We study the free boundary Euler equations in three spatial dimensions with surface tension. Under natural assumptions, we prove that solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity. The well-posedness of the free-boundary equations under the relevant hypothesis for the study of the limit is also established. This is joint work with David G. Ebin.
The free boundary Euler equations with large surface tension Sponsored by the Meyer Fund
Functions such as can always be integrated (with respect to ) in ``elementary terms" involving logarithms and exponentials. But not unless . A more interesting example is , which can be done also for . In 1981 James Davenport claimed that an arbitrary such algebraic can be integrated for at most finitely many special complex values (unless it can be integrated for a general value of ). Umberto Zannier and I dare to hope for a full proof in the next couple of years; but for now I will content myself with a general discussion of the problem together with some of the key concepts involved in settling significant special cases.
The unlikelihood of integrability in elementary terms
Mar. 18, 2014 2pm (MATH 350)
Lie Theory
Eric Marberg (Stanford)
X
A vector species is a functor from the category of finite sets with bijections to vector spaces; informally, one can view this as a sequence of -modules, one for each natural number . A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. In this talk I will discuss the structure theory of Hopf monoids exhibiting certain strong forms of self-duality, and explain some results which can be viewed as species analogues of Zelevinsky's classification of positive self-dual Hopf algebras. As applications, I will describe how such results can be used to provide a simple way of constructing and understanding the relationships between various Hopf algebras attached to towers of unipotent groups, recently considered in the literature.
Strongly self-dual Hopf monoids in species with applications to combinatorial Hopf algebras Sponsored by the Meyer Fund
Mar. 18, 2014 3pm (Math 220)
Algebraic Geometry
Maksym Fedorchuk (Boston College)
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I will discuss a GIT problem concerning stability of syzygy points of canonical curves, and explain its solution for the first syzygy point in the generic case (joint with Deopurkar and Swinarski). I will also explain connections with Green's conjecture and applications related to the minimal model program for the moduli space of curves.
The free boundary Euler equations with large surface tension