We discuss algebras over a field with the unusual property that all of their subalgebras are noetherian. We discuss some of the general results one can prove about such algebras. Some well-known algebras associated to elliptic curves turn out to have this property, and we discuss these examples in detail.

A corollary of the Erd\H{o}s-Stone theorem is that, for any $0 \leq
\alpha < 1$, graphs with density greater than $\alpha$ contain an
(arbitrarily) large subgraph of density at least $\alpha+c$ for some
fixed $c = c(\alpha)$, so long as the graph itself is sufficiently
large. This phenomenon is known as a jump at $\alpha$. Erd\H{o}s
conjectured that similar statements should hold for hypergraphs, and
multigraphs where each edge can appear with multiplicity at most $q$,
for $q \geq 2$ fixed. Brown, Erd\H{o}s, and Simonovits answered this
conjecture in the affirmative for $q=2$, that is for multigraphs where
each edge can appear at most twice. R\"{o}dl answered the question in

We will show that smooth complete solutions to the Ricci flow of
uniformly bounded curvature are analytic in time in the interior of
their interval of existence. The analyticity is a consequence of
classical Bernstein-Bando-Shi type estimates on the temporal and
spatial derivatives of the curvature tensor, and offers an alternative proof of the unique continuation of solutions to the Ricci flow. As a further application of these estimates, we will show that, under the above global hypotheses, about any interior space-time point (x0, t0), there exist local coordinates x on a neighborhood U of x0 in which the representation of the metric is real-analytic in both x and t on some cylinder over U.

Biological membranes are composed of (among other things) hundreds of different lipids, which are believed to segregate into fluid rafts, which may be relevant to processes like virus assembly. I'll talk about the spherical cow version of cells, synthetic membranes with three components (saturated and unsaturated lipids and cholesterol), which also segregate into two fluid phases. Membranes are also particularly interesting from a physical standpoint because they have both two- and three-dimensional hydrodynamic behavior ("quasi-2D"), with many strange features, such as diffusion coefficients of membrane rafts being effectively independent of their size. These quirks are characteristic of many interfacial fluids, and also appear in thin layers of liquid crystals and protein films at the air-water interface. I'll show some continuum stochastic simulations of membrane domains and phase separation, discuss new ways of measuring membrane viscosity, and suggest why some well-known dynamical scaling laws can change their exponents or even break down for phase separation in membranes. If there's time, I will also discuss how the dynamics of protein diffusion can be altered by coupling to the lipid membrane.

A sequence $(a_1,..., a_n)$ of positive integers is a parking function if its nondecreasing rearrangement $(b_1 \leq ... \leq b_n)$ satisfies $b_i < i + 1$ for all $i$. Parking functions were introduced by Konheim and Weiss to study a hashing problem in computer science, but have since received a great deal of attention in algebraic combinatorics. We will define two new objects attached to any (finite, real, irreducible) reflection group which generalize parking functions and deserve to be called parking spaces. We present a conjecture (proved in some cases) which asserts a deep relationship between these constructions. This is joint work with Drew Armstrong at the University of Miami and Vic Reiner at the University of Minnesota.