(Marta Lewicka)

Variational methods have been extensively used in the past decades to rigorously derive nonlinear models in the description of thin elastic films. In this context, natural growth or differential swelling-shrinking lead to models where an elastic body aims at reaching a space-dependent metric. We will describe the effect of such, generically incompatible, prestrain metrics on the singular limits' bidimensional models. We will discuss metrics that vary across the specimen in both the midplate and the thin (transversal) directions. We will also cover the case of the oscillatory prestrain, exhibit its relation to the non-oscillatory case via identifying the effective metrics, and discuss the role of the Riemann curvature tensor in the limiting models.

(Shankar Venkataramani)

Using the bidimensional models for pre-strained Elasticity, that Marta will discuss in her talk, I will discuss some contrasts between the mechanics of thin objects with non-negative curvature (plates, spherical shells, etc) and the mechanics of hyperbolic sheets, i.e. soft/thin objects with negative curvature. I will motivate the need for new "geometric" methods for discretizing the relevant equations, and present some of our preliminary work in this direction.

This is joint work with Toby Shearman and Ken Yamamoto.

The moments of characteristic polynomials play a central role in Random Matrix Theory.  They appear in many applications, ranging from quantum mechanics to number theory.  The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.

Let $p$ be a fixed prime number and let $SCVF_p$ be the theory of separably closed non-trivially valued fields of
characteristic $p$. In the talk, we will see that, in many ways, the step from $ACVF_{p,p}$ to $SCVF_p$ is not more
complicated than the one from $ACF_p$ to $SCF_p$.

At a basic level, this is true for quantifier elimination (Delon), for which it suffices to add parametrized $p$-coordinate
functions to any of the usual languages for valued fields. It follows that all completions are NIP.

At a more sophisticated level, in finite degree of imperfection, when a $p$-basis is named or when one just works with
Hasse derivations, the imaginaries of $SCVF_p$ are not more complicated than the ones in $ACVF_{p,p}$, i.e., they are
classified by the geometric sorts of Haskell-Hrushovski-Macpherson. The latter is proved using prolongations. One may
also use these to characterize the stable part and the stably dominated types in $SCVF_p$, and to show metastability.

The study of isolated systems has been vastly successful in the context of vanishing cosmological constant, Λ=0. However, there is no physically useful notion of asymptotics for the universe we inhabit with Λ>0.  The full non-linear framework is still under development, but some interesting results at the linearized level have been obtained. I will focus on the conceptual subtleties that arise at the linearized level and discuss the quadrupole formula for gravitational radiation as well as some recent developments.