cohomology is generally unwieldy, we show that addition admits a natural geometric interpretation for the volume classes of tame hyperbolic manifolds of infinite volume and bounded geometry: the volume classes of singly degenerate manifolds sum to the volume classes for manifolds with many degenerate ends. It turns out that this generates the linear dependencies among volume classes, giving a complete description of the algebraic structure of some geometrically defined subspaces of bounded cohomology. We will indicate some problems left open by this discussion and give some suggestions for future directions. Definitions, background, and geometric aspects of hyperbolic manifolds homotopy equivalent to a closed surface will be reviewed, but we assume some familiarity with hyperbolic geometry.

The Spectrum of Simplicial Volume- Nicolaus Heuer (Cambridge)

Abstract: Simplicial volume was first introduced by Gromov to study the minimal volume of manifolds. Since then it has emerged as an active research field with a wide range of applications. In dimensions two and three, the set of possible values of simplicial volume may be fully computed using geometrization, but is hardly understood in higher dimensions. In joint work with Clara Löh (University of Regensburg), we show that the set of simplicial volumes in higher dimensions is dense in the non-negative reals. I will also discuss how the exact set of simplicial volumes in dimension four or higher may look like.

Computability of the Minimal Genus on Second Homology- Thorben Kastenholz (University of Bonn)

Abstract: Surface representatives of second homology classes can be used to give geometric invariants for second homology classes, the most prominent examples are the genus and the Euler characteristic. In this talk I will explain why determining the minimal genus of a given homology class is in general undecidable, and how to compute it for a large class of "negatively-curved" spaces including 2-dimensional CAT(-1)-complexes. This will need a normal Form result proven by me and Mark Pedron, that extends a theorem by Edmonds on maps between surfaces.

Quasimorphisms on diffeomorphism groups- Richard Webb (University of Manchester)

Abstract: I will explain how to construct an unbounded quasimorphism on the group of isotopically-trivial diffeomorphisms of a surface of positive genus. As a corollary the commutator length and fragmentation norm are both (stably) unbounded, which solves a problem of Burago--Ivanov--Polterovich. The proof uses a new hyperbolic graph on which these groups act by isometries, which is inspired by techniques from mapping class groups. This is joint work with Jonathan Bowden and Sebastian Hensel.

Homeomorphism and mapping class groups:

Surjective homomorphisms from surface braid groups- Lei Chen (Caltech)

Abstract: In this talk, I will present that any surjective homomorphism from the surface braid group to a torsion-free hyperbolic group factors through a forgetful map. This extends and gives a new proof of an earlier result of the author which works only for free groups and surface groups.

Inverse limits of covering spaces- Curtis Kent (Brigham Young University)

Abstract: Many techniques that have proved effective in the study of finitely generated groups at first approach seem insufficient for the study of uncountable groups, e.g. big mapping class groups or fundamental groups of complicated spaces. We will discuss how to use inverse limits to study such groups through approximation by finitely generated groups. Specifically we will show that inverse limits of covering spaces, when path-connected, satisfy a Galois type correspondence and how this leads to a splitting of the first homology for many spaces.

Universal bounds for torsion generating sets of mapping class groups- Justin Lanier (Georgia Tech)

Abstract: I showed with Margalit that the conjugacy class of any periodic element is a generating set for Mod(S_g), as long as g is at least 3 and the element isn’t trivial or a hyperelliptic involution. Since Mod(S_g) is finitely generated, these infinite torsion generating sets can always be whittled down to finite generating sets. Is there a universal upper bound on the number of conjugates required, independent of g and the element chosen? In general, the answer is no. However, for elements with order at least 3 there is a universal upper bound; our proof shows that 60 conjugates always suffice. I will record two short talks: a first talk that describes the history and context of this result and that gives an overview of the proof strategy, and then a second talk that goes into some of the nuances of the proof.

Analogs of the curve graph for infinite type surfaces- Alexander Rasmussen (Yale)

Abstract: The curve graph of a finite type surface is a crucial tool for understanding the algebra and geometry of the corresponding mapping class group. Many of the applications that arise from this relationship rely on the fact that the curve graph is hyperbolic. We will describe actions of mapping class groups of infinite type surfaces on various graphs analogous to the curve graph. In particular, we will discuss the hyperbolicity of these graphs, some of their quasiconvex subgraphs, properties of the corresponding actions, and applications to bounded cohomology.

The 'what' and 'why' of framed mapping class groups- Nick Salter (Columbia)

Abstract: The 'what': a framed mapping class group is the stabilizer of an isotopy class of vector field on a surface. Despite being infinite-index subgroups, Aaron Calderon and I have shown that these are generated by simple collections of finitely many Dehn twists. The 'why': Many natural families of Riemann surfaces are also equipped with the extra data of a vector field. This is true of families of translation surfaces (strata), as well as the family of Milnor fibers of an isolated plane curve singularity. Understanding the behavior of these families thus requires an understanding of the framed mapping class group. Work on the framed mapping class group and on strata is joint with Aaron Calderon, and work on plane curve singularities is joint with Pablo Portilla Cuadrado.

Convex projective geometry:

Codimension-1 Flats in Convex Projective Geometry- Martin Bobb (Michigan)

Abstract: Convex projective manifolds generalize hyperbolic manifolds while allowing for some similarities to non-positively curved spaces, and some interesting deformation theory. In this lecture we will discuss the structure of codimension-1 flats in compact convex projective manifolds.

Rank one phenomena in convex projective geometry- Mitul Islam (Michigan)

Abstract: The goal of this talk is to develop analogies between rank one non-positive curvature/ CAT(0) and convex projective geometry. We will introduce the notion of rank one automorphisms of properly convex domains and characterize them as contracting group elements. We will prove that a discrete rank one automorphism group is either virtually cyclic or acylindrically hyperbolic. This leads to some applications like computation of space of quasimorphisms, counting of closed geodesics, and genericity results.

