The "singularity theorems" of the 1960s, demonstrated that large enough celestial bodies, or collections of such bodies, would, collapse gravitationally, to what are referred to as "singularities", where the equations and assumptions of Einstein's classical theory of general relativity cannot be mathematically continued. These singularities are normally expected to lie deep within what are now referred to as black holes, and would, themselves, not be observable from the outside. Nevertheless, their presence is regarded as fundamentally problematic for classical physics and it is argued that a quantum theory of gravity would be needed to resolve the issue.
Similar arguments (largely developed by Stephen Hawking) apply also to the "Big-Bang" picture of the origin of the universe, showing, again, the inevitability of a "singular" structure of such an initial state. However, a puzzling yet fundamental distinction between these two types of singularity is found, deeply connected with the 2nd law of thermodynamics, according to which the "randomness" in the universe increases with time. It is hard to see how any ordinary procedures of "quantization" of the gravitational field can resolve this problem,
Nevertheless, a deeper understanding of the special nature of the Big Bang can be obtained by examining it from the perspective of conformal geometry, according to which the Big-Bang singularity, unlike those in black holes, becomes non-singular. In conformal geometry, big and small become equivalent, and the Big Bang may be taken as conformally non-singular. Moreover, the extremely hot and dense Big Bang is conformally similar to the extremely cold and rarefied remote future, so that our Big Bang can be regarded as the conformal continuation of a previous "cosmic aeon", leading to the picture of conformal cyclic cosmology (CCC) whereby the entire universe consists of a succession of such cosmic aeons, each of whose big bang is the conformal continuation of the remote expanding future of a previous aeon. Certain strong observational signals, provide some remarkable support for this CCC picture.
By the request of the speaker, the recording of the talk is not public.