Speaker: Martin Bridson (University of Oxford & Clay Math Institute)
Title: Commensurator rigidity for automorphism groups of free groups
Abstract: The natural map from Aut(\( F_n \)) to its abstract commensurator is an isomorphism if \( n \) is at least \( 3 \), and the image is a subgroup of finite index if \( n=2 \). The proof relies, among other things, on an understanding of the ways in which a direct product of free groups can be embedded in Aut(\( F_n \)). This is joint work with Ric Wade.

Speaker: Danny Calegari (University of Chicago)
Title: Sausages and Butcher Paper
Abstract: For each \( q>1 \) the Shift Locus of degree \( q \) is the space of monic depressed degree \( q \) polynomials in one complex variable for which every critical point is in the attracting basin of infinity. We give an explicit description of the Shift Locus as a (combinatorial) building whose pieces turn out to be homeomorphic to affine complex algebraic varieties.

Speaker: Ruth Charney (Brandeis University)
Title: Outer Space for Right-Angled Artin Groups
Abstract: Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups. Thus, their (outer) automorphism groups range from Out(\( F_n \)) to \( \text{GL}(n,\mathbb{Z}) \). These two classes of groups have been extensively studied from a geometric viewpoint, using the action of \( \text{GL}(n,\mathbb{Z}) \) on homogeneous space on the one hand, and the action of Out(\( F_n \)) on Culler-Vogtmann's "Outer Space" on the other. I will talk about joint work with Bregman and Vogtmann in which we intertwine these two ideas to construct an analogue of Outer Space for arbitrary RAAGs.

Speaker: Pallavi Dani (Louisiana State University)
Title: Commensurability between right-angled Coxeter and Artin groups
Abstract: A common theme in geometric group theory is to try to understand when two groups are quasi-isometric (or "geometrically close") and when they are commensurable (or "algebraically close"), and when these two notions coincide. Davis-Januszkiewicz showed that every right-angled Artin group (RAAG) is commensurable to some right-angled Coxeter group (RACG). It is not hard to see that the reverse is far from true, prompting the question: Which RACGs are commensurable to RAAGs? I will talk about joint work with Ivan Levcovitz which explores this. In particular, we show that for a large class of RACGs, being commensurable and being quasi-isometric to a RAAG are equivalent. The talk will be aimed at a broad audience.

Speaker: David Futer (Temple University)
Title: Infinitely many virtual geometric triangulations
Abstract: Since the pioneering work of Thurston on hyperbolic geometry, and of Neumann-Zagier on volume and triangulations, it has been believed that every cusped hyperbolic \( 3 \)-manifold should admit a decomposition into a union of positively oriented ideal tetrahedra. Somewhat shockingly, the question of whether such a geometric triangulation exists is still open today. Luo, Schleimer, and Tillmann proved that geometric ideal triangulations of this sort exist in some finite cover of every cusped \( 3 \)-manifold. We extend their result by showing that every cusped hyperbolic \( 3 \)-manifold has a finite cover admitting an infinite trivalent tree of geometric ideal triangulations. Furthermore, every sufficiently long Dehn filling of this cover also admits infinitely many geometric ideal triangulations.

The proof involves a mixture of geometric constructions and subgroup separability tools. One of the separability tools is a new theorem about separating a peripheral subgroup from every conjugate of a coset. I will try to give a glimpse into both the geometry and the subgroup separability. This is joint work with Emily Hamilton and Neil Hoffman.

Speaker: David Gabai (Princeton University)
Title: Knotted \( 3 \)-balls in \( S^4 \) and knotted \( 3 \)-spheres in \( S^1 \times S^3 \)
Abstract: We demonstrate codimension-\( 1 \) knotting in \( S^4 \) and \( S^1 \times S^3 \). That is, there are \( 3 \)-balls with boundary the standard \( 2 \)-sphere in \( S^4 \), which are not isotopic rel boundary to the standard \( 3 \)-ball and there are non separating \( 3 \)-spheres in \( S^1 \times S^3 \) not isotopic to \( \text{pt.} \times S^3 \). The latter induces diffeomorphisms of \( S^1 \times S^3 \) that are homotopic to \( \text{id} \) but not isotopic to \( \text{id} \). (Joint work with Ryan Budney)

Speaker: Craig Hodgson (University of Melbourne)
Title: The meromorphic 3D-index and its asymptotics
Abstract: The meromorphic 3D-index of Garoufalidis-Kashaev is a new topological invariant of orientable 3-manifolds with toroidal boundary, defined as a state integral of Turaev-Viro type on ideal triangulations. The domain of integration can be regarded as a component of the space of circle-valued angle structures studied by Luo. In this talk, we discuss the definition and predicted asymptotic behaviour of this invariant as the quantisation parameter \( q \) tends to 1. The conjectured asymptotic approximation depends on classical invariants, including hyperbolic volumes of representations of the fundamental group into PSL(2,C), as well as a new "beta invariant" associated to PSL(2,R) representations. This is based on joint work with Andrew Kricker and Rafael Siejakowski.