Moduli space of unmarked convex projective surfaces- Zhe Sun (Luxembourg)

Abstract: Mirzakhani found a beautiful recursive formula to compute the volume of the moduli space of Riemann surfaces. We discuss the possible similar recursive formula where the Riemann surfaces are replaced by the convex projective surfaces. We investigate the boundedness of projective invariants, area, and many other notions that are uniformly related to each other and we show one of these bounded subsets has polynomially bounded Goldman symplectic volume.

Dynamics of a representation:

Proper actions on symplectic groups and their Lie algebras- Jean-Philippe Burelle (Sherbrooke)

Abstract: Danciger-Guéritaud-Kassel developed a theory of proper actions on PSL(2,R) (anti-de Sitter space) and its Lie algebra sl(2,R) (Minkowski space) using length contraction/expansion properties. They applied this machinery to obtain several results on the structure of anti-de Sitter and flat Lorentzian manifolds in dimension 3, including proofs of the tameness of Margulis space times and of the crooked plane conjecture. I will show how part of this theory generalizes to symplectic groups Sp(2n,R) and their Lie algebras sp(2n,R). This is joint work with Fanny Kassel.

Growth of quadratic forms under Anosov subgroups- León Carvajales (Universidad de la República & Sorbonne)

Abstract: For positive integers p and q we define a counting problem in the (pseudo-Riemannian symmetric) space of quadratic forms of signature (p,q) on R^{p+q}. This is done by associating to each quadratic form o a totally geodesic copy of the Riemannian symmetric space of PSO(o) inside the Riemannian symmetric space of PSL_{p+q}(R), and by looking at the orbit of this geodesic copy under the action of a discrete subgroup of PSL_{p+q}(R). We then present some contributions to the study of this counting problem for Anosov subgroups of PSL_{p+q}(R).

Topological dynamics of the Weyl chamber flow- Nguyen-Thi Dang (Heidelberg)

Abstract: Let G be a connected, real linear, semi-simple Lie group without compact factors and D be a discrete subgroup. Let K be a maximal compact subgroup, A a Cartan subgroup for which Cartan decomposition holds. Consider an action D\G on the right by a one-parameter subgroup of A. As a family of topological dynamical systems, when do all orbits diverge ? What are the interesting sets for the dynamics ? Are there dense orbits in those sets ? Is there topological mixing ? When D is a cocompact lattice, there are no diverging orbits and every flow is exponentially mixing by a result of Howe-Moore. Assume now that D is only Zariski dense in G. For SO(n,1)⁰, the corresponding flow is the frame flow which factors over the geodesic flow of a hyperbolic manifold. Mixing properties of such flows in convex-cocompact hyperbolic manifolds were obtained by Winter in 2016 and improved recently by Winter-Sarkar. In 2017, Maucourant-Schapira proved topological mixing when D is Zariski dense. In this talk, I'll focus in the case where G is a higher rank split simple Lie group and D a Zariski dense subgroup. The existence of Weyl chambers in A allows to define regular Weyl chamber flows as those parametrised by an element of the interior a Weyl chamber. The Benoist cone, which contains all the information about the spectrum of D will then give us a necessary condition for the existence of non-diverging orbits. Finally I'll explain a topological mixing criteria for regular flows.

Limits of subgroups:

Chabauty Limits of the Diagonal Subgroup in SL(n,Q_p)- Arielle Leitner (Weizmann)

Abstract: A conjugacy limit group is a limit of a sequence of conjugates of the positive diagonal subgroup C in SL(n) in the Chabauty topology. In low dimensions, there are finitely many limits up to conjugacy, and we explain why there are more limits over Q_p than over R. In higher dimensions there are infinitely many limits up to conjugacy. We can understand limits of C by understand how to go to infinity in the building (you won't need to know what a building is for this talk, we'll explain the geometry with low dimensional examples). We use the geometry of the building to classify limits of C. This is joint work with C. Ciobotaru and A. Valette. The hidden agenda of this talk is to convince you that Q_p is friendly, and things that we do over R and C can work over Q_p as well.

Sequences of Hitchin representations of Tree-Type- Giuseppe Martone (Michigan)

Abstract: Let S be an oriented surface of genus greater than 1. The Teichmuller space of S can be described as a connected component of the space of representations of the fundamental group of S into the Lie group PSL(2,R). The Hitchin component generalizes this classical picture to the Lie group PSL(d,R). Hitchin representations are a prominent subject of study in the field of Higher Teichmuller theory. Motivated by classical work of Thurston, one wishes to understand the asymptotic behavior of sequences of Hitchin representations. In this talk we describe non-trivial sufficient conditions on a diverging sequence of Hitchin representations so that its limit can be described as an action on a tree. In other words, we single out sequences whose asymptotic behavior is similar to diverging sequences in the Teichmuller space. Our non-trivial conditions are given in terms of Fock-Goncharov coordinates on moduli spaces of positive tuples of flags.

Curves and surfaces:

Counting simple closed multi-geodesics on hyperbolic surfaces with respect to the lengths of individual components- Francisco Arana-Herrera (Stanford)

Abstract: In her thesis, Mirzakhani showed that on any closed hyperbolic surface of genus g, the number of simple closed geodesics of length at most L is asymptotic to a polynomial in L of degree 6g-6. Wolpert conjectured that analogous results should hold for more general countings of multi-geodesics that keep track of the lengths of individual components. In this talk we will present a proof of this conjecture which combines techniques and results of Mirzakhani as well as ideas introduced by Margulis in his thesis.

