The rotation curves of spiral galaxies (i.e. the velocities of stars around galaxy centers)
provide one of the strongest lines of evidence establishing the need for dark matter in the Universe. An elaborate paradigm
for dark matter has subsequently developed, known as LCDM, in which the Universe consists of Cold Dark Matter, such that it
is non-relativistic at the time of recombination (i.e., when neutral hydrogen atoms first formed). Simulations of LCDM structure
formation can recreate the large-scale structure of the Universe extremely well.
However, an important question is whether this cosmic paradigm provides a satisfactory description
on smaller scales, i.e. that of individual spiral galaxies.
A complete explanation of spiral galaxies in LCDM requires a comprehensive theory of galaxy
formation. This remains lacking. Indeed, there are a number of lingering problems on small (i.e. galaxy) scales.
One of the most crucial tests for LCDM comes from the estimates of galaxy mass distributions.
High-resolution N-body simulations of LCDM structure formation predict that the central density profiles of dark matter halos
should rise steeply at small radii. However, observations of rotation curves of late-type disk galaxies and dwarf galaxies,
have shown that quite often, mass distributions with lower than predicted densities or with constant density cores, are preferred.
This is known as the cusp/core problem. Another problem for LCDM cosmology is the Tully-Fisher zero-point problem, which refers
to the fact that standard models cannot reproduce the observed relation between galaxy luminosity and circular velocity without
over-producing the number density of galaxies at fixed luminosity.
In this keynote talk, I will address these issues and introduce some new ideas and methods that
are being developed to solve them.