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  1. ABSTRACT We present new measurements of the spatial distribution and kinematics of neutral hydrogen in the circumgalactic and intergalactic medium surrounding star-forming galaxies at z ∼ 2. Using the spectra of ≃3000 galaxies with redshifts 〈z〉 = 2.3 ± 0.4 from the Keck Baryonic Structure Survey, we assemble a sample of more than 200 000 distinct foreground-background pairs with projected angular separations of 3–500 arcsec and spectroscopic redshifts, with 〈zfg〉 = 2.23 and 〈zbg〉 = 2.57 (foreground, background redshifts, respectively.) The ensemble of sightlines and foreground galaxies is used to construct a 2D map of the mean excess $\rm{H\,{\small I}}$$\rm Ly\,\alpha$ optical depth relative to the intergalactic mean as a function of projected galactocentric distance (20 ≲ Dtran/pkpc ≲ 4000) and line-of-sight velocity. We obtain accurate galaxy systemic redshifts, providing significant information on the line-of-sight kinematics of $\rm{H\,{\small I}}$ gas as a function of projected distance Dtran. We compare the map with cosmological zoom-in simulation, finding qualitative agreement between them. A simple two-component (accretion, outflow) analytical model generally reproduces the observed line-of-sight kinematics and projected spatial distribution of $\rm{H\,{\small I}}$. The best-fitting model suggests that galaxy-scale outflows with initial velocity vout ≃ 600 km s$^{-1}\,$ dominate the kinematics of circumgalactic $\rm{H\,{\small I}}$ out to Dtran ≃ 50 kpc, whilemore »$\rm{H\,{\small I}}$ at Dtran ≳ 100 kpc is dominated by infall with characteristic vin ≲ circular velocity. Over the impact parameter range 80 ≲ Dtran/pkpc ≲ 200, the $\rm{H\,{\small I}}$ line-of-sight velocity range reaches a minimum, with a corresponding flattening in the rest-frame $\rm Ly\,\alpha$ equivalent width. These observations can be naturally explained as the transition between outflow-dominated and accretion-dominated flows. Beyond Dtran ≃ 300 pkpc (∼1 cMpc), the line-of-sight kinematics are dominated by Hubble expansion.« less