Short answers, pulled from the story.
Nāgārjuna established the Madhyamaka tradition sometime in the first three centuries CE. Scholars cannot pinpoint his exact birth date or place of residence, though Joseph Walser suggests he lived in Amarāvatī within the east Deccan region.
His core philosophical work remains the Mūlamadhyamakakārikā, known as Root Verses on the Middle Way. This text serves as the foundational document for the entire school and explicitly cites the Advice to Kātyāyana sutra in chapter 15.
This concept refers to the idea that dharmas are empty of svabhava, meaning they lack intrinsic nature or inherent existence. Emptiness itself is also empty, having no independent existence or reference to a transcendental reality beyond phenomenal reality.
The early transmission involved influential figures such as Śāntarakśita who lived from 725 to 788 CE and his students Haribhadra and Kamalashila. The Kadampa figures Atisha and Dromtön later taught Madhyamaka using works by Bhāviveka and Candrakīrti.
Svātantrika states that conventional phenomena have conventional essential existence without ultimately existing essence and uses autonomous syllogisms. Prāsañgika holds not necessary for proponent and opponent to use same kind of valid cognition and shows by reductio ad absurdum that any positive assertion regarding phenomena must be regarded as merely conventional.