Standard methods for approximating the solution of time-dependent PDEs typically produce sequential time-stepping algorithms, which are not optimally efficient on today's highly-parallel supercomputers. Spacetime finite element methods have been developed over the last few years as an alternative approach which can harness the massive parallelism of modern computing platforms. In this talk, I will give an overview of what spacetime finite element methods are and how they differ from traditional methods. I will then discuss some a priori error estimates for these methods, as well as strides towards a posteriori error estimators which are reliable and efficient.

For integers $k\geq 1$ and $n\geq 2k+1$, the \emph{Kneser graph} $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the \emph{odd graphs}. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. The proof is based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words. As a byproduct, we obtain a new proof of the so-called middle levels conjecture. This is joint work with Torsten Mütze and Jerri Nummenpalo.

From Helmholtz to vaping hipsters, the dynamics of vortex filaments, i.e. fluids with vorticity concentrated along a smooth curve, has been a topic of significant interest in fluid dynamics. The global well-posedness of vortex filaments with small circulation follows from the theory of mild solutions of the 3d Navier-Stokes equations at critical regularity. However, for filaments with large circulation these results no longer apply. In this talk we discuss a proof of well-posedness (in a suitable sense) for vortex filaments of arbitrary circulation. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of the 3d Navier-Stokes without additional symmetry assumptions. This is joint work with Jacob Bedrossian and Pierre Germain.

Let $A$ denote a non-constant ordinary abelian surface over a
global function field (of characteristic $p > 2$) with good reduction
everywhere. Suppose that $A$ does not have real multiplication by any real
quadratic field with discriminant a multiple of $p$. Then we prove that
there are infinitely many places modulo which $A$ is isogenous to the
product of two elliptic curves. This is joint work with Davesh Maulik and
Yunqing Tang.