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Tyler Foster

Publications and source records attributed to Tyler Foster.

At least 19 recordsLinked to original sources
arXiv2023

Adic tropicalizations and cofinality of Gubler models

We introduce adic tropicalizations for subschemes of toric varieties as limits of Gubler models associated to polyhedral covers of the ordinary tropicalization. Our main result shows that Huber's adic analytification of a subscheme of a toric variety is naturally isomorphic to the inverse limit of its adic tropicalizations, in the category of locally topologically ringed spaces. The key new technical idea underlying this theorem is cofinality of Gubler models, which we prove for projective schemes and also for more general compact analytic domains in closed subschemes of toric varieties. In addition, we introduce a G-topology and structure sheaf on ordinary tropicalizations, and show that Berkovich analytifications are limits of ordinary tropicalizations in the category of topologically ringed topoi.

math.AG↗
arXiv2022

A New Distance to the Supernova Remnant DA 530 Based on HI Absorption of Polarized Emission

Supernova remnants (SNRs) are significant contributors of matter and energy to the interstellar medium. Understanding the impact and the mechanism of this contribution requires knowledge of the physical size, energy, and expansion rate of individual SNRs, which can only come if reliable distances can be obtained. We aim to determine the distance to the SNR DA 530 (G93.3+6.9), an object of low surface brightness. To achieve this, we used the Dominion Radio Astrophysical Observatory Synthesis Telescope and the National Radio Astronomy Observatory Very Large Array to observe the absorption by intervening HI of the polarized emission from DA 530. Significant absorption was detected at velocities $-28$ and -67 km/s (relative to the local standard of rest), corresponding to distances of 4.4 and 8.3 kpc, respectively. Based on the radio and X-ray characteristics of DA 530, we conclude that the minimum distance is 4.4$^{+0.4}_{-0.2}$ kpc. At this minimum distance, the diameter of the SNR is 34$^{+4}_{-1}$ pc, and the elevation above the Galactic plane is 537$^{+40}_{-32}$ pc. The $-67$ km/s absorption likely occurs in gas whose velocity is not determined by Galactic rotation. We present a new data processing method for combining Stokes $Q$ and $U$ observations of the emission from an SNR into a single HI absorption spectrum, which avoids the difficulties of the noise-bias subtraction required for the calculation of polarized intensity. The polarized absorption technique can be applied to determine distances to many more SNRs.

astro-ph.GA↗
arXiv2019

Canadian Investigations of the Interstellar Medium

The interstellar medium mediates galactic evolution as the reservoir of material for future star formation and the repository of energy and matter output by stellar processes. Canadians have played leading roles in ISM science for decades. The Canadian Galactic Plane Survey identified a wealth of small-scale structure in H I emission as well as self-absorption and in the structure of polarized emission. These observations demonstrated that no phase of the ISM, including the transition from atomic gas to star formation, can be understood in isolation. Canadians have also played leading roles in the characterization of dust with Planck and balloon-borne telescopes. Canadians have also used pulsar scintillometry to measure structure in the ISM at the smallest scales, below 1 AU. The 2020s offer many opportunities for ISM science in Canada. A major but cost-effective upgrade to the Synthesis Telescope with broadband (400-1800 MHz) single-pixel feeds would enable broadband polarimetry as well as wide-area, arcminute surveys of radio recombination lines. The next generation of balloon-borne telescopes will investigate magnetic fields and dust properties. Large single dishes, particularly the Green Bank Telescope, remain essential for our understanding of the diffuse structure which characterizes the ISM. Very long baseline interferometry capability enables parallax measurements of pulsars and masers and for further progress in scintillometry. ISM astronomers will continue to participate in cosmological experiments including CHIME and CHORD. Protecting quiet radio frequency interference environments will be ever more important as broadband observations are ever more central to ISM science. Computational capability is essential both for numerical work and for handling the observational data.

