Snap-through buckling is a type of instability in which an elastic object rapidly jumps from one state to another, just as an umbrella flips upwards in a gust of wind. While snap-through under dry, mechanical loads has already been harnessed in engineering to generate fast motions between two states, the mechanisms underlying snapping in bulk fluid flows remain relatively unexplored. In this talk we demonstrate how elastic snap-through may be used to passively control fluid flows at low Reynolds number, in contrast to some pre-existing valves that rely on active control. We study viscous flow through a channel in which one of the bounding walls is an elastic arch. By performing experiments at the macroscopic scale, we show that snap-through of the arch rapidly changes the channel from a constricted to an unconstricted state, increasing the hydraulic conductivity by up to an order of magnitude. We also observe nonlinear pressure-flux characteristics away from snapping due to the coupling between the driving flow and elasticity. This behaviour is confirmed by a mathematical model that also shows the device may readily be scaled down for microfluidic applications. Finally, we demonstrate that such a device may be used to create a fluidic analogue of a fuse: the fluid flux through a channel may not rise above a given value.

# Past Forthcoming Seminars

Current technological progress has raised concerns about automation of tasks performed by workers resulting in job losses. Previous studies have used machine learning techniques to compute the automation probability of occupations and thus, studied the impact of automation on employment. However, such studies do not consider second-order effects, for example, an occupation with low automation probability can have a surplus of labor supply due to similar occupations being automated. In this work, we study such second-order effects of automation using a network approach. In our network – the Job Space – occupations are nodes and edges link occupations which share a significant amount of work activities. By mapping employment, automation probabilities into the network, and considering the movement of workers, we show that an occupation’s position in the network may be crucial to determining its employment future.

**I will discuss recent developments in the study of scattering amplitudes in Einstein-Yang-Mills theory. At tree level we find new structures at higher order collinear limits and novel connections with amplitudes in Yang-Mills theory using the CHY formalism. Finally I will comment on unitarity based observations regarding one-loop amplitudes in the theory. **

In this presentation, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size ε, and its asymptotic behavior as the parameter ε vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to ε. This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light about the homogenization of the voltage potential uε caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possibly negative) conductivity a surrounded by a dielectric medium, with unit conductivity.

The cyclic surgery theorem of Culler, Gordon, Luecke, and Shalen implies that any knot in S^3 other than a torus knot has at most two nontrivial cyclic surgeries. In this talk, we investigate the weaker notion of SU(2)-cyclic surgeries on a knot, meaning surgeries whose fundamental groups only admit SU(2) representations with cyclic image. By studying the image of the SU(2) character variety of a knot in the “pillowcase”, we will show that if it has infinitely many SU(2)-cyclic surgeries, then the corresponding slopes (viewed as a subset of RP^1) have a unique limit point, which is a finite, rational number, and that this limit is a boundary slope for the knot. As a corollary, it follows that for any nontrivial knot, the set of SU(2)-cyclic surgery slopes is bounded. This is joint work with Raphael Zentner.

Abstract: In this talk we describe an invariance principle for a class of non-homogeneous martingale random walks in $\RR^d$ that can be recurrent or transient for any dimension $d$. The scaling limit, which we construct, is a martingale diffusions with law determined uniquely by an SDE with discontinuous coefficients at the origin whose pathwise uniqueness may fail. The radial component of the diffusion is a Bessel process of dimension greater than 1. We characterize the law of the diffusion, which must start at the origin, via its excursions built around the Bessel process: each excursion has a generalized skew-product-type structure, in which the angular component spins at infinite speed at the start and finish of each excursion. Defining a Riemannian metric $g$ on the sphere $S^{d−1}$, different from the one induced by the ambient Euclidean space, allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) as a time-changed Browninan motion with drift on the Riemannian manifold $(S^{d−1}, g)$. In particular, this provides a multidimensional generalisation of the Pitman–Yor representation of the excursions of Bessel process with dimension between one and two. Furthermore, the density of the stationary law of the angular component with respect to the volume element of $g$ can be characterised by a linear PDE involving the Laplace–Beltrami operator and the divergence under the metric $g$. This is joint work with Nicholas Georgiou and Andrew Wade.

The interplay of minimum degree and 'structural properties' of large graphs with a given forbidden subgraph is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum $\alpha$ such that every $n$-vertex $F$-free graph $G$ with minimum degree $>\alpha n$ has a homomorphic image $H$ of bounded size (independent of $n$), which is $F$-free as well. Without the restriction of $H$ being $F$-free we recover the definition of the chromatic threshold, which was determined for every graph $F$ by Allen et al. The homomorphism threshold is less understood and we present recent joint work with O. Ebsen on the homomorphism threshold for odd cycles.

In this talk two different methods for constructing complete steady and expanding Ricci solitons of cohomogeneity one will be discussed. The first is based on an estimate on the growth of the soliton potential and holds for large classes of cohomogeneity one manifolds. The second approach is specific to the two summands case and uses a Lyapunov function. This method also carries over to the Einstein case and as an application, a simplified construction of B\"ohm's Einstein metrics of positive scalar curvature on spheres will be explained.

We propose a new splitting algorithm to solve a class of quasilinear PDEs with convex and quadratic growth gradients.

By splitting the original equation into a linear parabolic equation and a Hamilton-Jacobi equation, we are able to solve both equations explicitly.

In particular, we solve the associated Hamilton-Jacobi equation by the Hopf-Lax formula,

and interpret the splitting algorithm as a stochastic Hopf-Lax approximation of the quasilinear PDE.

We show that the numerical solution will converge to the viscosity solution of the equation.

The upper bound of the convergence rate is proved based on Krylov's shaking coefficients technique,

while the lower bound is proved based on Barles-Jakobsen's optimal switching approximation technique.

Based on joint work with Shuo Huang and Thaleia Zariphopoulou.

We consider the one parameter mirror families W of the Calabi-Yau 3-folds with Picard-Fuchs equations of hypergeometric type. By mirror symmetry the even D-brane masses of orginial Calabi-Yau manifolds M can be identified with four periods with respect to an integral symplectic basis of $H_3(W,\mathbb{Z})$ at the point of maximal unipotent monodromy. We establish that the masses of the D4 and D2 branes at the conifold are given by the two algebraically independent values of the L-function of the weight four holomorphic Hecke eigenform with eigenvalue one of $\Gamma_0(N)$. For the quintic in $\mathbb{P}^4$ it this Hecke eigenform of $\Gamma_0(25)$ was as found by Chad Schoen. It was discovered by de la Ossa, Candelas and Villegas that its coefficients $a_p$ count the number of solutions of the mirror quinitic at the conifold over the finite number field $\mathbb{F}_p$ . Using the theory of periods and quasi-periods of $\Gamma_0(N)$ and the special geometry pairing on Calabi-Yau 3 folds we can fix further values in the connection matrix between the maximal unipotent monodromy point and the conifold point.