The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-ℓ mapping class group, which is a fundamental finite-index subgroup of the mapping class group. One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed.
In 1992 Hitchin discovered distinguished components of the PSL(d,R) character variety for closed surface groups pi_1S and asked for an interpretation of those components in terms of geometric structures. Soon after, Choi-Goldman identified the SL(3,R)-Hitchin component with the space of convex projective structures on S. In 2008, Guichard-Wienhard identified the PSL(4,R)-Hitchin component with foliated projective structures on the unit tangent bundle T^1S. The case d \ge 5 remains open, and compels one to move beyond projective geometry to flag geometry.
We begin with a survey of some Floer-theoretic knot concordance and homology cobordism invariants. Building on these ideas, we describe a new family of homology cobordism invariants and give a new proof that there are no 2-torsion elements with Rokhlin invariant 1. This is joint work in progress with Irving Dai, Matt Stoffregen, and Linh Truong.
The Poincaré metric on the unit disc $\mathbb{D} \subset \mathbb{C}$, known for its invariance under all biholomorphisms (bijective holomorphic maps) of $\mathbb{D}$, is one of the most fundamental Riemannian metrics in differential geometry.
We present a novel example of a Lorentzian manifold-with-boundary featuring a dramatic degeneracy in its deterministic and causal properties known as “causal bubbles” along its boundary. These issues arise because the regularity of the Lorentzian metric is below Lipschitz and fit within the larger framework of low regularity Lorentzian geometry. Although manifolds with causal bubbles were recently introduced in 2012 as a mathematical curiosity, our example comes from studying the fundamental equations of fluid mechanics and shock singularities which arise therein.
We revisit the classical problem of constructing a developable surface along a given Frenet curve $\gamma$ in space. First, we generalize a well-known formula, introduced in the literature by Sadowsky in 1930, for the Willmore energy of the rectifying developable of $\gamma$ to any (infinitely narrow) flat ribbon along the same curve. Then we apply the direct method of the calculus of variations to show the existence of a flat ribbon along $\gamma$ having minimal bending energy. Joint work with Simon Blatt.
In 1976, Thurston decidedly showed that symplectic geometry and Kähler geometry were strictly distinct by providing the first example of a compact symplectic manifold which is not symplectomorphic to any Kähler manifold. Since this example, first studied by Kodaira, much work has been done in explicating the difference between algebraic manifolds such as affine and projective varieties, complex manifolds such as Stein and Kähler manifolds, and general symplectic manifolds.
There are many bizarre examples in the world of smooth 4-manifolds. We will describe some of these examples and discuss a tool from gauge theory that may be used to compute some of the invariants that distinguish these weird examples.
Thompson links are links arising from elements of the Thompson group. They were introduced by Vaughan Jones as part of his effort to construct a conformal field theory for every finite index subfactor. In this talk I will first talk about Jones' construction of Thompson links. Then I will talk about grid diagrams and introduce a notion of half grid diagrams to give an equivalent construction of Thompson links and further associate with each Thompson link a canonical Legendrian type.
A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a given contact manifold? Contact surgery number is a measure of this complexity. In this talk, I will discuss this notion of complexity along with some examples. This is joint work with Marc Kegel.