Speaker: Svetlana Katok (The Pennsylvania State University)
Title: Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups
Abstract: Given a closed, orientable surface of constant negative curvature and genus \( g \geq 2 \), we study a family of generalized Bowen–Series boundary maps and their two dynamical invariants: the topological entropy and the measure-theoretic entropy with respect to their smooth invariant measure. Each such map is defined for a particular fundamental polygon and a particular multi-parameter. We prove two strikingly different results: rigidity of topological entropy and flexibility of measure-theoretic entropy. The topological entropy is constant in this family and depends only on the genus of the surface. We give an explicit formula for this entropy and show that it stays constant both within our parameter space and within the Teichmüller space of the surface. We obtain an explicit formula for the measure-theoretic entropy that only depends on the genus of the surface and the perimeter of the \( (8g-4) \)-sided fundamental polygon, and prove that it varies in the Teichmüller space and takes all values between 0 and maximum that is achieved on the surface which admits a regular fundamental \( (8g-4) \)-gon, and stays constant on a subset of the parameter space.

The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof - valid only for certain multi-parameters - uses the realization of the geodesic flow on the surface as a special flow over the natural extension of the boundary map.

This is joint work with Adam Abrams and Ilie Ugarcovici.

Speaker: Matthias Kreck (Bonn University)
Title: Differentiable manifolds and quadratic forms - revisited
Abstract: This is the title of a lovely book by two of my mathematical heroes: Friedrich Hirzebruch and Walter Neumann. I will add a little chapter to it by building a bridge from \( 3 \)-manifolds to unimodular lattices in \( \mathbb{R}^n \) via self dual binary codes. This is based on older work with Volker Puppe and more recent work.

Speaker: Christopher Leininger (Rice University)
Title: Billiards, geometry, and symbolic coding
Abstract: Given a polygon in the Euclidean or hyperbolic plane, its billiard flow on the tangent bundle has trajectories that describe the paths of particles in the polygon (the billiard trajectories) traveling along straight lines and "bouncing" off the sides. Labeling the sides of the polygon the billiard flow determines a symbolic coding we call the bounce spectrum, which is the set of biinfinite sequences of labels corresponding to the sides encountered by all trajectories. A natural question asks the extent to which the bounce spectrum determines the shape of the polygon. For both Euclidean and hyperbolic polygons, there are nontrivial constructions of polygons with the same bounce spectrum that are not isometric/similar. In this talk, I'll describe these constructions, and then results of joint work with Duchin, Erlandsson, and Sadanand stating that these are in fact the only ways in which non-isometric/non-similar polygons can have the same bounce spectrum.

Speaker: Feng Luo (Rutgers University)
Title: Weyl problem in the hyperbolic \( 3 \)-space and the Koebe circle domain conjecture
Abstract: The classical Weyl problem (in the hyperbolic geometric setting) asks if every complete hyperbolic surface of genus zero can be isometrically embedded in to the hyperbolic \( 3 \)-space as a convex surface and the Koebe circle domain conjecture states if every domain is conformal to a domain whose boundary components are circles or points. We show that these two problems are closely related. This is a joint work with Tianqi Wu.

Speaker: Alex Margolis (Vanderbilt University)
Title: 3-manifolds, Leighton's theorem and rigidity
Abstract: A theorem of Leighton says that two finite graphs with isomorphic universal covers have a common finite cover. In group theoretic terms, Leighton's theorem says that any two uniform lattices in the automorphism group of a tree are weakly commensurable. We introduce the class of graphically discrete groups, a large class of groups that includes nilpotent groups, most lattices in semisimple Lie groups, mapping class groups and fundamental groups of closed connected irreducible 3-manifolds. We then prove a Leighton-style rigidity theorem for free products of torsion-free graphically discrete groups, showing that if two such groups are uniform lattices in the same locally compact group, then they are abstractly commensurable. This result applies to fundamental groups of closed connected non-prime 3-manifolds. Joint with Sam Shepherd, Emily Stark and Daniel Woodhouse.

Speaker: Robert Meyerhoff (Boston College)
Title: Hyperbolic 3-Manifold Stories
Abstract: A brief historical outline of selected results on hyperbolic 3-manifolds will be given, leading up to recent research. Topics: low-volume manifolds, hyperbolic Dehn Filling spaces, Chern-Simons invariant.

