A Beltrami field on an open set U in R^3 is a vector field u on U satisfying the conditions
curl(u) = f*u, div(u) = 0,
where f is a smooth function on U. When f is constant, it is straightforward to show via Cartan-Kahler that local, real analytic solutions to these equations depend on 2 arbitrary functions of 2 variables; in particular, when f=0, solutions have the form u = grad(h), where h is an arbitrary harmonic function on U.
When f is non-constant, the story is more complicated. Encisco and Peralta-Salas have shown that "Beltrami fields with a non-constant proportionality factor are rare," in fact, there are no solutions at all unless f satisfies an explicit (but very complicated) differential equation.
In this work (joint with Stephen Preston and Taylor Klotz), we seek to gain a better understanding of Beltrami fields with non-constant proportionality factor. Using a moving frames approach, we prove the following results regarding local solutions in a neighborhood of any point in U where f and grad(f) are both nonzero:
(1) If the level surfaces of f are totally umbilic (i.e., open subsets of planes or spheres), then there are no local solutions unless the level sets of f are contained in either parallel planes or concentric spheres, in which case local solutions depend on 2 functions of 1 variable.
(2) If the level surfaces of f contain no umbilic points, then there is at most a 3-dimensional space of local solutions. We also give an explicit example with a 2-dimensional space of local solutions.
Unfortunately, the question of precisely which functions f admit solutions remains computationally intractable. A Cartan-Kahler argument suggests that the functions f admitting solutions should depend on 3 functions of 2 variables, so the tantalizing question remains: which functions f admit solutions, and how many solutions does each such function admit?
Beltrami fields with non-constant proportionality factor via moving frames
Oct. 25, 2016 1pm (MATH 220)
James Mitchell (St Andrews, UK)
X
In this talk I will outline recent progress in the application of computational group theory to the problem of effectively computing a finite semigroup. Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which thJaese results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and -matrix semigroups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate its ideal structure, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.
Computing with finite semigroups Sponsored by the Meyer Fund
Oct. 25, 2016 2pm (MATH 220)
James Mitchell (St Andrews, UK)
X
In this talk I will discuss the problem of finding minimally generated dense subgroups of the groups of automorphisms of some interesting combinatorial structures, such as the natural numbers, the random graph, the rational numbers with the usual order, and the so-called countable ultrahomogeneous graphs.
Approximating permutations and automorphisms Sponsored by the Meyer Fund