"How to get an awesome postdoc position"

"A multi-scale integrative model of uterine fluid dynamics"

We present a model of intra-uterine fluid flow in a sagittal cross-section of the uterus by inducing peristalsis in a 2D channel.  This is an integrative multiscale computational model that takes as input fluid viscosity, passive tissue properties of the uterine channel and a prescribed wave of membrane depolarization.  This voltage pulse is coupled to a model of calcium dynamics inside a uterine smooth muscle cell, which in turn drives a kinetic model of myosin phosphorylation governing contractile muscle forces.  Using the immersed boundary method, these muscle forces are communicated to a fluid domain to simulate the contractions which occur in a human uterus.  An analysis of the effects of model parameters on the flow properties and emergent geometry of the peristaltic channel will be presented.

"The Risch-Norman Algorithm for Symbolic Indefinite Integration"

The Risch-Norman algorithm is a heuristic method for symbolically computing indefinite integrals that is conceptually simpler than the famous Risch algorithm, but lacks the ability to prove that a given integral is non-elementary.  It is, however, much better suited to certain extensions allowing it to handle a much larger class of functions.  In this talk the basic algorithm will be outlined, and, if time permits, some extensions will be presented.

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Mardi Gras

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"Designing Modern Numerical Methods Using Analytical Properties of PDEs"

Our understanding of the fundamental processes of the natural world is based on a large extent on partial differential equations (PDEs) that describe a variety of phenomena in physical, astrophysical, geophysical, meteorological, biological, chemical, financial, social and other scientific areas.  Applications are abundant and widespread.  A few examples include: shallow water and internal waves in ocean or atmosphere modeling, shock waves and rarefaction waves in gas dynamics, geophysical flows associated with tsunamis, volcanoes, debris flows, etc., porous media flows, e.g. water or petroleum under the earth, blood flow in tissue and bone, models of chemotaxis, computer vision and computational anatomy.

While these applications vary greatly, there is a common mathematical structure to the equations which arise from all of these applications.  A powerful set of computational techniques have been developed over the past several decades to compute accurate solutions to these problems.  In this talk, I will explore the basic mathematical properties of the underlying PDEs and explain how these properties can be used to develop modern numerical methods.

"Schoeberl, Wallis and Cylindrical Algebraic Decomposition"

Recently Stefan Gerhold and Manuel Kauers introduced a new approach for algorithmically proving inequalities involving a discrete parameter. The proof then proceeds by induction along this discrete parameter, where the proof of the induction step is carried out using cylindrical algebraic decomposition (CAD). This is the first-and to this day only-practical method available for applying computer algebra to proving special functions inequalities. In this talk we introduce this approach and present two recent results obtained by it. First a direct application: the proof of a positivity conjecture by Joachim Schoeberl that arose in the construction of a new, stable interpolation operator for high order finite element methods. Second a variant of the Gerhold/Kauers approach: an automated way to sharpen classical estimates such as Wallis' inequality on approximating sequences for pi. The latter is joint work with Peter Paule.

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"From Wood to Chrome"

This talk is going to give an overview of some techniques possible with modern programmable graphics hardware to simulate surface materials. As an end-of-the-semester talk it is going to focus on illustrating the possibilities (along with pictures and live demonstrations) rather than on the theory