L3

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.

Global symmetries greatly enrich the phase diagram of quantum many-body systems. As well as symmetry-breaking phases, symmetry-protected topological (SPT) phases have symmetric ground states that cannot be connected to a trivial state without a phase transition. There can also be symmetry-enriched critical points between these phases of matter. I will demonstrate these phenomena in phase diagrams constructed using the N-state chiral clock family of spin chains.  [Based on joint work with Paul Fendley and Abhishodh Prakash.]

The chiralization in the title denotes a certain procedure which turns cluster X-varieties into q-W algebras. Many important notions from cluster and q-W worlds, such as mutations, global functions, screening operators, R-matrices, etc emerge naturally in this context. In particular, we discover new bosonizations of q-W algebras and establish connections between previously known bosonizations. If time permits, I will discuss potential applications of our approach to the study of 3d topological theories and local systems with affine gauge groups. This talk is based on a joint project with J. Shiraishi, J.E. Bourgine, B. Feigin, A. Shapiro, and G. Schrader.

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.

Liquid crystal elastomers are rubbery solids containing molecular LC rods that align along a common director. On heating, the alignment is disrupted, leading to a substantial (~50%) contraction along the director.  In recent years, there has been a great deal of interest in fabrication LCE sheets with a bespoke alignment pattern. On heating, these patterns generate  corresponding patterns of contraction that can morph a sheet into a bespoke curved surface such as a cone or face. Moreover, LCEs can also be activated by light, either photothermally or photochemically, leading to similarly large contractions. Stimulation by light also introduces an important new possibility: using spatio-temporal patterns of illumination to morph a single LCE sample into a range of different surfaces. Such stimulation can enable non-reciprocal actuation for viscous swimming or pumping, and control over the whole path taken by the sheet through shape-space rather than just the final destination. In this talk, I will start by with an experimental example of a spatio-temporal pattern of illumination being used to actuate an LCE peristaltic pump. I will then introduce a second set of experiments, in which a monodomain sheet morphs first into a cone, an anti-cone and then an array of cones upon exposure to different patterns of illumination. Finally, I will then discuss the general problem of how to choose a pattern of illumination to morph a director-patterned sheet into an arbitrary surface, first analytically for axisymmetric cases, then numerically for low symmetry cases. This last study exceeds our current experimental capacity, but highlights how, with full spatio-temporal control over the stimulation magnitude, one can choreograph an LCE sheet to undergo almost any pattern of morphing.