On the number of rich lines in truly high dimensional sets

We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemer\'edi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. This conjecture was shown to hold in $\mathbb{R}^3$ in the seminal work of Guth and Katz \cite{GK10} and was also recently proved over $\mathbb{R}^4$ (under some additional restrictions) \cite{SS14}. For the special case of arithmetic progressions ($r$ collinear points that are evenly distanced) we give a bound that is tight up to low order terms, showing that a $d$-dimensional grid achieves the largest number of $r$-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree $r-2$ Veronese embedding takes $r$-collinear points to $r$ linearly dependent images. Hence, each collinear $r$-tuple of points, gives us a dependent $r$-tuple of images. We then use the design-matrix method of \cite{BDWY12} to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set

Point-curve incidences in the complex plane

We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in ${\mathbb R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big)$. We establish the slightly weaker bound $O_\varepsilon\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in ${\mathbb C}^2$ with $k$ degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ${\mathbb C}$.Comment: The proof was significantly simplified, and now relies on the Picard-Lindelof theorem, rather than on foliation

Exploration of Parameter Spaces in a Virtual Observatory

Like every other field of intellectual endeavor, astronomy is being revolutionised by the advances in information technology. There is an ongoing exponential growth in the volume, quality, and complexity of astronomical data sets, mainly through large digital sky surveys and archives. The Virtual Observatory (VO) concept represents a scientific and technological framework needed to cope with this data flood. Systematic exploration of the observable parameter spaces, covered by large digital sky surveys spanning a range of wavelengths, will be one of the primary modes of research with a VO. This is where the truly new discoveries will be made, and new insights be gained about the already known astronomical objects and phenomena. We review some of the methodological challenges posed by the analysis of large and complex data sets expected in the VO-based research. The challenges are driven both by the size and the complexity of the data sets (billions of data vectors in parameter spaces of tens or hundreds of dimensions), by the heterogeneity of the data and measurement errors, including differences in basic survey parameters for the federated data sets (e.g., in the positional accuracy and resolution, wavelength coverage, time baseline, etc.), various selection effects, as well as the intrinsic clustering properties (functional form, topology) of the data distributions in the parameter spaces of observed attributes. Answering these challenges will require substantial collaborative efforts and partnerships between astronomers, computer scientists, and statisticians.Comment: Invited review, 10 pages, Latex file with 4 eps figures, style files included. To appear in Proc. SPIE, v. 4477 (2001

Quantitative chemical tagging, stellar ages and the chemo-dynamical evolution of the Galactic disc

The early science results from the new generation of high-resolution stellar spectroscopic surveys, such as GALAH and the Gaia-ESO survey, will represent major milestones in the quest to chemically tag the Galaxy. Yet this technique to reconstruct dispersed coeval stellar groups has remained largely untested until recently. We build on previous work that developed an empirical chemical tagging probability function, which describes the likelihood that two field stars are conatal, that is, they were formed in the same cluster environment. In this work we perform the first ever blind chemical tagging experiment, i.e., tagging stars with no known or otherwise discernable associations, on a sample of 714 disc field stars with a number of high quality high resolution homogeneous metal abundance measurements. We present evidence that chemical tagging of field stars does identify coeval groups of stars, yet these groups may not represent distinct formation sites, e.g. as in dissolved open clusters, as previously thought. Our results point to several important conclusions, among them that group finding will be limited strictly to chemical abundance space, e.g. stellar ages, kinematics, colors, temperature and surface gravity do not enhance the detectability of groups. We also demonstrate that in addition to its role in probing the chemical enrichment and kinematic history of the Galactic disc, chemical tagging represents a powerful new stellar age determination technique.Comment: 12 pages, 9 figures, accepted for publication in Monthly Notices of the Royal Astronomical Society (MNRAS

SrCu2(BO3)2Sr Cu_2 (BO_3)_2: A Unique Mott Hubbard Insulator

We discuss the recently discovered system $Sr Cu_2 (BO_3)_2$, a realization of an exactly solvable model proposed two decades earlier. We propose its interpretation as a Mott Hubbard insulator. The possible superconducting phase arising from doping is explored, and its nature as well as its importance for testing the RVB theory of superconductivity are discussed.Comment: 18 pages,7 figures, Based on Invited Talk at 16th Nishinomiya Yukawa memorial symposium, Nov 13-14, Nishinomiya. To appear in Progress in Theoretical Physics, Suppliment 145(2002

Type-Ia Supernova Remnant Shell At Z=3.5Z=3.5 Seen In The Three Sightlines Toward The Gravitationally Lensed Qso B1422+231

Using the Subaru 8.2m Telescope with an IRCS Echelle spectrograph, we obtained high-resolution (R=10,000) near-infrared (1.01-1.38 \mu m) spectra of images A and B of the gravitationally lensed QSO B1422+231 (z=3.628) consisting of four known lensed images. We detected MgII absorption lines at z=3.54, which show a large variance of column densities (~ 0.3 dex) and velocities (~ 10 km/s) between the sightlines A and B with a projected separation of only 8.4h_{70}^{-1} pc at the redshift. This is the smallest spatial structure of the high-z gas clouds ever detected after Rauch et al. found a 20-pc scale structure for the same z=3.54 absorption system using optical spectra of images A and C. The observed systematic variances imply that the system is an expanding shell as originally suggested by Rauch et al. By combining the data for three sightlines, we managed to constrain the radius and expansion velocity of the shell (~ 50-100 pc, 130 km/s), concluding that the shell is truly a supernova remnant (SNR) rather than other types of shell objects, such as a giant HII region. We also detected strong FeII absorption lines for this system, but with much broader Doppler width than that of \alpha-element lines. We suggest that this FeII absorption line originates in a localized FeII-rich gas cloud that is not completely mixed with plowed ambient interstellar gas clouds showing other \alpha-element low-ion absorption lines. Along with the Fe richness, we conclude that the SNR is produced by an SNIa explosion.Comment: 16 pages, 15 figures, to be published in The Astrophysical Journa

Hot bottom burning and s-process nucleosynthesis in massive AGB stars at the beginning of the thermally-pulsing phase

We report the first spectroscopic identification of massive Galactic asymptotic giant branch (AGB) stars at the beginning of the thermal pulse (TP) phase. These stars are the most Li-rich massive AGBs found to date, super Li-rich AGBs with logE(Li)~3-4. The high Li overabundances are accompanied by weak or no s-process element (i.e. Rb and Zr) enhancements. A comparison of our observations with the most recent hot bottom burning (HBB) and s-process nucleosynthesis models confirms that HBB is strongly activated during the first TPs but the 22Ne neutron source needs many more TP and third dredge-up episodes to produce enough Rb at the stellar surface. We also show that the short-lived element Tc, usually used as an indicator of AGB genuineness, is not detected in massive AGBs which is in agreement with the theoretical predictions when the 22Ne neutron source dominates the s-process nucleosynthesis.Comment: Accepted for publication in Astronomy & Astrophysics Letters (7 pages, 5 figures and 1 table); final version (language corrected