Cutting and pasting along measured laminations- Aaron Calderon (Yale)

Abstract: One of the fundamental techniques of low-dimensional topology is cutting and pasting along embedded codimension 1 submanifolds. In surface theory these submanifolds are just simple closed curves, and cutting and pasting gives rise to Fenchel-Nielsen coordinates for Teichmüller space, normal forms for simple closed curves, and many other foundational results. The set of simple closed curves completes to the space of “measured laminations,” and in my first talk, I will summarize how to build coordinates for Teichmüller space by cutting and pasting along any lamination, generalizing the shear coordinates of Bonahon and Thurston. In my second talk, I will explain how these coordinates lead to an extension of Mirzakhani’s conjugacy between the earthquake and horocycle flows, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. This represents joint work with James Farre.

Volumes and filling collections of multicurves- Andrew Yarmola (Princeton)

Abstract: Consider a link L in a Seifert-fibered space N over a surface S of negative Euler characteristic. If the fiber-wise projection of L to S is a collection C of closed curves in minimal position, then N \ L is hyperbolic if and only if C is filling and N \ L is acylindrical. The behavior of vol(N \ L) in terms of the topology and geometry of C have been studied in recent years, but effective lower bounds have been elusive. In this talk we will focus on the case where C is a collection of simple closed curves. In the special case where N = PT(S) is the projectivized tangent bundle and L is the canonical lift of a pair of filling multicurves, we show that vol(N \ L) is quasi-isometric to the Weil-Petersson distance between the corresponding strata in Teichmuller space. In the more general setting, we show that vol(N \ L) is quasi-isometric to expressions involving distances in the pants graph whenever L is a stratified link. This is joint work with T. Cremaschi and J. A. Rodriguez-Migueles.

Arithmetic manifolds:

Embedded totally geodesic submanifolds in small volume hyperbolic manifolds- Michelle Chu (University of Illinois, Chicago)

Abstract: The smallest volume cusped hyperbolic 3-manifolds are arithmetic and contain many immersed but not embedded closed totally geodesic surfaces. In this talk we discuss nonexistence of codimension-1 closed embedded totally geodesic submanifolds in small volume hyperbolic manifolds of higher dimensions. This is joint work with Long and Reid.

Arithmetic manifolds and their geodesic submanifolds- Nicholas Miller (University of California Berkeley)

Abstract: It is a well known consequence of the Margulis dichotomy that when an arithmetic hyperbolic manifold contains one totally geodesic hypersurface, it contains infinitely many. Both Reid and McMullen have asked conversely whether the existence of infinitely many geodesic hypersurfaces implies arithmeticity of the corresponding hyperbolic manifold. In this talk, I will discuss recent results answering this question in the affirmative. In particular, I will describe how this follows from a general superrigidity style theorem for certain natural representations of fundamental groups of hyperbolic manifolds. I will also discuss a recent extension of these techniques into the complex hyperbolic setting, which requires the aforementioned superrigidity theorems as well as some theorems in incidence geometry. This is joint work with Bader, Fisher, and Stover.

Deligne-Mostow lattices and line arrangements in complex projective 2-space- Irene Pasquinelli (Institut de Mathématiques de Jussieu, Paris)

Abstract: In 1983, Hirzebruch considers some arrangements of complex lines in complex projective 2-space. Then he shows that a suitable branched cover ramified along the line arrangement is a complex hyperbolic manifold. This manifold turns out to be one of the well known Deligne-Mostow lattices. In the first part of this talk I will introduce you to the complex hyperbolic space, to its group of isometries and to the Deligne-Mostow lattices. Then I will tell you about Hirebruch's construction. In the second part, I will first explain how Hirzebruch's work has been generalised by Bartel, himself and Hoefer to all of the Deligne-Mostow lattices. Then I will tell you how, in a joint work with Elisha Falbel, we created an explicit dictionary between line arrangements and fundamental domains for the lattices. I will also explain one of the applications of this. In fact, we use this result to contribute to the problem of creating a complex analogue to the hybridisation construction.

Triangulations and complexity:

Stable commutator length in graphs of groups- Lvzhou (Joe) Chen (University of Chicago)

Abstract: The stable commutator length (scl) is a relative version of the Gromov-Thurston norm. For a given null-homologous loop L in a space X, its scl is the infimal complexity of surfaces bounding L, measured in terms of Euler characteristics. Surfaces realizing the minimal complexity are called extremal. They are pi_1-injective, and can only exist when scl is rational. We show that scl takes rational values for all loops in a space X if pi_1(X) is certain graphs of groups, inclusing Baumslag-Solitar groups. Moreover, there is a linear programming algorithm to compute scl.

Flows, Thurston norm, and surfaces: homology to isotopy- Michael Landry (Washington University)

Abstract: Let M be a closed hyperbolic 3-manifold. We begin by reviewing a classical picture due to Thurston, Fried, and Mosher in which a single pseudo-Anosov flow organizes the data of a fibered face F of the Thurston norm ball of M as well as certain nice surface representatives of homology classes lying in the cone over F. We then announce a new theorem which strengthens the above by collating all isotopy classes of incompressible surfaces representing classes in the cone over F. We explain that the result follows from a more general theorem, involving veering triangulations, which also applies to other faces of the Thurston norm ball (possibly nonfibered, possibly lower-dimensional).

Dehn filling and knot complements that irregularly cover- William Worden (Rice University)

Abstract: It is a longstanding conjecture of Neumann and Reid that exactly three knot complements can irregularly cover a hyperbolic orbifold---the figure-8 knot and the two Aitchison--Rubinstein dodecahedral knots. This conjecture, when combined with work of Boileau--Boyer--Walsh, implies a more recent conjecture of Reid and Walsh, which states that there are at most 3 knot complements in the commensurability class of any hyperbolic knot. We give a Dehn filling criterion that is useful for producing large families of knot complements that satisfy both conjectures. The work we will discuss is partially joint with Hoffman and Millichap, and partially joint with Chesebro, Deblois, Hoffman, Millichap, and Mondal.

