University of Washington

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...

We show how product decompositions of measures can detect directionality in sets. In order to show this we prove that sets of small measure are always contained in a "small" number of Lipschitz surfaces. We also discuss the higher co-dimensional...

Let f = f(z1, . . . , zn) be a holomorphic function defined on an open subset P ∈ U ⊂ Cn. The log canonical threshold of f at P is the largest s ∈ R such that |f|s is locally integrable at P. This invariant gives a sophisticated measure of the...

In the last two decades, there has been substantial progress in the study of nonlinear dispersive PDE thanks to the influx of ideas and tools from nonlinear Fourier and harmonic analysis, geometry and analytic number theory, to the existing...

The analy­sis of the sta­bil­ity and effi­ciency of algo­rithms for con­vex opti­miza­tion nat­u­rally leads to the study of con­di­tion num­bers. The Grass­mann con­di­tion, which is a geo­met­ric ver­sion of Renegar’s con­di­tion, is espe­cially...

The study of first-order methods has largely dominated research in continuous optimization for the last decade, yet still the range of problems for which efficient and easily-applicable first-order methods have been developed is surprisingly limited...

Many sys­tems involve the allo­ca­tion of rewards for achieve­ments, and these rewards pro­duce a set of incen­tives that in turn guide behav­ior. Such effects are vis­i­ble in many domains from every­day life, and they are increas­ingly form­ing a...

Random matrix theory studies the asymptotics of the spectral distributions of families of random matrices, as the sizes of the matrices tend to infinity. Derivation such asymptotics frequently requires analyzing the spectral properties of random...