Speaker: Paul Norbury (University of Melbourne)
Title: Enumerative geometry via the moduli space of super Riemann surfaces.
Abstract: Mumford initiated the calculation of many algebraic topological invariants over the moduli space of Riemann surfaces in the 1980s, and Witten related these invariants to two dimensional gravity in the 1990s. This viewpoint led Witten to a conjecture, proven by Kontsevich, that a generating function for intersection numbers on the moduli space of curves is a tau function of the KdV hierarchy which allowed their evaluation. In 2004, Mirzakhani produced another proof of Witten's conjecture via the study of Weil-Petersson volumes of the moduli space using hyperbolic geometry. In this lecture I will describe a different collection of integrals over the moduli space of Riemann surfaces whose generating function is a tau function of the KdV hierarchy. I will sketch a proof of this result that uses an analogue of Mirzakhani's argument applied to the moduli space of super Riemann surfaces - defined by replacing the field of complex numbers with a Grassmann algebra - which uses recent work of Stanford and Witten. This appearance of the moduli space of super Riemann surfaces to potentially solve a problem over the classical moduli space is deep and surprising.

Speaker: Helge Møller Pedersen (Universidade Federal do Ceara, Brazil)
Title: Lipschitz normally embedded singularities
Abstract: Any real or complex singularity \( (X, 0) \) is equipped with two natural metrics. The outer metric, which is the restriction of the ambient euclidean metric, and the inner metric, which is the metric associated with a riemannian metric on the germ. Up to bilipschitz equivalence these metrics does not depends on the choices of analytic embedding. The inner and outer metrics are in general not bilipschitz equivalent, and one says that \( (X, 0) \) is Lipschitz normally embedded if they are. In this talk we will give an overview of the subject and discuss the current state of the question about which singularities are Lipschitz normally embedded. From the beginning of the modern study done by Birbrair, Fernandes and Neumann, over our joint work with Neumann and Pichon on proving that minimal surfaces singularities are Lipschitz normally embedded, and our work with Kerner and Ruas on which matrix singularities are Lipschitz normally embedded, to work still in progress joint with Fantini, Pichon and Schober on surface singularities and with Langois on toric singularities.

Speaker: Anne Pichon (Aix Marseille University)
Title: Walter's contribution on Lipschitz geometry of singularities
Abstract: I will give an introduction to Lipschitz geometry of singularities and present the main results obtained by Walter and his collaborators during the last decade on Lipschitz classification of complex surface germs.

Speaker: Alan Reid (Rice University)
Title: Profinite rigidity, direct products and finite presentability
Abstract: A finitely generated residually finite group G is called profinitely rigid, if for any other finitely generated residually finite group H, whenever the profinite completions of H and G are isomorphic, then H is isomorphic to G. In this talk we will review what is known about this in the context of groups arising in low-dimensional geometry and topology. We will then discuss some recent work that constructs finitely presented groups that are profinitely rigid amongst finitely presented groups but not amongst finitely generated one.

Speaker: Jonathan Wahl (University of North Carolina, Chapel Hill)
Title: A characteristic number of a surface singularity pair
Abstract: The link \( \Sigma \) of a normal surface singularity \( (X,0) \) is a compact \( 3 \)-manifold constructed by plumbing using the graph \( \Gamma \) of a resolution of \(X \). Our 1990 JAMS paper defined a topological invariant \( I(X) \) from \( \Gamma \) with: (a) \( I(X)\in \mathbb{Q}_{\geq 0} \) ; (b) \( I(X)=0 \) iff \( X \) is log-canonical; (c) \( I(X) \) is ``characteristic'', multiplying by degree in unramified covers; (d) \( X \) Gorenstein (e.g., hypersurface), \( I(X)\neq 0 \) implies \( I(X)\geq 1/42 \), and \( \{I(X)|X \text{Gorenstein}\} \) satisfies the ACC. We introduce definitions now for the more general log (or orbifold) situation of a pair \( (X,\sum c_iC_i) \), where the \( C_i \) are curves on \( X \) (resp. knots in \( \Sigma \) ), and \( c_i\in [0,1] \) (usually \( c_i=1/n_i \) or \( 1-1/n_i \) ). Like \( I(X) \), it is based on the Zariski decomposition of some \( \mathbb{Q} \)-line bundles on a resolution of \( X \).

Speaker: Christian Zickert (University of Maryland, College Park)
Title: Polylogarithms, motives, and cluster algebras
Abstract: We discuss the relationship between polylogarithms, cluster algebras, and motivic cohomology. The story will be linked to Walter's work on the Chern-Simons class and hyperbolic 3-manifolds.