Volume:

Geodesics on hyperbolic surfaces and volumes of link complements in Seifert-fibered spaces- José Andrés Rodriguez Migueles (University of Helsinki)

Abstract: Let Γ be a link in a Seifert-fibered space M over a hyperbolic surface Σ that projects injectively to a collection of closed geodesics γ in Σ. When γ is filling, the complement of Γ in M admits a hyperbolic structure of finite volume. We give bounds of its volume in terms of topological invariants of (γ,Σ).

Uniform models for random 3-manifolds- Gabriele Viaggi (University of Heidelberg)

Abstract: As discovered by Thurston, hyperbolic 3-manifolds are abundant among all 3-manifolds. In many examples, the generic element in a family of 3-manifolds sharing a common combinatorial description admits such a hyperbolic structure. The family of random 3-manifolds (Dunfield and Thurston model) is one of these examples. The existence of a hyperbolic metric on such random objects has been established by Maher, exploiting the solution of the Geometrization conjecture by Perelman. In this talk, I will describe a more constructive approach to this result and give an explicit construction for the metric that only uses tools from the deformation theory of Kleinian groups. The metric obtained is explicit enough to allow the computation of geometric invariants such as volume and diameter. Joint with Peter Feller and Alessandro Sisto.

Hyperbolic Limits of Cantor sets complements in the Sphere- Franco Vargas Pallete (Yale University)

Abstract: In this talk we will show that if M is a hyperbolic manifold that embeds in S3 with no Z2 in \pi_1, then M can be approximated (in the geometric sense) by hyperbolic metrics on Cantor set complements in S3. This is joint work with Tommaso Cremaschi.

### Investigating the L-space conjecture: Jonathan Johnson (UT Austin), Siddhi Krishna (Boston College/ GA Tech/ Columbia), and Hannah Turner (UT Austin).

Through Floer homology:

From Floer homology to spectral theory, and hyperbolic geometry- Francesco Lin (Columbia)

Abstract: In the first part of the talk, I will review some spectral theory of three manifolds, and discuss its relation with Floer homology, and in particular L-spaces. In the second part of the talk, I will discuss how spectral theory on a hyperbolic three manifold can be understood in terms of natural geometric quantities. This is joint work with M. Lipnowski.

On the monopole Lefschetz number of finite order diffeomorphisms- Jianfeng Lin (UC San Diego)

Abstract: Let K be a knot in an integral homology 3-sphere Y, and Σ the corresponding n-fold cyclic branched cover. Assuming that Σ is a rational homology sphere (which is always the case when n is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of Σ. Our formula is motivated by a Witten-style conjecture relating the two gauge theoretic invariants of homology S1 cross S3s (the Furuta-Ohta invariant and the Casson-Seiberg-Witten invariant). As applications, we give a new obstruction (in terms of the Jones polynomial) for the branched cover of a knot in S3 being an L-space and we define a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers. This is a joint work with Danny Ruberman and Nikolai Saveliev.

L-spaces in instanton Floer homology- Steven Sivek (Imperial)

Abstract: Framed instanton homology

is a gauge-theoretic invariant which appears to coincide with the hat version of Heegaard Floer homology. Inspired by the notion of a Heegaard Floer L-space, we say that a rational homology sphere Y is an “instanton L-space" if the rank of is as small as possible, namely . In this talk I’ll summarize what is known about instanton L-spaces, especially those which arise as Dehn surgeries on knots in , and what this tells us about fundamental groups, A-polynomials of knots, and the L-space conjecture. Various parts of this are joint with Antonio Alfieri, John Baldwin, Irving Dai, and Raphael Zentner.

With Dehn Surgery:

L-space knots do not have essential Conway spheres- Tye Lidman (NC State)

Abstract: The properties of a knot are heavily governed by the essential surfaces that sit in the exterior. We will study a relation between essential planar surfaces in a knot exterior and knot Floer homology. This is joint work with Allison Moore (VCU) and Claudius Zibrowius (UBC).

Introduction to L-space links- Beibei Liu (Bonn)

Abstract: L-spaces are simplest 3-manifolds in terms of Heegaard Floer homology and L-space links are links such that all large surgeries are L-spaces. In this talk, we will concentrate 2-component L-space links which is a family of ``simple” links in the sense that their Alexander polynomials contain full information of the link Floer complex, and give explicit answers to questions relating to the link itself and its surgeries such as some detection results, sharp slice genus bounds and Thurston polytope.

Fibred knots, positivity and L-spaces- Filep Misev (Regensburg)

Abstract: Torus knots are lens space knots: they admit surgeries to lens spaces. This classical theorem has a modern analogue in terms of Floer homology: algebraic knots are L-space knots. I will present knots which do not admit L-space surgeries despite strikingly resembling algebraic knots and L-space knots in general. More precisely, we will see a method which allows to construct infinite families of knots of arbitrary fixed genus g > 1 which are all algebraically concordant to the torus knot T(2,2g+1) of the same genus and which are fibred and strongly quasipositive. Besides the study of L-spaces, these knots are of interest in the context of knot concordance, in particular Fox's slice-ribbon question, as well as Boileau-Rudolph's question, or Baker's conjecture, on the independence of strongly quasipositive fibred knots in the concordance group. Joint work with Gilberto Spano.

Non-left-orderable surgeries on iterated 1-bridge braids- Zipei Nie (Princeton)

Abstract: We prove that the L-space conjecture holds for those L-spaces obtained from Dehn surgery on knots which are closures of iterated 1-bridge braids, i.e., the braids obtained from satellite operations on 1-bridge braids. In the proof, we emphasize the power of fixed points in the Homeo_+(R) representation, and introduce property (D) to handle the satellite operation.

