Georgia Institute of Technology College of Sciences

Isometric embeddings between a domain manifold and a target manifold are differentiable maps f such that the pullback of the target metric h coincides with the metric g in the domain manifold. This problem can also be formulated as a non-linear PDE via $\nabla f^{\top} h \nabla f = g$. In the case of contact manifolds, it is additionally required that the embedding preserves a certain restriction on the tangent bundle.

We prove that the Nash iteration scheme can be quantified in order to construct infinitely many $C^{1,\alpha}$-isometric embeddings for contact manifolds. In this way, we extend an existing result regarding non-uniqueness for $C^1$ regularity. The strategy of the proof follows a paper by Conti, De Lellis and Szekelyhidi Jr. on the Riemannian case, which is built on the Nash-Kuiper scheme. The main difficulty in our case is to keep the additional linear constraint coming from the contact setting along the iteration procedure.

In the larger program of a quantitative analysis of isometric embeddings between sub-Riemannian manifolds, our result can be seen as an important first step. Another aspect is the flexibility of this convex integration method: the geometric constraint coming from the contact condition is just one special case of a (potentially large) class of admissible constraints, under which this scheme can still be applied.

 A central theme in algebraic geometry is understanding how much of the well-behaved theory of smooth varieties survives in the presence of singularities. Du Bois and rational singularities are among the most important classes studied in algebraic geometry due to their nice cohomological behavior. For instance, they preserve features like the Hodge decomposition and vanishing theorems one expects for smooth varieties.
Recently, motivated by developments in Hodge theoretic methods, there has been substantial interest in studying their higher analogs. This talk will survey recent developments connecting these notions to deformation properties and moduli theory, and applications to Calabi-Yau varieties. I will also report on recent joint work with Brian Nugent extending the theory of higher Du Bois and higher rational singularities to pairs — a framework that is both essential in modern birational geometry and a natural setting for studying Hodge theory for open varieties.

The Heegaard Floer d-invariant is a numerical invariant of rational homology spheres which is analogous to the Frøyshov h-invariant from Instanton theory. In this talk, we use Zemke’s recent isomorphism between lattice Floer and Heegaard Floer homology to compute the d-invariant for all rational homology spheres which arise as negative definite plumbed manifolds, verifying a 20 year old conjecture of Némethi. 

As a generalization of Hamiltonicity problems, one may consider partitioning the vertices of a (directed) graph into as few paths (or cycles) as possible. Ore's classical theorem gives a tight upper bound on the number of paths needed to cover an n-vertex graph, with imbalanced bipartite graphs showing that the bound is best possible. A conjecture of Magnant and Martin (2009) suggests that Ore's bound can be significantly improved for regular graphs. This is also connected to the famous linear arboricity conjecture and has attracted considerable attention in recent years, including a very recent result establishing the conjecture up to a factor of two. In this talk, I will discuss directed and oriented variants of this conjecture and present some results in these settings. Based mostly on joint work with Allan Lo and Viresh Patel.