  Visualizing the high-redshift Universe is difficult due to the dearth of
available data; however, the Lyman-alpha forest provides a means to map the
intergalactic medium at redshifts not accessible to large galaxy surveys.
Large-scale structure surveys, such as the Baryon Oscillation Spectroscopic
Survey (BOSS), have collected quasar (QSO) spectra that enable the
reconstruction of HI density fluctuations. The data fall on a collection of
lines defined by the lines-of-sight (LOS) of the QSO, and a major issue with
producing a 3D reconstruction is determining how to model the regions between
the LOS. We present a method that produces a 3D map of this relatively
uncharted portion of the Universe by employing local polynomial smoothing, a
nonparametric methodology. The performance of the method is analyzed on
simulated data that mimics the varying number of LOS expected in real data, and
then is applied to a sample region selected from BOSS. Evaluation of the
reconstruction is assessed by considering various features of the predicted 3D
maps including visual comparison of slices, PDFs, counts of local minima and
maxima, and standardized correlation functions. This 3D reconstruction allows
for an initial investigation of the topology of this portion of the Universe
using persistent homology.
