Let K be a local field with residue field of characteristic p>0. Our goal is to understand the cyclic extensions of K of degree a power of p. If K has characteristic 0 and contains a p^m-th primitive root of unity, then one can use class field theory and Kummer theory to construct a symbol which helps us to understand the ramification of cyclic extensions of degree p^m. If K has characteristic p, then one can construct a symbol, using class field theory and Artin-Schreier-Witt theory, which helps us to understand the cyclic extensions of degree p^m for any m. We will discuss both symbols in more detail and discuss methods for computing these symbols.

The pursuit of better understanding the universe of set theory V motivated an extensive study of definable inner models M whose goal is to serve as good approximations to V. A common property of these inner models is that they are contained in HOD, the universe of hereditarily ordinal definable sets. Motivated by the question of how ``close" HOD is to V, we consider various related forcing methods and survey known and new results. This is a joint work with Spencer Unger.

Dirichlet’s approximation theorem states that for every real number x there exist infinitely many rationals p/q with |x-p/q| < 1/q^2. If x is in the unit interval, then viewing rationals as algebraic numbers of degree 1, q is also the height of its primitive integer polynomial, where height means the maximum of the absolute values of the coefficients. This suggests a more general question: How well can real numbers be approximated by algebraic numbers of degree at most n, relative to their heights? We will discuss Wirsing’s conjecture which proposes an answer to this question and Schmidt and Davenport’s proof of the n = 2 case, as well as some open questions.