The authors use Hartree-Fock methods to provide a comprehensive understanding of the moir\'e Hubbard model, believed describe low-energy physics twisted bilayer dichalcogenide material WSe${}_{2}$. They present phase diagram, reveal previously unnoticed gauge symmetry, demonstrate region reentrant metal-insulator transition, and elucidate roles fermiology commensurability in transitions. Compar...

We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs re...

We prove several superrigidity results for isometric actions on metric spaces satisfying some convexity properties. First, we extend some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups. Second, we include the proof of an unpublished result on commensurability superrigidity due to Margulis. The proofs rely on certain noti...

We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs re...

We prove that the property of a free group endomorphism being irreducible is invariant ascending HNN extension it defines. This answers question posed by Dowdall-Kapovich-Leininger. further and atoroidal commensurability invariant. The invariance follows from an algebraic characterization extensions determines exactly when their defining endomorphisms are atoroidal; specifically, we show if onl...

This paper investigates a purely qualitative version of Savage’s theory for decision making under uncertainty. Until now, most representation theorems for preference over acts rely on a numerical representation of utility and uncertainty where utility and uncertainty are commensurate. Disrupting the tradition, we relax this assumption and introduce a purely ordinal axiom requiring that the Deci...

Fusion and inference from multiple and massive disparate data sources – the requirement for our most challenging data analysis problems and the goal of our most ambitious statistical pattern recognition methodologies – has many and varied aspects which are currently the target of intense research and development. One aspect of the overall challenge is manifold matching – identifying embeddings ...

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of PSL(2,C). We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between el...