Via orderability:

PSL(2,R) representations and left-orderablility of Dehn filling- Xinghua Gao (KIAS)

Constructing non-trival PSL(2,R) representations has been proven to be a useful tool for showing the left-orderablity of a 3-manifold group. In this talk, I will show how to use representations of the fundamental group of a knot complement to determine which Dehn filling of it has left-orderable fundamental group. In particular, I will compute slopes of left-orderable Dehn filling of a class of two-bridge knots as an example.

Promoting circular-orderability to left-orderability- Ty Ghaswala (UQAM)

I will present new necessary and sufficient conditions for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. I will then show how newly developed machinery can help us compute the obstruction spectrum in a variety of examples, including 3-manifold groups relevant to the L-space conjecture, and mapping class groups. I will finish the talk with new progress in answering the question of when the direct product of two circularly-orderable groups is circularly-orderable, a fundamental question which is frustratingly difficult to say anything about.

Left-orderability of 3-manifold groups and foliations of 3-manifolds- Ying Hu (Nebraska)

In this talk, we will discuss how the existence of certain nice dimension 1 and dimension 2 foliations of 3-manifolds can lead to the left-orderability of their fundamental groups. We will give some applications of these observations to cyclic branched covers of a knot. Limitations of the techniques will also be discussed. This is joint work with Steve Boyer and Cameron Gordon.

### Knots, surfaces, and 4-manifolds: Maggie Miller (Princeton University) and JungHwan Park (GA Tech).

Knots and concordances:

Unknotting with a single twist- Samantha Allen (Dartmouth College)

Abstract: Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. With this, a wider range of tools become available, including classical invariants and invariants arising from Heegaard Floer theory. Using these tools, if a knot can be unknotted with a single twist of linking number , we give restrictions on the genus, signature function, Upsilon function, and and invariants of in terms of and the sign of the twist. In this talk, I will discuss some of these restrictions, their implications, and some unanswered questions. This talk is based on joint work with Charles Livingston.

Characterising homotopy ribbon discs- Anthony Conway (MPIM Bonn)

Abstract: After reviewing some notions from knot concordance, we explore the following question: how many slice discs does a slice knot admit? This is joint work with Mark Powell.

Fox-Milnor Conditions for 1-solvable Knots and Links- Shawn Williams (Rice University)

Abstract: A well known result of Fox and Milnor states that the Alexander polynomial of slice knots factors as f(t)f(t-1) providing us with a useful obstruction to a knot being slice. In this talk, I will present a generalization of this result to certain localized Alexander polynomials of 1-solvable boundary links, and first order Alexander polynomials of 1-solvable knots.

Knotted surfaces:

Exotic slice disks and symplectic surfaces- Kyle Hayden (Columbia University)

Abstract: One approach to understanding the smooth topology of a 4-dimensional manifold is to study the embedded surfaces it contains. I'll construct new examples of "exotically knotted" surfaces in 4-manifolds, i.e. surfaces that are isotopic through ambient homeomorphisms but not through diffeomorphisms. We'll begin with simple examples of exotic disks in the 4-ball. Then we'll turn to the symplectic setting and exhibit new types of exotic phenomena among symplectic, holomorphic, and Lagrangian surfaces.

Doubly slice pretzels- Clayton McDonald (Boston College)

Abstract: A knot K in S^3 is slice if it is the cross section of an embedded sphere in S^4, and it is doubly slice if the sphere is unknotted. Although slice knots are very well studied, doubly slice knots have been given comparatively less attention. We prove that an odd pretzel knot is doubly slice if it has 2n+1 twist parameters consisting of n+1 copies of a and n copies of -a for some odd integer a. Combined with the work of Issa and McCoy, it follows that these are the only doubly slice odd pretzel knots. Time permitting, we might go over some preliminary results involving links as well.

Topologically embedding spheres in knot traces- Patrick Orson (Boston College)

Abstract: Knot traces are smooth 4-manifolds with boundary, that are homotopic to the 2-sphere, and obtained by attaching a 2-handle to the 4-ball along a framed knot in the 3-sphere. I will give a complete characterisation for when the generator of the second homotopy group of a knot trace can be represented by a locally flat embedded 2-sphere with abelian exterior fundamental group. The answer is in terms of classical and computable invariants of the knot. This result is directly analogous to the the result of Freedman and Quinn that says a knot with Alexander polynomial 1 is topologically slice, and can be used to exhibit new exotic 4-dimensional phenomena. This is a joint project with Feller, Miller, Nagel, Powell, and Ray.

Regular homotopy and Gluck twists- Hannah Schwartz (MPIM Bonn)

Abstract: Any 2-sphere K smoothly embedded in the 4-sphere is related to the unknotted sphere through a finite sequence of locally supported homotopies called finger moves and Whitney moves. We call the minimum number of finger moves (or equivalently Whitney moves) in any such a homotopy the "Casson-Whitney number" of the sphere K. In this talk, I will discuss joint work with Joseph, Klug, and Ruppik showing that if the Casson-Whitney number of K is equal to one, then the unknotted torus can be obtained by attaching a single 1-handle to K. I will also present an application of this result, from joint work with Naylor, that the Gluck twist of any sphere with Casson-Whitney number equal to one is diffeomorphic to the standard 4-sphere, and give some well-known families of 2-spheres for which this is the case.

4-manifolds:

Corks, involutions, and Heegaard Floer homology- Abhishek Mallick (Michigan State)

Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution of a 3-manifold over any homology ball that it may bound (rather than a particular contractible manifold). We utilize the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification of the integer homology cobordism group which takes into account involutions acting on homology spheres, and prove that this group admits an infinite rank subgroup generated by corks. Using our invariants, we also establish several new families of (strong) corks (This is joint work with Irving Dai and Matthew Hedden). This talk will be complementary to the talk given by Irving Dai (in the "Floer theory and low-dimensional topology" topic group).

