Torsion groups do not act on $2$–dimensional $\mathrm{CAT}(0)$ complexes

It is a well known theorem that if a finitely generated group $G$ acts on a tree without a global fixed point, then it has an element of infinite order. I will sketch the proof of this theorem and our generalisation to $2$–dimensional complexes that are $\mathrm{CAT}(0)$, that is, simply connected of metric nonpositive curvature. This is joint work with Sergey Norin and Damian Osajda.

Boundary Maps in Non-positive Curvature

One important tool geometric group theorists use to study groups with non-positive curvature is boundaries. Given a subgroup, one may ask if there is a nice map from the boundary of that subgroup into the boundary of the supergroup. In the classical setting of hyperbolic $3$–manifolds fibered over the circle, Cannon and Thurston showed that the inclusion of the fundamental group of the fiber (a hyperbolic surface) into the fundamental group of the $3$–manifold induces a continuous, surjective(!) map from the boundary of the surface group (a circle) to the boundary of hyperbolic $3$–space (a sphere). These maps are called Cannon-Thurston maps. Mj (Mitra) later generalized this to all hyperbolic groups with hyperbolic normal subgroups. In this talk I will explain this cannot be generalized in the context of $\mathrm{CAT}(0)$ groups. This is joint work with Benjamin Beeker, Giles Gardam, Radhika Gupta, and Emily Stark.

Taut sutured handlebodies as twisted homology products

A basic problem in the study of $3$–manifolds is to determine when geometric objects are of 'minimal complexity'. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called 'tautness'.

One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called 'rational homology product'. Most sutured manifolds do not have this form, but do always take the more general form of a 'twisted homology product', which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such?

We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples that, when restricted to solvable representations, the twisting representation cannot be 'too simple'.

On solitons for the $G_2$–Laplacian flow

For closed $G_2$-structures on $7$–manifolds, there is a natural analog of the Ricci-flow for Riemannian metrics. In this talk, after giving a brief introduction to the geometry of $G_2$–structures, I will discuss what we know about this flow, including short time existence, convergence, and the existence of solitons.

A higher-height lift of Rohlin's Theorem: on $\eta^3$ in hyperreal bordism

Rohlin's theorem on the signature of Spin $4$–manifolds can be restated in terms of the connection between real and complex $K$–theory given by homotopy fixed points. This comes from a bordism result about Real manifolds versus unoriented manifolds, which in turn, comes from a $C_2$–equivariant story . I'll describe a surprising analogue of this for larger cyclic $2$–groups, showing that the element eta cubed is never detected! In particular, for any bordism theory orienting these generalizations of Real manifolds, the three torus is always a boundary.

String diagrams in Topology, Geometry, and Analysis

I will introduce string diagrams for $2$–categories to illuminate connections between low dimensional topology, symplectic geometry, and the analysis of pseudoholomorphic curves. On the one hand, the construction of topological invariants (eg. Heegaard-Floer homology for $3$–manifolds) via symplectic geometry can be understood in terms of isomorphisms in a symplectic $2$–category. Here the crucial algebraic identity corresponds to strip shrinking in a string diagram, which represents an adiabatic limit between different types of elliptic PDEs. On the other hand, a novel singularity formation in this adiabatic limit is naturally represented as generalized string diagram, which in turn gives rise to new algebraic structures.

Flexible Weinstein manifolds

Weinstein manifolds are a class of symplectic manifolds defined by having a handlebody decomposition compatible with their symplectic geometry, examples include all cotangent bundles and all smooth affine varieties. Among these are a special class: flexible Weinstein manifolds, which are characterized by being the "geometrically trivial" type, so that a unique flexible Weinstein manifold exists in each diffeomorphism type. In this talk we discuss their properties and various applications, going from their discovery in 2012 to today.

