# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Classical work of Jeffery from 1922 established how at low Reynolds number, ellipsoids in steady shear flow undergo periodic motion with non-uniform rotation rate, termed 'Jeffery orbits'. I will present two problems where Jeffery orbits play a critical role in understanding the transport and aggregation of rod-shaped organisms. I will discuss the trapping of motile chemotactic bacteria in high shear, and the sedimentation rate of negatively buoyant plankton.

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals, and they are central to the theory of rough paths in stochastic analysis. For some special families of curves, such as polynomial paths and piecewise-linear paths, their parametrized signature tensors trace out algebraic varieties in the space of all tensors. We introduce these varieties and examine their fundamental properties, while highlighting their intimate connection to the problem of recovering a path from its signature. This is joint work with Peter Friz and Bernd Sturmfels.

Stochastic analysis on a Riemannian manifold is a well developed area of research in probability theory.

We will discuss some recent developments on stochastic analysis on a manifold whose Riemannian metric evolves with time, a typical case of which is the Ricci flow. Familiar results such as stochastic parallel transport, integration by parts formula, martingale representation theorem, and functional inequalities have interesting extensions from

time independent metrics to time dependent ones. In particular, we will discuss an extension of Beckner’s inequality on the path space over a Riemannian manifold with time-dependent metrics. The classical version of this inequality includes the Poincare inequality and the logarithmic Sobolev inequality as special cases.

I will prove that the knot Floer homology group

HFK-hat(K, g-1) for a genus g fibered knot K is isomorphic to a

variant of the fixed points Floer homology of an area-preserving

representative of its monodromy. This is a joint work with Gilberto

Spano.

Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDES. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, and outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.

I will present a procedure for perturbatively constructing the field content of gravitational theories from a convolutive product of two Yang-Mills theories. A dictionary "gravity=YM * YM" is developed, reproducing the symmetries and dynamics of the gravity theory from those of the YM theories. I will explain the unexpected, yet crucial role played by the BRST ghosts of the YM system in the construction of gravitational fields. The dictionary is expected to develop into a solution-generating technique for gravity.

We prove that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$. This result improves on a 30-year old bound of F\"uredi and Kahn, and is within a $\log^{o(1)}\Delta$ factor of optimal. We prove this result via the notion of boxicity. The \emph{boxicity} of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $\Delta$ has boxicity at most $\Delta\log^{1+o(1)} \Delta$, which is also within a $\log^{o(1)}\Delta$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $\Theta(\sqrt{g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a $O(1)$ factor. This is joint work with Alex Scott (arXiv:1804.03271).