Why general relativity does not admit enough observables

One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity. Some of the core technical and epistemological difficulties come from the fact that General Relativity is fundamentally a geometric theory and, as such, it ought to be invariant under change of coordinates by the arbitrary element of the diffeomorphism group Diff(M) of the ambient manifold M.

Let's count things

Arithmetic statistics is an area devoted to counting a wide range of objects of algebraic interest, such as polynomials, fields, and elliptic curves.  Fueled by the interplay of analysis and number theory, we'll count polynomials and number fields, which though basic objects of study in number theory, are quite difficult to actually count.  How often does a random polynomial fail to have full Galois group?  How many number fields are there?  We will address both of these questions today.

The Apportionment Problem for the U.S House of Representatives

We focus on the history and the mathematics of the apportionment problem for the US House of Representatives.  An apportionment is a function from {1, 2, ..., s} (where s is the number of states) to the positive integers, A(i), so that the sum of the A(i) is H, the house size.  Today s=51 (including the District of Columbia) and H=435, although s and H have had many values since 1790. Based on census data, one can compute the fair share of representatives for state i, call it f_i, which might turn out to be 4.69435.