Local picture and level-set percolation of the Gaussian free field on a large discrete torus

For $d \geq 3$ we obtain an approximation of the zero-average Gaussian free field on the discrete $d$-dimensional torus of large side length $N$ by the Gaussian free field on $\mathbb Z^d$, valid in boxes of roughly side length $N - N^\delta$ with $\delta \in (\frac12,1)$. As an implication, the level sets of the zero-average Gaussian free field on the torus can be approximated by the level sets of the Gaussian free field on $\mathbb Z^d$. This leads to a series of applications related to level-set percolation. We show that level sets of the zero-average Gaussian free field on the torus for levels $h > h_\star$ (where $h_\star$ denotes the critical value for level-set percolation of the Gaussian free field on $\mathbb Z^d$) with high probability contain no connected component of volume comparable to the total volume of the torus. Moreover, level sets with $h < h_\star$ with high probability contain a connected component of (extrinsic) diameter comparable to the torus diameter $N$. We also show that level sets of the zero-average Gaussian free field on the torus for levels $h$ above a second critical parameter $h_{\star\star}(\geq h_\star)$, again defined via the Gaussian free field on $\mathbb Z^d$, with high probability only contain connected components negligible in their size when compared to the size of the torus. Similar results have been obtained by A. Teixeira and D. Windisch in [Comm. Pure Appl. Math., 64(12):1599-1646, 2011] and J. \v{C}ern\'y and A. Teixeira in [Ann. Appl. Probab., 26(5):2883-2914, 2016] for the vacant set of simple random walk on a large discrete torus with the help of random interlacements on $\mathbb Z^d$, introduced by A.-S. Sznitman in [Ann. of Math. (2), 171(3):2039-2087, 2010].Comment: 23 pages, to appear in Stochastic Processes and their Application

The High Angular Resolution Multiplicity of Massive Stars

We present the results of a speckle interferometric survey of Galactic massive stars that complements and expands upon a similar survey made over a decade ago. The speckle observations were made with the KPNO and CTIO 4 m telescopes and USNO speckle camera, and they are sensitive to the detection of binaries in the angular separation regime between 0.03" and 5" with relatively bright companions (Delta V < 3). We report on the discovery of companions to 14 OB stars. In total we resolved companions of 41 of 385 O-stars (11%), 4 of 37 Wolf-Rayet stars (11%), and 89 of 139 B-stars (64%; an enriched visual binary sample that we selected for future orbital determinations). We made a statistical analysis of the binary frequency among the subsample that are listed in the Galactic O Star Catalog by compiling published data on other visual companions detected through adaptive optics studies and/or noted in the Washington Double Star Catalog and by collecting published information on radial velocities and spectroscopic binaries. We find that the binary frequency is much higher among O-stars in clusters and associations compared to the numbers for field and runaway O-stars, consistent with predictions for the ejection processes for runaway stars. We present a first orbit for the O-star Delta Orionis, a linear solution of the close, apparently optical, companion of the O-star Iota Orionis, and an improved orbit of the Be star Delta Scorpii. Finally, we list astrometric data for another 249 resolved and 221 unresolved targets that are lower mass stars that we observed for various other science programs.Comment: 76 pages, 6 figures, 11 table

The Goodwillie tower for S^1 and Kuhn's theorem

We analyze the homological behavior of the attaching maps in the 2-local Goodwillie tower of the identity evaluated at S^1. We show that they exhibit the same homological behavior as the James-Hopf maps used by N. Kuhn to prove the 2-primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie tower of S^1 at the prime 2.Comment: v2: 23 pages, clarified exposition in many parts, to appear in AG

Reified valuations and adic spectra

We revisit Huber's theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have been forced to contain the real numbers. This yields reified adic spectra which provide a framework for an analogue of Huber's theory compatible with Berkovich's construction of nonarchimedean analytic spaces. As an example, we extend the theory of perfectoid spaces to this setting.Comment: v5: refereed versio