Department of Mathematics Colloquium
Abstract: A famous and difficult theorem of Szemeredi asserts that every subset of the integers of positive density will contain arbitrarily long arithmetic progressions; this theorem has had four different proofs (graph-theoretic, ergodic, Fourier analytic, and hypergraph-theoretic), each of which has been enormously influential, important, and deep. It had been conjectured for some time that the same result held for the primes (which of course have zero density). I shall discuss recent work with Ben Green obtaining this conjecture, by viewing the primes as a subset of the almost primes (numbers with few prime factors) of positive relative density. The point is that the almost primes are much easier to control than the prime themselves, thanks to sieve theory techniques such as the recent work of Goldston and Yildirim. To "transfer" Szemeredi's theorem to this relative setting requires that one borrow techniques from all four known proofs of Szemeredi's theorem, and especially from the ergodic theory proof.
Lefschetz Center for Dynamical Systems Seminar
Abstract: Spiral waves are planar patterns that arise in many natural systems. Part of their fascination is due to the intriguing instabilities, such as meandering and drifting, core and far-field breakup, and spatio-temporal period doubling, that they exhibit. Among the challenges for theoretical studies of spirals is the task of relating these instabilities to spiral spectra. In this talk, I will present recent results that predict certain aspects of spiral spectra and outline how these can be used to explain some of the observed instabilities of spiral waves.
SPECIAL SEMINAR -- JOB CANDIDATE
Abstract: I will give an overview of coherent structures in one and two space dimensions. These structures can be thought of as interfaces between spatially periodic patterns: A prominent example are planar spiral waves. Coherent structures have been observed in many biological, chemical and physical experiments. In this talk, I will discuss their shape and robustness, put forward a classification that distinguishes different classes of coherent structures, and give an overview of the underlying techniques such as spatial dynamics and spectral theory used to carry out these analyses.
Center for Fluid Mechanics Seminar
Abstract: Insects, like birds and fish, locomote via interactions between fluids and flapping wings and fins. Their motion is governed by the Navier-Stokes equation coupled to moving boundaries. In this talk, I will first describe how dragonflies fly: their wing motions and the flows and forces they generate. I will then consider insects in several species and discuss three questions: 1) Is insect flight optimal? 2) How does the efficiency of flapping flight compare to classical fixed-wing flight? 3) How might aerodynamic effects have influenced the evolution of insect flight?
Brown Analysis Seminar
Abstract: We start with an overview of the wavelet representation and connect it with the Laplacian pyramidal representation. We then discuss some of existing challenges in the wavelet/pyramidal representations, especially in high dimensions. Among the new methodologies for representing data on regular grids, which are hybrids of the wavelet and the pyramidal representations, two methodologies will be discussed in detail. The first one is called the L-CAMP and it leads to extremely local wavelet systems in high dimensions. The other methodology leads to (non redundant) biorthogonal systems that allow more directionalities than standard separable wavelet systems in high dimensions. Both representations are computed and inverted by fast algorithms, and they are accompanied by solid mathematical theory that reveals their performance in terms of the maximal level of smoothness that is accurately encoded by the representations.
PDE Seminar
<--- 2007 Index