Short answers, pulled from the story.
Most scholars believe this initial split occurred after Ashoka's death in the third century BCE. The unified Buddhist community fractured into two distinct groups following the reign of Emperor Ashoka.
Sri Lanka stands as the sole surviving home of Theravada Buddhism, the only extant mainstream non-Mahayana school. This tradition traces its lineage directly to the Vibhajyavadins who arrived on the island during Ashoka's reign circa 240 BCE.
Nagarjuna founded the Madhyamaka tradition, also called the Middle Way or Emptiness-view, around the second century CE. His followers developed distinct sub-schools including Svatantrika and Prasangika branches led by thinkers like Bhaviveka and Chandrakirti.
B. R. Ambedkar delivered a historic speech on the 14th of October 1956 at Deekshabhoomi in Nagpur during his conversion ceremony. This event launched the Navayana or Dalit Buddhist movement which reinterpreted Buddhism for social welfare rather than individual liberation alone.
A separate movement emerged in the fourth century AD when Asanga and Vasubandhu established the Yogacara school. Known variously as Vijnanavada or Cittamatra, this tradition emphasized consciousness-only doctrines.