L3

TBA

TBA

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.

We study supersymmetric Berry connections of 2d N = (2,2) gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin-Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arises naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a deformation of the Cherkis–Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with Kähler target X, we show that the difference modules are related to deformations of the equivariant quantum cohomology of X.

In geophysical and astrophysical flows, we are often interested in understanding the impact of internal waves on the non-wavelike flow. For example, oceanic internal waves generated at the surface and the seafloor transfer energy from the large scale flow to dissipative scales, thereby influencing the global ocean state. A primary challenge in the study of wave-flow interactions is how to separate these processes – since waves and non-wavelike flows can vary on similar spatial and temporal scales in the Eulerian frame. However, in a Lagrangian flow-following frame, temporal filtering offers a convenient way to isolate waves. Here, I will discuss a recently developed method for evolving Lagrangian mean fields alongside the governing equations in a numerical simulation, and extend this theory to allow effective filtering of waves from non-wavelike processes.