A strong operator topology adiabatic theorem

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.Comment: 15 pages, no figure

Integrable RCS as a proposed replacement for Fermilab Booster

Integrable optics is an innovation in particle accelerator design that potentially enables a greater betatron tune spread and damps collective instabilities. An integrable rapid-cycling synchrotron (RCS) would be an effective replacement for the Fermilab Booster, as part of a plan to reach multi-MW beam power at 120 GeV for the Fermilab high-energy neutrino program. We provide an example integrable lattice with features of a modern RCS - dispersion-free drifts, low momentum compaction factor, superperiodicity, chromaticity correction, bounded beta functions, and separate-function magnets.Comment: arXiv admin note: substantial text overlap with arXiv:1703.0095

Bridge trisections of knotted surfaces in 4--manifolds

We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4--manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside $\mathbb{CP}^2$. Using these examples, we prove that there exist exotic 4--manifolds with $(g,0)$--trisections for certain values of $g$. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams.Comment: 17 pages, 5 figures. Comments welcom

Characterizing Dehn surgeries on links via trisections

We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of the standard four-sphere that are unlikely to be standard. Finally, we give an analog of the Casson- Gordon Rectangle Condition for trisections that can be used to obstruct reducibility of a given trisection.Comment: 15 pages, 4 color figures. Comments welcome