Between Republic Park on fourth street and the new AMLI tower on third street sits a run-down parking lot with a tumultuous history. For many years, the lot has been owned by the Austin Museum of Art (AMOA), a local art institution split between a storefront location on Congress Avenue and a mansion near 35th street and Mount Bonnell.Over the years, AMOA has worked hard to build a permanent home on the downtown lot that has become one of the most choice central Austin parcels. First, they proposed a free-standing museum which was scratched after the technology . . .