The organized movement of intracellular material is part of the functioning of cells and the development of organisms. These flows can arise from the action of molecular machines on the flexible, and often transitory, scaffoldings of the cell. Understanding phenomena in this realm has necessitated the development of new simulation tools, and of new coarse-grained mathematical models to analyze and simulate. In that context, I'll discuss how a symmetry-breaking "swirling" instability of a motor-laden cytoskeleton may be an important part of the development of an oocyte, modeling active material in the spindle, and what models of active, immersed polymers tell us about chromatin dynamics in the nucleus.

The organized movement of intracellular material is part of the functioning of cells and the development of organisms. These flows can arise from the action of molecular machines on the flexible, and often transitory, scaffoldings of the cell. Understanding phenomena in this realm has necessitated the development of new simulation tools, and of new coarse-grained mathematical models to analyze and simulate. In that context, I'll discuss how a symmetry-breaking "swirling" instability of a motor-laden cytoskeleton may be an important part of the development of an oocyte, modeling active material in the spindle, and what models of active, immersed polymers tell us about chromatin dynamics in the nucleus.

Some proteins (in their folded form) are classified as being knotted.

The function of the knotting is mysterious since knotting seemingly

would make the folding process unnecessarily complicated.  To

function, proteins need to fold quickly and reproducibly, and

misfolding can have catastrophic results.  For example, Mad Cow

disease and the human analog, Creutzfeldt-Jakob disease, come from

misfolded proteins.

Traditionally, knotting is only defined for closed curves, where the

topology is trapped in the loop.  However, proteins have free ends, as

well as most of the objects that humans consider as being knotted

(like shoelaces and strings of lights).  Defining knotting in open

curves is tricky and ambiguous.  We consider some definitions of

knotting in open curves and see how one of these definitions is used

to characterize the knotting in proteins.

In this talk, I will discuss a wide range of mechanical systems,
including Hoberman’s sphere, Euler’s disk, a sliding cylinder, the
Dynabee, BB-8, and Littlewood’s hoop, and the research they inspired.
Studies of the dynamics of the cylinder ultimately led to a startup
company while studying Euler’s disk led to sponsored research with a
well-known motorcycle company.

This talk is primarily based on research performed with a number of
former students over the past three decades. including Prithvi Akella,
Antonio Bronars, Christopher Daily-Diamond, Evan Hemingway, Theresa
Honein, Patrick Kessler, Nathaniel Goldberg, Christine Gregg, Alyssa
Novelia, and Peter Varadi over the past three decades.

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 $\pi_3(TMF)=\mathbb{Z}/24$.