astro-ph.GA↗
arXiv2019

HI observations of IC 10 with the DRAO synthesis telescope

HI observations of the nearby blue compact dwarf galaxy IC 10 obtained with the Dominion Radio Astrophysical Observatory synthesis telescope (DRAO), for a total integration of $\sim$1000 hours, are presented. We confirm the NW faint 21 cm HI emission feature discovered in GBT observations. The HI feature has an HI mass of 4.7 $\times 10^{5}$ M$_{\odot}$, which is only $\sim$ 0.6$\%$ of the total HI mass of the galaxy (7.8 $\times 10^{7}$ M$_{\odot}$). In the inner disk, the rotation curve of IC 10 rises steeply, then flattens until the last point where it rises again, with a maximum velocity of 30 km s$^{-1}$. Based on our mass models, the kinematics of the inner disk of IC 10 can be described without the need of a dark matter halo. However, this does not exclude the possible presence of dark matter on a larger scale. It is unlikely that the disturbed features seen in the outer HI disk of IC 10 are caused by an interaction with M 31. Features seen from our simulations are larger and at lower surface density than can be reached by current observations. The higher velocity dispersions seen in regions where several distinct HI features meet with the main core of IC 10 suggests that there is ongoing accretion.

astro-ph.GA↗
arXiv2017

HI Kinematics and Mass Distribution of Messier 33

A new deep HI survey of the galaxy Messier 33 is presented, based on observations obtained at the Dominion Radio Astrophysical Observatory. We observe a perturbed outer gas distribution and kinematics in M33, and confirm the disk warping as a significant kinematical twist of the major axis of the velocity field, though no strong tilt is measured, in agreement with previous work. Evidence for a new low brightness HI component with anomalous velocity is reported. It harbours a large velocity scatter, as its kinematics both exceeds and lags the rotation of the disk, and leaks in the forbidden velocity zone of apparent counter-rotation. The observations also reveal wide and multiple peak HI profiles which can be partly explained by crowded orbits in the framework of the warp model. Asymmetric motions are identified in the velocity field, as possible signatures of a lopsided potential and the warp. The mass distribution modeling of the hybrid Halpha-HI rotation curve favours a cuspy dark matter halo with a concentration in disagreement with the LambdaCDM dark halo mass-concentration relationship. The total mass enclosed in 23 kpc is 8 10^10 Msol, of which 11% are stars and gas. At the virial radius of the cuspy halo, the resulting total mass is 5 10^11 Msol, but with a baryonic mass fraction of 2% only. This strongly suggests a more realistic radius encompassing the total mass of M33 well smaller than the virial radius of the halo, maybe comparable to the size of the HI disk.

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arXiv2017

Non-Archimedean and tropical theta functions

We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud--Bosch--L\"utkebohmert theory of non-Archimedean theta functions for abelian varieties with semi-abelian reduction.

math.AG↗
arXiv2017

Galois actions on analytifications and tropicalizations

This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and Zharkov. We also prove two results which say that under minimal hypotheses on a variety $X_{0}$ over a Henselian field $K_{0}$, Galois-equivariant tropicalizations carry all of the arithmetic structure of $X_{0}$. Namely: (1) The Galois-orbit of any point of $X_{0}$ valued in the separable closure of $K_{0}$ is reproduced faithfully as a Galois-set inside some Galois-equivariant tropicalization of our variety. (2) The Berkovich analytification of $X_{0}$ over the separable closure of $K_{0}$, equipped with its canonical Galois-action, is the inverse limit of all Galois-equivariant tropicalizations of our variety.

math.AG↗
arXiv2017

Primes in short intervals on curves over finite fields

We prove an analogue of the Prime Number Theorem for short intervals on a smooth projective geometrically irreducible curve of arbitrary genus over a finite field. A short interval "of size E" in this setting is any additive translate of the space of global sections of a sufficiently positive divisor E by a suitable rational function f. Our main theorem gives an asymptotic count of irreducible elements in short intervals on a curve in the "large q" limit, uniformly in f and E. This result provides a function field analogue of an unresolved short interval conjecture over number fields, and extends a theorem of Bary-Soroker, Rosenzweig, and the first author, which can be understood as an instance of our result for the special case of a divisor E supported at a single rational point on the projective line.

