finite element methods are numerical methods that approximate solutions to
pdes using piecewise polynomials on a mesh representing the problem
domain. adaptive finite element methods are a class of finite element
methods that selectively refine specific elements in the mesh based on
their predicted error. in order to establish the viability of an afem, it
is essential to know whether or not that method can be proven to converge.
in this talk i will present a general framework that would establish
convergence for an afem and apply the framework to specific problems.

special values of zeta functions have been around for a very long time, and yet are still quite mysterious. once the basic theory of the riemann zeta function has been established, i will introduce the dedekind zeta function along with ties between special values and algebraic k-theory.

processor sharing is a mathematical idealization of round-robin scheduling algorithms commonly used in computer time-sharing. it is a fundamental example of a non-head-of-the-line service discipline. for such disciplines, it is typical that any markov description of the system state is infinite dimensional. due to this, measure-valued stochastic processes are becoming a key tool used in the modeling and analysis of stochastic network models operating under various non-head-of-the-line service disciplines.

in this talk, we discuss a new approach to studying the asymptotic behavior of fluid model solutions (formal functional law of large numbers limits) for critically loaded processor sharing queues. for this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. this approach is developed with idea that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under protocols naturally described by measure-valued processes.

we show that the $\phi$-selmer ranks of twists of an elliptic curve $e$ with a point of order two are distributed like the ranks of random groups in a manner consistent with the philosophy underlying the cohen-lenstra heuristics.

if $e$ has a point of order two, then the distribution of $dim_{\mathbb{f}_2} \operatorname{sel}_\phi{(e^d/{\mathbb{q}})} - dim_{\mathbb{f}_2} \operatorname{sel}_{\hat\phi}{(e^{\prime d}/{\mathbb{q}})}$ tends to the discrete normal distribution $\mathcal{n}(0,\frac{1}{2} \log \log x)$ as $x \rightarrow \infty$. we consider the distribution of $dim_{\mathbb{f}_2} \operatorname{sel}_\phi{(e^d/{\mathbb{q}})} - dim_{\mathbb{f}_2} \operatorname{sel}_{\hat\phi}{(e^{\prime d}/{\mathbb{q}})}$ has a fixed value $u$.

we show that for every $r$, the limiting probability that $dim_{\mathbb{f}_2} \operatorname{sel}_\phi{(e^d/{\mathbb{q}})}= r$ is given by an explicit constant $\alpha_{r,u}$ introduced in cohen and lenstra's original work on the distribution of class groups.