Core: Yes. There are yoga poses to target just about every core muscle. Want to tighten those love handles? Then prop yourself up on one arm and do a side plank. To really burn out the middle of your abs, you can do boat pose, in which you balance on your "sit bones" (the bony prominences at the base of your pelvic bones) and hold your legs up in the air.
Tantra is a range of esoteric traditions that began to arise in India no later than the 5th century CE.[note 17] George Samuel states, "Tantra" is a contested term, but may be considered as a school whose practices appeared in mostly complete form in Buddhist and Hindu texts by about 10th century CE. Tantric yoga developed complex visualizations which included meditation on the body as a microcosm of the cosmos. They included also the use of mantras, pranayama, and the manipulation of the subtle body, including its nadis and cakras. These teachings on cakras and Kundalini would become central to later forms of Indian Yoga.
"...[T]here is the cultivation of meditative and contemplative techniques aimed at producing what might, for the lack of a suitable technical term in English, be referred to as 'altered states of consciousness'. In the technical vocabulary of Indian religious texts such states come to be termed 'meditations' ([Skt.:] dhyāna / [Pali:] jhāna) or 'concentrations' (samādhi); the attainment of such states of consciousness was generally regarded as bringing the practitioner to deeper knowledge and experience of the nature of the world." (Gethin, 1998, p. 10.)
Equipment required? No. You don't need any equipment because you'll rely on your own body weight for resistance. But you'll probably want to use a yoga mat to keep you from sliding around in standing poses, and to cushion you while in seated and lying positions. Other, optional equipment includes a yoga ball for balance, a yoga block or two, and straps to help you reach for your feet or link your hands behind your back.