  The 2-dimensional electron system at the interface between LaAlO$_{3}$ and
SrTiO$_{3}$ has several unique properties that can be tuned by an externally
applied gate voltage. In this work, we show that this gate-tunability extends
to the effective band structure of the system. We combine a magnetotransport
study on top-gated Hall bars with self-consistent Schr\"odinger-Poisson
calculations and observe a Lifshitz transition at a density of
$2.9\times10^{13}$ cm$^{-2}$. Above the transition, the carrier density of one
of the conducting bands decreases with increasing gate voltage. This surprising
decrease is accurately reproduced in the calculations if electronic
correlations are included. These results provide a clear, intuitive picture of
the physics governing the electronic structure at complex oxide interfaces.