A relative invariant of smooth 4-manifolds- Hyunki Min (Georgia Tech)

Abstract: We define a polynomial invariant of a smooth and compact 4-manifold with connected boundary by modifying an invariant of closed 4-manifolds from Heegaard Floer homology. Using this invariant, we show that there exist infinitely many exotic fillings of 3-manifolds with non-vanishing contact invariant. This is a joint work with John Etnyre and Anubhav Mukherjee.

Standardizing Some Low Genus Trisections- Jesse Moeller (University of Nebraska-Lincoln)

Abstract: Trisections are a novel way to study smooth 4-manifold topology reminiscent of Heegaard splittings of a 3-manifolds; a surface, together with families of embedded curves, determines a 4-manifold up to diffeomorphism. Given a specific genus, we would like to know which 4-manifolds have trisection diagrams inhabiting a surface with this genus. For genus one and two, this is known. In this talk, we will introduce a family of seemingly complicated genus three trisection diagrams and demonstrate that they are, in fact, connected sums of well understood diagrams.

Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines - Sümeyra Sakalli (MPIM Bonn)

Abstract: It is known that all minimal complex surfaces of general type have exactly one (Seiberg-Witten) basic class, up to sign. Thus, it is natural to ask if one can construct smooth 4-manifolds with one basic class. First, Fintushel and Stern built simply connected, spin, smooth, nonsymplectic 4-manifolds with one basic class. Next, Fintushel, Park and Stern constructed simply connected, noncomplex, symplectic 4-manifolds with one basic class. Later Akhmedov constructed infinitely many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds with nontrivial Seiberg-Witten invariants. Park and Yun also gave a construction of simply connected, nonspin, smooth, nonsymplectic 4-manifolds with one basic class. All these manifolds were obtained via knot surgeries, blow-ups and rational blow-downs. In this talk, we will first review some main concepts and recent techniques in symplectic 4-manifolds theory. Then we will construct minimal, simply connected and symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines by the so-called star surgeries, and by using complex singularities. We will show that our manifolds have exotic smooth structures and each of them has one basic class. We will also present a completely geometric way of constructing certain configurations of Kodaira’s singularities in the rational elliptic surfaces, without using any monodromy arguments.

Ribbon homology cobordisms- Mike Wong (Louisiana State University)

Abstract: A cobordism between 3-manifolds is ribbon if it has no 3-handles. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we describe a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies, and some applications. This is joint work with Aliakbar Daemi, Tye Lidman, and Shea Vela-Vick.

Spaces of translation surfaces:

In the moduli space of Abelian differentials, big invariant subvarieties come from topology!- Paul Apisa (Yale University)

Abstract: It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action can also be described as the group action generated by scalar multiplication and Teichmuller geodesic flow. By work of McMullen in genus two, and Eskin, Mirzakhani, and Mohammadi in general, given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety. While McMullen classified these orbit closures in genus two, little is known in higher genus. In the first part of the talk, I will describe the Mirzakhani-Wright boundary of an invariant subvariety (using mostly pictures) and a new result about reconstructing an orbit closure from its boundary. In the second part of the talk, I will define the rank of an invariant subvariety - a measure of size related to dimension - and explain why invariant subvarieties of rank greater than g/2 are loci of branched covers of lower genus Riemann surfaces. This will address a question of Mirzakhani. No background on Teichmuller theory or dynamics will be assumed. This material is work in progress with Alex Wright.

Translation surfaces with multiple short saddle connections- Ben Dozier (Stony Brook University)

Abstract: The SL_2(R) action on strata of translation surfaces allows us to answer many questions about the straight-line flow on individual translation surfaces (and this flow is in turn closely connected to billiards on rational polygons). By the pioneering work of Eskin-Mirzakhani, to understand dynamics on strata one is led to study "affine" measures. It is natural to ask about the interaction between volumes of certain subsets of surfaces and the geometric properties of the surfaces. I will discuss a proof of a bound on the volume, with respect to any affine measure, of the locus of surfaces that have multiple independent short saddle connections. This is a strengthening of the regularity result proved by Avila-Matheus-Yoccoz. A key tool is the new smooth compactification of strata due to Bainbridge-Chen-Gendron-Grushevsky-Moller, which gives a good picture of how a translation surface can degenerate.

What does your average translation surface look like?- Anja Randecker (Heidelberg University)

Abstract: Almost every talk on translation surfaces starts with a double pentagon or with an octagon. But are these n-gons generic examples of translation surfaces? We will look at this question from the point of view of the geometry of the translation surfaces. Specifically, we prove an upper bound for the average diameter (resp. covering radius) in a stratum of translation surfaces of large genus. The first part of my talk assumes only basic knowledge about translation surfaces, the second part assumes more background to give a sketch of the proof. Both are based on joint work with Howard Masur and Kasra Rafi (available at https://arxiv.org/abs/1809.10769).

Billiards and dilation surfaces

Periodic billiard paths on regular polygons- Diana Davis (Swarthmore College)

Abstract: Mathematicians have understood periodic billiards on the square for hundreds of years, and my collaborator Samuel Lelièvre and I have understood them on the regular pentagon for about five years now. During the COVID-19 pandemic, I have been in France, working with Samuel to extend our understanding to all regular polygons with an odd number of sides. In this talk, I'll briefly explain results and techniques for the square and pentagon, and then show lots of nice pictures of billiards on polygons with more than 5 sides, that we have created recently.

