In a joint work with Peter Ozsvath we have developed algebraic invariants for knots using a family of bordered knot algebras. The goal of this lecture is to review these constructions and discuss some of the latest developments.

Rigidity theorems prove that a group's geometry determines its algebra, typically up to virtual isomorphism. Motivated by rigidity problems, we study graphically discrete groups, which impose a discreteness criterion on the automorphism group of any graph the group acts on geometrically. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds. We will present new examples, proving this property is not a quasi-isometry invariant. We will discuss action rigidity for free products of residually finite graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions $n\le 5$. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic.

The main tool is Bestvina - Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

Consider simple rank-one Lie groups $SO(n, 1)$, $SU(n, 1)$ and $Sp(n ,1)$ ($n>1$). They are the isometry groups of real, complex and quaternionic hyperbolic spaces respectively.

By a result of Kostant, the trivial representation of $Sp(n ,1)$ is isolated in the space of irreducible unitary representations on Hilbert spaces. That is, $Sp(n ,1)$ has Kazhdan’s property (T) which is equivalent to the vanishing of 1st cohomology of the group in all unitary representations. This is in contrast to the case of $SO(n ,1)$ and $SU(n ,1)$ where they have the Haagerup approximation property, a strong negation of property (T).

This dichotomy between $SO(n ,1)$, $SU(n ,1)$ and $Sp(n ,1)$ disappears when we consider so-called uniformly bounded representations on Hilbert spaces. By a result of Cowling in 1980’s, the trivial representation of $Sp(n ,1)$ is no longer isolated in the space of uniformly bounded representations. Moreover, there is a uniformly bounded representation of $Sp(n ,1)$ with non-zero first cohomology group.

The goal of this talk is to describe these facts.

I will talk about joint work with Andrew Lobb related to Toeplitz's square peg problem, which asks whether every (continuous) Jordan curve in the Euclidean plane contains the vertices of a square. Specifically, we show that every smooth Jordan curve contains the vertices of a cyclic quadrilateral of any similarity class. I will describe the context for the result and its proof, which involves symplectic geometry in a surprising way.

Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) for which cocycles are 2-dimensional (supersymmetric) field theories. Properties of these field theories lead to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from the field theory point of view. In this talk, we will describe a map from 2-dimensional field theories to a cohomology theory that approximates TMF. This map affords a cocycle description of certain torsion classes. In particular, we will explain how a choice of anomaly cancelation for the supersymmetric sigma model with target $S^3$ determines a cocycle representative of the generator of $\pi_3(TMF)=\mathbb{Z}/24$.

Topologically speaking, left-orderable countable groups are precisely those countable groups that embed into the group of orientation preserving homeomorphisms of the real line. A recent advancement in the theory of left-orderable groups is the discovery of finitely generated left-orderable simple groups by Hyde and Lodha. We will discuss a construction that extends this result by showing that every countable left-orderable group is a subgroup of such a group. We will also discuss some of the algebraic, geometric, and computability properties that this construction bears. The construction is based on novel topological and geometric methods that also will be discussed. The flexibility of the embedding method allows us to go beyond the class of left-orderable groups as well. Based on a joint work with Markus Steenbock.

In this talk we will introduce Leary and Minasyan's CAT(0) but not biautomatic groups as lattices in a product of a Euclidean space and a tree.  We will then investigate properties of general lattices in that product space.  We will also consider a construction of lattices in a Salvetti complex for a right-angled Artin group and a Euclidean space.  Finally, if time permits we will also discuss a "hyperbolic Leary–Minasyan group" and some work in progress with Motiejus Valiunas towards an application.

The Hull--Strominger system is a system of non-linear PDEs on heterotic string theory involving a pair of Hermitian metrics $(g,h)$ on a six dimensional manifold $M$. One of these equations dictates the metric $g$ on $M$ to be conformally balanced. We will begin the talk by giving a description of the geometry of cohomogeneity one manifolds and SU(3)-structures. Then, we will look for solutions to the Hull--Strominger system in the cohomogeneity one setting. We show that a six-dimensional simply connected cohomogeneity one manifold under the almost effective action of a connected Lie group $G$ admits no $G$-invariant balanced non-Kähler SU(3)-structures. This is a joint work with F. Salvatore.