math.NT↗
arXiv2016

Logarithmic Picard groups, chip firing, and the combinatorial rank

Illusie has suggested that one should think of the classifying group of $M_X^{gp}$-torsors on a logarithmically smooth curve $X$ over a standard logarithmic point as a logarithmic analogue of the Picard group of $X$. This logarithmic Picard group arises naturally as a quotient of the algebraic Picard group by lifts of the chip firing relations of the associated dual graph. We connect this perspective to Baker and Norine's theory of ranks of divisors on a finite graph, and to Amini and Baker's metrized complexes of curves. Moreover, we propose a definition of a combinatorial rank for line bundles on $X$ and prove that an analogue of the Riemann-Roch formula holds for our combinatorial rank. Our proof proceeds by carefully describing the relationship between the logarithmic Picard group on a logarithmic curve and the Picard group of the associated metrized complex. This approach suggests a natural categorical framework for metrized complexes, namely the category of logarithmic curves.

math.AG↗
arXiv2015

Degenerations of toric varieties over valuation rings

We develop a theory of multi-stage degenerations of toric varieties over finite rank valuation rings, extending the Mumford--Gubler theory in rank one. Such degenerations are constructed from fan-like structures over totally ordered abelian groups of finite rank. Our main theorem describes the geometry of successive special fibers in the degeneration in terms of the polyhedral geometry of a system of recession complexes associated to the fan.

math.AG↗
arXiv2015

D-structures and derived Koszul duality for unital operad algebras

Generalizing a concept of Lipshitz, Ozsv\'ath and Thurs-ton from Bordered Floer homology, we define $D$-structures on algebras of unital operads, which can also be interpreted as a generalization of a seemingly unrelated concept of Getzler and Jones. This construction gives rise to an equivalence of derived categories, which can be thought of as a unital version of Koszul duality using non-unital Quillen homology. We also discuss a multi-sorted version of the construction, which provides a framework for unifying the known algebraic contexts of Koszul duality.

math.KT↗
arXiv2015

Introduction to adic tropicalization

This is an expository article on the adic tropicalization of algebraic varieties. We outline joint work with Sam Payne in which we put a topology and structure sheaf of local topological rings on the exploded tropicalization. The resulting object, which blends polyhedral data of the tropicalization with algebraic data of the associated initial degenerations, is called the adic tropicalization. It satisfies a theorem of the form "Huber analytification is the limit of all adic tropicalizations." We explain this limit theorem in the present article, and illustrate connections between adic tropicalization and the curve complexes of O. Amini and M. Baker.

math.AG↗
arXiv2015

Hahn analytification and connectivity of higher rank tropical varieties

We show that the tropicalization of a connected variety over a higher rank valued field is a path connected topological space. This establishes an affirmative answer to a question posed by Banerjee. Higher rank tropical varieties are studied as the images of "Hahn analytifications", introduced in this paper. A Hahn analytification is a space of valuations on a scheme over a higher rank valued field. We prove that the Hahn analytification is related to higher rank tropicalization by means of an inverse limit theorem, extending well-known results in the non-Archimedean case. We also establish comparison results between the Hahn analytification and the Huber and Berkovich analytifications, as well as the Hrushovski-Loeser stable completion.

math.AG↗
arXiv2014

Rational Connectivity and Analytic Contractibility

Let k be an algebraically closed field of characteristic 0, and let f be a morphism of smooth projective varieties from X to Y over the ring k((t)) of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the Berkovich analytifications of X and Y are homotopy equivalent. Two important consequences of this result are that the homotopy type of the Berkovich analytification of any smooth projective variety X over k((t)) is a birational invariant of X, and that the Berkovich analytification of a rationally connected smooth projective variety over k((t)) is contractible.

math.AG↗
arXiv2014

G141.2+5.0, a new Pulsar Wind Nebula discovered in the Cygnus Arm of the Milky Way

