Who is Adi Shankara?
Sri K Pattabhi Jois belonged to the Hoysala Brahmin caste, a subset of the popular Smarta Brahmins whose root teacher was Adi Shankara (788-820 CE). Adi Shankara is considered to be one of the most revered Hindu philosophers and theologians. His works establish and consolidate the doctrine of Advaita Vedanta, or non-dual Vedanta.
Advaita Vedanta is often called a monistic system of thought, in that it claims to explain the diversity of existence in terms of a single reality or substance. The word "Advaita" thus essentially refers to the identity of the Self (Atman) and the Whole (Brahman). Advaita Vedanta says that the one unchanging entity (Brahman) alone exists. Much as the ocean's waves have no existence in separation from the ocean, changing entities do not have absolute existence of their own. In his words, “Brahman or the Absolute is alone real; this world is unreal; and the Jiva or the individual soul is non-different from Brahman." In short, the conditioned reality that our ego-structured minds and bodies can comprehend is an illusion—through the direct experience of the ever-transforming nature of everything from ourselves to the world in which we live, we may begin to identify not with that which must always change, but rather with that which will always remain. Advaita Vedanta thus proposes the theory of Maya, in which the universe is likened to "a trick of a magician." Adi Shankara and his followers see this as a consequence of their basic premise that Brahman alone is real. Their theory of Maya emerges from their belief in experiential reality of the absolute consciousness 'Brahman'—in other words, it is only through direct knowledge that one may realize Brahman.
Shankara travelled across the Indian subcontinent to propagate his philosophy through discourses and debates with other thinkers. He established the importance of monastic life as sanctioned in the Upanishads and Brahma Sutra, in a time when the prevailing Mimamsa school of Hindu philosophy established strict ritualism and ridiculed monasticism.
Adi Shankara's opponents accused him of teaching Buddhism in the garb of Hinduism, because his non-dualistic ideals were radical for the Hindu philosophy of his time. Hinduism, of course, is a religion of many cults and different deities. The claim of Shankaracharya is that though Hinduism consists of several cults there should be no conflict among them, as they are all manifestations of the same whole of absolute consciousness that is Brahman. There is, however, a marked difference in Shankara’s approach to non-duality and the Buddhist doctrine of absolute emptiness: while both perspectives agree on the illusive nature of the conditioned world, Advaita Vedanta professes that it is the unceasing presence of pure consciousness that is the nature of true reality (Self). The Buddhist doctrine of shunyata identifies the dependently arisen nature of material phenomena as devoid of permanent and eternal substance (non-self). For more on the similarities and difference between different theories of liberation, click here.
How do his teachings relate to our practice?
Well, to begin with, we invoke Shankaracharya every morning as we chant the Ashtanga mantra. What we call the Ashtanga Mantra is really two verses from different sources. The first is a verse from the “Yoga Taravali” by Shankaracharya, while the second verse is from a longer prayer to Patanjali. In the “Yoga Taravali,” Shankara poetically and metaphorically summarizes the highest teachings of yoga, explaining the different stages of yoga and how to reach the highest state, that of Raja Yoga.
vande gurūṇāṃ caraṇâravinde
I honour the lotus-feet of the gurus,
those peerless feet through which the understanding of the delight of one’s self is manifested,
performing like a snake-charmer,
for the quelling of the delusions from the poison of worldly existence.
That is, by honoring the two lotus-feet of the plurality of gurus, we hope to quiet the poisonous effects of samsara, or conditioned experience, because it is through these feet that we may gain access to the experiential reality of the Absolute. We begin our daily practices thus by paying homage to the lineage of teachers that offer access to an experience beyond the prison of worldly existence.
I conclude with two other salient verses from Shankara’s text Aparokshanubhuti that explicitly address some specific techniques that we cultivate in Ashtanga Vinyasa. As lines 114 and 115 read:
114. That [Brahman] which is the root of all existence and on which the restraint of the mind is based is called the restraining root [mulabandha] which should always be adopted since it is fit for raja-yogins.
115. Absorption in the uniform Reality should be known as the equipoise of the limbs [dehasamya]. Otherwise, mere straightening of the body like that of a dried-up tree is no equipoise.
Whatever our religious or spiritual identifications, the physical yoga of our morning practice situates us on a threshold of the possible and known and the seeming impossible and inaccessible. Through our practice, we may get a sense, a fleeting taste, of the idea that our own subtle architecture contains a physical and psychological capability that is both beyond our gross anatomy and located at the very root of it. This is the gift of non-duality, whatever its specific doctrine may be: the absolute IS the conditioned. The gross is the subtle. A physical practice that points itself towards this awareness, then, may offer the practitioner an experience of embodied equipoise, a samasthiti, that allows for an absorption into something beyond our egoic architecture.
 Translation by Isaac Murchie.