University of Washington

Let X be a connected smooth projective variety over Q. If X has a Q point, then X must have local points, i.e. points over the reals and over the p-adic completions Q_p. However, local solubility is often not sufficient. Manin showed that quadratic...

I will describe some scary moments that have occurred in the history of elliptic curve cryptography almost from its inception; talk about some of the ironies of controversies in cryptography -- such as RSA's role as defender of strong encryption; and...

Cryptography is the mathematical science of keeping secrets and maintaining trust. Once an arcane study of primarily military significance, over the past several decades the advent of the internet has invigorated this science. There are now results...

Representation theory of finite groups seeks to understand functions on the group known as irreducible characters. A group like GL(n,F_q), the group of invertible square matrices with entries in a finite field F_q, originates in algebraic geometry...

We introduce a notion of what it means for one graph to be a good spectral approximation of another, and prove that every graph can be well-approximated by a graph with few edges.We ask how well a given graph can be approximated by a sparse graph...

A typical measure of risk in finance must take into account of the uncertainty of probability model itself (called Knightian uncertainty). Nonlinear expectation and the corresponding non-linear distributions provides a deep and powerful tool...

The rings studied by students in most first-year algebra courses turn out to have what's known as the "Invariant Basis Number" property: for every pair of positive integers m and n, if the free left R-modules RRm and RRn are isomorphic, then m = n...

In the last 30 years representations of (infinite) Galois groups have played an increasingly important role in number theory. Indeed, arithmetic objects such as the Diophantine equation y2=x3+x2+1 or xn+yn=zn often have attached Galois...