We report the discovery of the new pulsar wind nebula (PWN) G141.2+5.0 in data observed with the Dominion Radio Astrophysical Observatory's Synthesis Telescope at 1420 MHz. The new PWN has a diameter of about 3.5', which translates at a distance of 4.0 kpc to a spatial extent of about 4 pc. It displays a radio spectral index of $\alpha\approx -0.7$, similar to the PWN G76.9+1.1. G141.2+5.0 is highly polarized up to 40% with an average of 15% in the 1420 MHz data. It is located in the centre of a small spherical HI bubble, which is expanding at a velocity of 6 km/s. The bubble's systemic velocity is -53 km/s and could be the result of the progenitor star's mass loss or the shell-type SNR created by the same supernova explosion in a highly advanced stage. The systemic velocity of the bubble shares the velocity of HI associated with the Cygnus spiral arm, which is seen across the 2nd and 3rd quadrants and an active star-forming arm immediately beyond the Perseus arm. A kinematical distance of 4+/-0.5 kpc is found for G141.2+5.0, similar to the optical distance of the Cygnus arm (3.8+/-1.1 kpc) and its kinematic HI distance (4.4+/-0.6 kpc). G141.2+5.0 represents the first radio PWN discovered in 17 years and the first SNR discovered in the Cygnus spiral arm, which is in stark contrast with the Perseus arm's overwhelming population of shell-type remnants.

astro-ph.GA↗
arXiv2013

A Relation Between the Warm Neutral and Ionized Media Observed in the Canadian Galactic Plane Survey

We report on a comparison between 21 cm rotation measure (RM) and the optically-thin atomic hydrogen column density (N_HI) measured towards unresolved extragalactic sources in the Galactic plane of the northern sky. HI column densities integrated to the Galactic edge are measured immediately surrounding each of nearly 2000 sources in 1-arcminute 21 cm line data, and are compared to RMs observed from polarized emission of each source. RM data are binned in column-density bins 4x10^20 cm^-2 wide, and one observes a strong relationship between the number of hydrogen atoms in a 1 cm^2 column through the plane and the mean RM along the same line-of-sight and path length. The relationship is linear over one order of magnitude (from 0.8-14x10^21 atoms cm^-2) of column densities, with a constant RM/N_HI -23.2+/-2.3 rad m^-2/10^21 atoms cm^-2, and a positive RM of 45.0+/-13.8 rad m^-2 in the presence of no atomic hydrogen. This slope is used to calculate a mean volume-averaged magnetic field in the 2nd quadrant of ~1.0+/-0.1 micro-Gauss directed away from the Sun, assuming an ionization fraction of 8% (consistent with the WNM). The remarkable consistency between this field and =1.2 micro-Gauss found with the same RM sources and a Galactic model of dispersion measures suggests that electrons in the partially ionized WNM are mainly responsible for pulsar dispersion measures, and thus the partially-ionized WNM is the dominant form of the magneto-ionic interstellar medium.

astro-ph.GA↗
arXiv2013

Two Radio Supernova Remnants Discovered in the Outer Galaxy

We report on the discovery of two supernova remnants (SNRs) designated G152.4-2.1 and G190.9-2.2, using Canadian Galactic Plane Survey data. The aims of this paper are, first, to present evidence that favours the classification of both sources as SNRs, and, second, to describe basic parameters (integrated flux density, spectrum, and polarization) as well as properties (morphology, line-of-sight velocity, distance and physical size) to facilitate and motivate future observations. Spectral and polarization parameters are derived from multiwavelength data from existing radio surveys carried out at wavelengths between 6 and 92cm. In particular for the source G152.4-2.1 we also use new observations at 11cm done with the Effelsberg 100m telescope. The interstellar medium around the discovered sources is analyzed using 1-arcminute line data from neutral hydrogen (HI) and 45-arcsecond 12CO(J=1-0). G152.4-2.1 is a barrel shaped SNR with two opposed radio-bright polarized flanks on the North and South. The remnant, which is elongated along the Galactic plane is evolving in a more-or-less uniform medium. G190.9-2.2 is also a shell-type remnant with East and West halves elongated perpendicular to the plane, and is evolving within a low-density region bounded by dense neutral hydrogen in the North and South, and molecular (12CO) clouds in the East and West. The integrated radio continuum spectral indices are -0.65+/-0.05 and -0.66+/-0.05 for G152.4-2.1 and G190.9-2.2 respectively. Both SNRs are approximately 1 kpc distant, with G152.4-2.1 being larger (32x30 pc in diameter) than G190.9-2.2 (18x16 pc). These two remnants are the lowest surface brightness SNRs yet catalogued at 5x10^-23 W m^-2 Hz^-1 sr^-1.

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