An invitation to dilation surfaces- Selim Ghazouani (University of Warwick)

Abstract: In this video talk, I will introduce dilation surfaces, their moduli spaces and related foliations on surfaces and then try to give some motivation for a range of open problems.

You can “hear” the shape of a polygonal billiard table- Chandrika Sadanand (University of Illinois Urbana Champaign)

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge-itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

Curves, paths, and counting:

Counting meanders and square-tilings on surfaces- Eduard Duryev (Institut de mathématiques de Jussieu in Paris)

Abstract: Meander is a homotopy class of a pair of transversal simple closed curves on a sphere. They appear, in particular, as natural enumeration of polymer foldings. Statistics of meanders as the number of intersection grows is one question about meanders people have been interested in. We will show how meanders are related to a particular class of square tilings on surfaces and explain an approach to this more general counting problem that uses ribbon graphs and intersection theory on moduli space.

Extremal length systole of the Bolza surface- Didac Martinez-Granado (Indiana University Bloomington)

Abstract: In Bers-Teichmueller theory there is a rich interaction between the hyperbolic geometry and the conformal geometry, as a result of the uniformization theorem. If the length notion in the hyperbolic side of this dichotomy is the hyperbolic length, on the conformal side the counterpart is extremal length, a conformal invariant notion of homotopy classes of curves. For a choice of hyperbolic structure, there is an essential curve of minimal extremal length and one of minimal hyperbolic length. We call both the curve and its length, respectively, the extremal length and hyperbolic length systoles. These are functions on moduli space. Bounds for maximal values of the hyperbolic length systole for some geni are known, and for genus 2, its absolute maximum is the Bolza surface, a triangle surface. In this talk we show that the Bolza surface realizes a local maximum of the extremal length systole, and compute its value: square root of 2. This is work in progress with Maxime Fortier Bourque and Franco Vargas Pallete.

Gaps of saddle connection directions for some branched covers of tori- Anthony Sanchez (University of Washington Seattle)

Abstract: Translation surfaces given by gluing two identical tori along a slit have genus two and two cone-type singularities of angle 4 \pi. There is a distinguished set of trajectories called saddle connections that are the straight lines trajectories between cone points. We can associate a holonomy vector in the plane to each saddle connection whose components are the horizontal and vertical displacement of the saddle connection. How random is the planar set of holonomy of saddle connections? We study this question by computing the gap distribution for slopes of saddle connections for these and other related classes of translation surfaces.

The topology and geometry of random square-tiled surfaces- Sunrose Thapa Shrestha (Tufts University)

Abstract: A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk, we will consider a randomizing model for STSs based on permutation pairs and see how to use it to compute the genus distribution. We will also look at holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. We will obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.

### Quantum invariants and low-dimensional topology: Carmen Caprau (Fresno State) and Christine Ruey Shan Lee (University of South Alabama).

New quantum invariants and other connections:

Sequence of Undetectable Nonquasipositive Braids - Elaina Aceves (University of Iowa)

When is a quasipositive transverse knot, Hedden and Plamenevskaya proved that , where is the Ozsv\'ath-Szab\'o concordance invariant and is the self-linking number of the knot. We construct a sequence of braids whose closures are nonquasipositive knots where we add more negative crossings as we progress through the sequence. Therefore, the knots formed from the sequence become more nonquasipositive as the sequence progresses. Furthermore, we conjecture that the knots obtained from this sequence behave like quasipositive knots in that they uphold the equality . Because of this property, the nonquasipositive knots obtained from our sequence are undetectable by transverse invariants like Ozsv\'ath and Szab\'o's from Heegaard Floer homology and Plamenevskaya's from Khovanov homology.

On framing changes of links in -manifolds- Dionne Ibarra (George Washington University)

Abstract: We show that the only way of changing the framing of a link by ambient isotopy in an oriented -manifold is when the manifold admits a properly embedded non-separating . We will illustrate the change in framing by the Dirac trick then relate the results to the framing skein module. Coauthors: Rhea Palak Bakshi, Gabriel Montoya-Vega, Jozef Przytycki, and Deborah Weeks.

Holonomy invariants of links and Holonomy invariants from quantum sl_2 and torsions- Calvin McPhail-Snyder (University of California-Berkeley)

Abstract: To describe geometric information about a space X, we can equip it with a representation \rho: \pi_1(X) \to G, where G is a group. Quantum invariants using this extra data are called quantum holonomy invariants or homotopy quantum field theories. In this talk, I will give some motivation for this idea (including connections to volume conjectures) and discuss how to modify the usual Reshetikhin-Turaev construction to get holonomy invariants of links in S^3.

Blanchet, Geer, Patureau-Mirand, and Reshetikhin have constructed a holonomy invariant for G = SL_2(C) using the quantum group U_q(sl_2) at a root of unity. In this talk, I will give an overview of their construction, then discuss my recent work showing how to interpret their invariant in terms of twisted Reidemeister torsion.

How to make quantum groups easier? and Quantum knot invariants according to Alexander- Roland van der Veen (University of Groningen)

Joint work with Dror Bar-Natan (Toronto)

Abstract: Quantum groups such as U_q sl_2 often appear at the foundations of most quantum knot invariants. And usually this results in forbidding lists of generators, relations and formulas. Does it have to be this way? One way out is to pass to representations but then all computations will usually grow exponentially. (Try computing the Jones polynomial of a 50 crossing knot). In this talk I would like to propose another way out. Will modify the algebra itself and then work with it through Gaussian expressions. In fact one can also understand much of the way quantum groups are built from a purely topological standpoint but that is the subject of another talk.