Quantum evolution in a magnetic field and the tail behaviour of Weyl sums

We will discuss the time evolution of a "box" function according to the Schrödinger equation in the presence of a magnetic field. This problem naturally leads to the study of a $1$–parameter family of integral operators, interpolating between the identity and the Fourier transform. We provide an asymptotic estimate for the $L^\infty$–norm of these Fourier-like multipliers when the box is approximated by slightly more regular functions as the size of the approximation tends to zero. As an application, we obtain an improved bound for the tail of the limiting distribution arising from quadratic Weyl sums. Their limiting distribution is obtained using homogeneous dynamics, along with a dynamical partition of unity, based on earlier work with J. Marklof. Joint work with J. Griffin and T. Osman.

Substitution Sequences and Tilings: Spectral and Numerical Considerations

Really, this talk could have been titled "Stuff I've thought about since my PhD generally themed along the dynamics of $1$–$D$ and $2$–$D$ substitution dynamics: What has worked and what has not." Consider it a general survey of the spectral properties of discrete Schrödinger operators arising in models of quasicrystals with some mathematical autobiography mixed in. In addition to presenting some of my results in the area, I will also talk about some projects I've undertaken that did not result in publication. Moreover, I'll share some of the ways I've managed to involved undergraduate students in my work.

Periodic paths on the pentagon

Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I'll explain my joint work with Samuel Lelièvre on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising "dense but not equidistributed" behavior.

Pure braids and link concordance

If one considers the set of m-component based links in $\mathbb{R}^3$ with a $4$–dimensional equivalence relationship on it, called concordance, one can form a group called the link concordance group, $C^m$. Questions in concordance are important in for classification questions in topological and smooth $4$–manifolds It is well known that the link concordance group contains the isotopy class of pure braid with m strands, $P_m$. That is, two braids are concordant if and only if they are isotopic! In the late 90's Tim Cochran, Kent Orr, and Peter Teichner defined a filtration of the knot/link concordance group called the $n$–solvable filtration. This filtration gives a way to approximate whether a link is trivial in the group. We discuss the relationship between pure braids and the n-solvable filtration as well as various other more geometrically defined filtrations coming from gropes and Whitney towers. This is joint work with Aru Ray and Jung Hwan Park.

Infinite Volume Rigidity and Bounded Cohomology

The volume of a closed hyperbolic $3$–manifold $M$ is an invariant of its fundamental group, by Mostow Rigidity. The fundamental group of $M$ acts by isometries in many different ways on hyperbolic space, but only one is the action corresponding to the unique hyperbolic structure of $M$, its hyperbolization. We associate a number to any action, called its volume; the (finite) Volume Rigidity Theorem says that this numerical invariant picks out the hyperbolization of $M$ among all actions of its fundamental group on hyperbolic space. In contrast to situation for finite volume manifolds, infinite volume hyperbolic $3$–manifolds admit many different hyperbolizations. We will explain how to use bounded cohomology, a tool that we will define (with examples) in the talk, to formulate a volume rigidity theorem in the setting that $M$ has infinite volume.

Convergence of normalized Betti numbers in nonpositive curvature

We will show that if $M_n$ is any sequence of distinct, finite volume manifolds that are quotients of a fixed, higher rank irreducible symmetric space $X$ of noncompact type, then the normalized Betti numbers $b_k(M_n) / \mathrm{vol}(M_n)$ converge to the $L^2$–Betti numbers of $X$. This is a joint paper from late 2018 with Abert, Bergeron and Gelander, generalizing a 2012 result of the four of us with Nikolov, Raimbault and Samet. The key ingredient here is a probabilistic notion of convergence of Riemannian manifolds. Related results for arbitrary irreducible symmetric spaces, and possibly for general manifolds of nonpositive curvature, will also be discussed.

From curves to currents

Bonahon introduced the space of geodesic currents, a space that contains all closed weighted closed curves as a dense subspace (generalizing measured laminations, which contain weighted simple closed curves). We give a simple criterion for when a function on weighted curves extends to a continuous function on geodesic currents; the key restriction is that it is monotone under smoothing a crossing. This puts all known results of functions extending to geodesic currents in a unified framework.

As a corollary of this result and work of Rafi-Souto, we can, for instance, give asymptotics for the number of curves of a given topological type with bounded extremal length. (For hyperbolic length, the corresponding result is a theorem of Mirzakhani.)

This is joint work with Didac Martinez-Granado.