Department of

Mathematics


Seminar Calendar
for events the day of Monday, December 5, 2005.

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Monday, December 5, 2005

3:00 pm in 2005 MEL,Monday, December 5, 2005

The Development of Hydrodynamically Unstable Flames

Moshe Matalon   [email] (Dept. of Mechanical Engineering, Northwestern University)

Abstract: The hydrodynamic instability, discovered independently in theoretical analyses by Darrieus and Landau over half a century ago, has many ramifications in combustion. The appearance of sharp folds and creases on the flame front of freely propagating flames, and the wrinkling observed over the surface of large spherically expanding flames are direct manifestations of this instability. The scale of the wrinkles in such circumstances is typically much larger than the characteristic cell size of the more conventional cellular flames, which result from diffusive-thermal instabilities. Furthermore, the wrinkled flame is seen to accelerate as it propagates further and may turn into a turbulent flame when reaching a sufficiently large size. The nonlinear development of hydrodynamically unstable flames has primarily relied on analytical and numerical investigation of simplified models, such as the weakly-nonlinear Michelson-Sivashinsky equation which assumes that the density change across the flame is relatively small. In this work we examine the flame evolution using a fully nonlinear model which places no restriction on the thermal expansion parameter. The computations are carried out within the context of the hydrodynamic theory, which contains fewer parameters than the full governing equations, yet permits description of multi-dimensional flames with sufficient accuracy over a wider range of conditions.

4:00 pm in 245 Altgeld Hall,Monday, December 5, 2005

Murphy's Law in algebraic geometry: Badly-behaved moduli spaces

Ravi Vakil (Stanford University)

Abstract: We consider the question: ``How bad can the deformation space of an object be?'' (Alternatively: ``What singularities can appear on a moduli space?'') The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad.'' We show this for a number of important moduli spaces. More precisely, up to smooth parameters, every singularity that can be described by equations with integer coefficients appears on moduli spaces parameterizing: smooth projective surfaces (or higher-dimensional manifolds); smooth curves in projective space (the space of stable maps, or the Hilbert scheme); plane curves with nodes and cusps; stable sheaves; isolated threefold singularities; and more. The objects themselves are not pathological, and are in fact as nice as can be. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. I will begin by telling you what ``moduli spaces'' and ``deformation spaces'' are. The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. This talk is intended to be (mostly) comprehensible to a broad audience. Host: Sheldon Katz