Abstract: From the point of view of universal invariants the Alexander polynomial is the most fundamental quantum invariant. It appears as the one loop contribution in the perturbative expansion of the Chern-Simons integral regardless of what gauge group one starts with. This suggests that one should be able to use Alexander to gain insight into more complicated quantum invariants and conversely presents the challenge of generalizing the many nice properties of Alexander to a wider context. I will make these points concrete by reproving and extending the formula for the Alexander polynomial from a Seifert surface towards the universal quantum sl_2 invariant.

Link homologies and categorification:

sln-homology theories obstruct ribbon concordance- Nicolle Gonzalez (University of California-Los Angeles)

Abstract: In a recent result, Zemke showed that a ribbon concordance between two knots induces an injective map between their corresponding knot Floer homology. Shortly after, Levine and Zemke proved the analogous result for ribbon concordances between links and their Khovanov homology. In this talk I will explain joint work with Caprau-Lee-Lowrance-Sazdanovic and Zhang where we generalize this construction further to show that a link ribbon concordance induces injective maps between sln -homology theories for all n \geq 2.

Khovanov-Rozansky homology of torus links (and beyond)- Matthew Hogancamp (Northeastern University)

Abstract: Throughout the past decade, torus links and their homological invariants have been the subject of numerous fascinating conjectures connecting link homology to seemingly distant areas of algebraic geometry and and representation theory (see work of various subsets of Cherednik, Gorsky, Negut, Oblomkov, Rasmussen, Shende). Many of these conjectures are now proven by "computing both sides" (the link homology side this was done by myself and Anton Mellit, both independently and jointly). In this talk I will discuss the main technique for computing (by hand!) Khovanov-Rozansky's HOMFLY-PT homology introduced by myself and Ben Elias, and its application to link homology. The message I hope to convey is that this technique is strikingly simple to use, and is useful in a wide variety of settings.

Khovanov homology detects T(2,6)- Gage Martin (Boston College)

Abstract: Khovanov homology is a combinatorially defined link homology theory. Due to the combinatorial definition, many topological applications of Khovanov homology arise via connections to Floer theories. A specific topological application is the question of which links Khovanov homology detects. In this talk, we will give an overview of Khovanov homology and link detection, mention some of the connections to Floer theoretic data used in detection results, and show that Khovanov homology detects the torus link T(2,6).

Generalizing Rasmussen's s-invariant, and applications- Michael Willis (University of California-Los Angeles)

I will discuss a method to define Khovanov and Lee homology for links in connected sums of copies of S1x S2. This allows us to define an -invariant that gives genus bounds on oriented cobordisms between links. I will discuss some applications to surfaces in certain 4-manifolds, including a proof that the s-invariant cannot detect exotic B4's coming from Gluck twists of the standard B4. All of this is joint work with Ciprian Manolescu, Marco Marengon, and Sucharit Sarkar.

Hyperbolic volume and the volume conjecture:

Splice-unknotting and crosscap numbers.- Thomas Kindred (University of Nebraska at Lincoln)

Abstract: Ito-Takimura recently introduced the splice-unknotting number of a knot. This diagrammatic invariant provides an upper bound for a knot's crosscap number, with equality in the alternating case. Using results of Kalfagianni-Lee, this equality leads to corollaries regarding hyperbolic volume and the Jones polynomial.

Families of fundamental shadow links realized as links in S3 - Sanjay Kumar (Michigan State University)

Abstract: In 2015, Chen and Yang provided evidence that the asymptotics of the Turaev-Viro invariant of a hyperbolic -manifold evaluated at the root of unity have growth rates given by the hyperbolic volume. This has been proven by Belletti, Detcherry, Kalfagianni, and Yang for an infinite family of hyperbolic links in connect sums of known as the fundamental shadow links. In this talk, I will present examples of links in satisfying the Turaev-Viro invariant volume conjecture through homeomorphisms with complements of fundamental shadow links along with an application towards the conjecture posed by Andersen, Masbaum, and Ueno (AMU conjecture).

Volume conjecture, geometric decomposition and deformation of hyperbolic structures (I) and (II)- Ka Ho Wong (Texas A & M University)

Abstract: The Chen-Yang volume conjecture of the Turaev-Viro invariant is a new topic in quantum topology. It has been shown that the -th Turaev-Viro invariant for a link complement can be expressed as a sum of norm squared of the colored Jones polynomial of the link evaluated at . This leads to the study of the asymptotics for the -th colored Jones polynomials of links evaluated at -th root of unity, with a fixed limiting ratio of to . In the first talk, I will recall the definition of the colored Jones polynomials and discuss how the asymptotics of the colored Jones polynomials of the Whitehead link is related to the (not necessarily complete) hyperbolic structures on its complement. Then, in the second talk, we will focus on some satellite links whose complements have more than one hyperbolic piece in the geometric decomposition, and relate the asymptotics of their colored Jones polynomials to the geometric structures on the geometric pieces.

Computational approaches:

Developing software for skein computations in knot complements- Rachel Marie Harris (Texas Tech University)

Abstract: Skein manipulations prove to be computationally intensive due to the exponential nature of skein relations. The purpose of this project is to construct an automated tool to generate a library of examples for use in testing new conjectures in Chern-Simons theory.

Finding Structure in Polynomial Invariants using Data Science- Jesse Levitt (UCLA)

Abstract: Authors: Pawel Dlotko, Mustafa Hajij, Jesse Levitt (presenter), Radmila Sazdanovic Abstract: Using Principal Component Analysis and Topological Data Analysis we analyze the distributions of the knot polynomials in coefficient space. These tools prove useful for both distinguishing how well different invariants separate the knots into distinct families and for how these families suggest correlations between different knot invariants. We focus on how the Ball Mapper of P. Dlotko, an exploratory data analysis tool that builds graphs from high dimensional clouds of data using just a radius measure, confirms and further illuminates substructure in this data. This includes some specific ways in which the s-invariant and signature differ through the distribution of the Alexander and Jones polynomials.