L3

Slender elastic filaments with intrinsic helical geometry are encountered in a wide range of physical and biological settings, ranging from coil springs in engineering to bacteria flagellar filaments. The equilibrium configurations of helical filaments under a variety of loading types have been well studied in the framework of the Kirchhoff rod equations. These equations are geometrically nonlinear and so can account for large, global displacements of the rod. This geometric nonlinearity also makes a mathematical analysis of the rod equations extremely difficult, so that much is still unknown about the dynamic behaviour of helical rods under external loading.

An important class of simplified models consists of 'equivalent-column' theories. These model the helical filament as a naturally-straight beam (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Such theories have long been used in engineering to describe the free vibrations of helical coil springs, though their validity remains unclear, particularly when distributed forces and moments are present. In this talk, we show how such an effective theory can be derived systematically from the Kirchhoff rod equations using the method of multiple scales. Importantly, our analysis is asymptotically exact in the small-wavelength limit and can account for large, unsteady displacements. We then illustrate our theory with two loading scenarios: (i) a heavy helical rod deforming under its own weight; and (ii) axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, as well as yielding analytical insight into their tensile instabilities.

I will exposit some recent joint work with Paul Larson and Grigor Sargsyan that uses higher models of the Axiom of Determinacy---models with nontrivial structure above $\Theta$, the least ordinal which is not the surjective image of the reals---to show that instances of the fundamental problem of inner model theory, the iterability conjecture, consistently fail.

In recent joint work with Pablo Cubides Kovacsics and Jinhe Ye on beautiful pairs in the unstable context, the amalgamation property (AP) for the class of global definable types plays a key role. In the talk, we will first indicate some important cases in which AP holds, and we will then present the construction of examples of theories, obtained in joint work with Rosario Mennuni, where AP fails.

We prove that any adjoint Chevalley group over an arbitrary commutative ring is regularly bi-interpretable with this ring. The same results hold for central quotients of arbitrary Chevalley groups and for Chevalley groups with bounded generation.
Also, we show that the corresponding classes of Chevalley groups (or their central quotients) are elementarily definable and even finitely axiomatizable.