I will discuss the reliability of large networks of coupled oscillators in response to fluctuating inputs. The networks considered are quite generic. In this talk, I view them as idealized models from neuroscience and borrow some of the associated language. Reliability is the opposite of trial-to-trial variability; a system is reliable if a signal elicits identical responses upon repeated presentations. I will address the problem on two levels: neuronal reliability, which concerns the behavior of individual neurons (or oscillators) embedded in the network, and pooled-response reliability, which measures total outputs from subpopulations. The effects of network structure, cell heterogeneity and noise on reliability will be discussed. Our findings are based largely on dynamical systems ideas (with a slight statistical mechanics flavor) and are supported by simulations. This is joint work with Kevin Lin and Eric Shea-Brown.

Let k be a local field, and let K be a separable quadratic field extension of k. It is known that an irreducible complex representation π_1 of the unitary group G_1 = U_n(k) has a multiplicity free restriction to the subgroup G_2 = U{n−1}(k) fixing a non-isotropic line in the corresponding Hermitian space over K. More precisely, if π_2 is an irreducible representation of G_2 , then π = π_1 ⊗ π_2 is an irreducible representation of the product G = G_1 G_2 which we can restrict to the subgroup H = G_2 , diagonally embedded in G. The space of H-invariant linear forms on π has dimension ≤ 1.

In this talk, I will use the local Langlands correspondence and some number theoretic invariants of the Langlands parameter of π to predict when the dimension of H-invariant forms is equal to 1, i.e. when the dual of π_2 occurs in the restriction of π_1 . I will also illustrate this prediction with several examples, including the classical branching formula for representations of compact unitary groups. This is joint work with Wee Teck Gan and Dipendra Prasad.

An overall picture of the theory Ricci flow for the general audience.

We say a region of space is "cloaked" with respect to electromagnetic measurements if its contents -- and even the existence of the cloak -- are inaccessible to such measurements. One recent proposal for such cloaking takes advantage of the coordinate-invariance of Maxwell's equations. As usually presented, this scheme uses a singular change of variables. That makes the mathematical analysis subtle, and the practical implementation difficult. This talk examines the correctness and robustness of the change-of-variable-based scheme, for scalar waves modelled by Helmholtz's equation, drawing on joint work with Onofrei, Shen, Vogelius, and Weinstein. The central idea is to use a less-singular change of variables. The quality of the resulting "approximate cloak" can be assessed by studying the detectability of a small inclusion in an otherwise uniform medium. We show that a small inclusion can be made nearly undetectable (regardless of its contents) by surrounding it with a suitable lossy layer.

Lagrangian coherent structures (LCS) are best described as moving curves in a fluid that separate particles that have qualitatively different trajectories. For instance, particles that circulate in an ocean bay have a separate behavior from particles that go on by the bay and don't get caught up in the circulation. Interestingly, these two classes of particles are separated by a sharp, but moving curve. Similar structures are found in Hurricanes: which particle are going to get swept up in the Hurricane and which don't? Likewise in Jellyfish, some particles enter the underbelly of the jellyfish and bring nutrients, while others are swept downstream to help propel the jellyfish. The way blood flows over a clot, as revealed by LCS can indicate whether or not the clot is dangerous. This lecture will give examples of this sort, explain how the LCS are computed and are connected with other mathematical constructions, such as Smale horseshoes in dynamical systems.