Gromov-Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov-Witten invariants are also invariant under appropriately defined logarithmic modifications.

We contribute to the foundations of tropical geometry with a view towards formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.Our approach to tropical geometry permits tropical moduli problems-moduli of curves or otherwise-to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

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Since Jun Li's original definition, several other definitions of expanded pairs and expanded degenerations have appeared in the literature. We explain how these definitions are related and introduce several new variants and perspectives. Among these are the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneation formulas.

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We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: J. Li's original approach, B. Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory without expansions due to Gross-Siebert and Abramovich-Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov-Witten invariants associated to all four of these theories are identical.

The field of microfluidics is undergoing rapid growth in terms of new device and system development.Among the many methods of fabricating microfluidic devices and systems, surface micromachining is relatively underrepresented due tw difficulties in the introduction of fluids into the very small channels produced, packaging problems, and difficulties in device and system characterization. The potential advantages of using surface micromachining includti compatibility with the existing integrated circuit tool set, integration of electronic sensing and actuation with microfluidics, and fluid volume minimization. In order to explore these potential advantages we have developed first generation surface micromachined microfluidic devices (channels) using an adapted pressure sensor fabrication procms to produce silicon nitride channels, and the SUMMiT process to produce polysilicon channels. The channels were characterized by leak testing and flow rate vs. pressure measurements. The fabrication processes used and results of these tests are reported in this paper.