Many constructions in algebraic geometry require one to choose a point
outside a countable union of subvarieties. Over $\C$ this is always
possible. Over a countable field, a countable union of subvarieties
can cover all the closed points. Let $k$ be a finitely generated
field of characteristic zero and let $\kbar$ be an algebraic closure.
Let $A$ be a semiabelian variety defined over $k$, and let $\End(A)$
be the ring of endomorphisms of $A$ over $\kbar$. Let $X\subset A$ be
a subvariety of smaller dimension. We show that $\Union_{f\in
\End(A)} f(X(\kbar))$ does not equal $A(\kbar)$. Bogomolov and
Tschinkel show that the above is false for $k$ equal to an algebraic
closure of a finite field, and use the result to show that on any
Kummer surface over such $k$, the union of all rational curves covers
all of the closed points. We give further examples of such problems.

Self assembly is the idea of creating a system whose component parts spontaneously assemble into a structure of interest. In this talk I will outline our research program aimed at creating self-assembled structures out of very small spheres, that bind to each other on sticking. The talk will focus on

(i) some fundamental mathematical questions in finite sphere packings (e.g. how do the number of rigid packings grow with N, the number of spheres);

(ii) algorithms for self assembly (e.g. suppose the spheres are not identical, so that every sphere does not stick to every other; how to design the system to promote particular structures);

(iii) physical questions (e.g. what is the probability that a given packing with N particles forms for a system of colloidal nanospheres); and

(iv) comparisons with experiments on colloidal nanospheres.

Let k be a local field, and let K be a separable quadratic field extension of k. It is known that an irreducible complex representation π_1 of the unitary group G_1 = U_n(k) has a multiplicity free restriction to the subgroup G_2 = U{n−1}(k) fixing a non-isotropic line in the corresponding Hermitian space over K. More precisely, if π_2 is an irreducible representation of G_2 , then π = π_1 ⊗ π_2 is an irreducible representation of the product G = G_1 G_2 which we can restrict to the subgroup H = G_2 , diagonally embedded in G. The space of H-invariant linear forms on π has dimension ≤ 1.

In this talk, I will use the local Langlands correspondence and some number theoretic invariants of the Langlands parameter of π to predict when the dimension of H-invariant forms is equal to 1, i.e. when the dual of π_2 occurs in the restriction of π_1 . I will also illustrate this prediction with several examples, including the classical branching formula for representations of compact unitary groups. This is joint work with Wee Teck Gan and Dipendra Prasad.