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.

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.

Extending results of Van Der Vlugt,
I shall derive a new non-trivial upper bound for the dimension of trace
codes connected to algebraic-geometric codes. Furthermore, I will deduce
their true dimension if certain conditions are satisfied. Finally,
potential areas of improvement and other related results will be outlined.

How many ways can an integer n be expressed as a sum of positive integers? This question is the cornerstone of Partition Theory, and it is surprisingly difficult to answer. For example, if we let p(n) be the number of these expressions of n, even the parity of p(n) remains something of a mystery, despite the fact that it has been studied for over a century. In particular, although empirical evidence (the first several million values) seems to indicate that Po(N) = [the number of odd values of p(n) up to N] is asymptotic to N/2, no one has even been able to show that Po(N) is larger than the square root of N for N sufficiently large. Many advances in discovering the mod 2 behavior of p(n) have been made over the past several years, and most of them have required properties of l-adic Galois representations and the theory of modular forms. However, one lower bound for Po(N) (which was the state-of-the-art for a brief period) was proven using only elementary generating function techniques and results from classical analytic number theory. In this talk, we develop the history of the mystery, and we prove the latter lower bound. The talk will be aimed at the partition theoretically uninitiated, and a great deal of background will be provided.