"Before" the Big Bang was the Universe really compressed into a mathematical point? A couple of weeks ago a teacher of mine (I'm taking mathematics) was giving a final inspiring lecture about how fundamental Math is to every possible universe. During the lecture though, he said that what the current physics theories tell us is that the Universe originated from a point. Not a really small point, but an actual mathematical point, one with 0 dimensions. I've heard the concept of a singularity before, but even in theory, can it happen in Nature?
EDIT: My question is not the same as Did the Big Bang happen at a point?. I'm not interested in how the Universe expanded, I want to know if the Universe was compressed into an actual point.
 A: General Relativity predicts that singularities happen, and that the universe began at a singularity which is, in principle, visible to us.  In some other physically-reasonable cases where GR predicts singularities they are 'censored' by event horizons.  I believe it is an open question whether they are so censored in all other physically-plausible situations although people think (hope) they are.
However this is a different question as to whether singularities actually happen.  General Relativity is a classical (non-quantum) theory and so it more-or-less can't be correct in some cases, one of which is the situation very close to the singularities it predicts.  To know what actually happens in these cases we would need a quantum theory of gravity which we do not have (and are probably not very close to having).
I think a reasonably common view would be that the prediction of singularities is an indication of the failure of the theory, not an indication that singularities actually exist.
So in summary: our current best theory of gravity says that yes, the universe began at a singularity, but it is probably wrong about this.
A: To complete tfb's answer about quantum gravity, General Relativity is simply not applicable beyond a certain regime when you evolve back in time the expansion of the universe all the way towards the singularity. At some point (at the Planck scale) we know that quantum effects start to dominate over classical GR effects and we don't control these yet. Actually the classical notion of space-time itself fails in this regime.
An active area of research is to seek for traces of these quantum effects in the Cosmic Microwave Background.
Edit: comment on classical space-time.
Generally in quantum gravity there is the notion that space-time is emergent. If you try to probe the structure of space-time at smaller and smaller distances, you need to use higher and higher energies (uncertainty principle). But at some point (around the Planck scale) you have so much energy in so little volume that you create a black-hole. This leads to the belief that the classical picture of space-time can not be valid at the Planck scale and could only be an "emergent" description that a fundamental theory of quantum gravity should provide.
A related notion is that gravity is non-renormalizable as a field theory, which leaves two options within the standard quantum field theory paradigm: either the theory has a "non-trivial ultrviolet fixed point" (scenario of asymptotic safety) or it has different degrees of freedom at high energies than at low energies. (at low energies the "degress of freedom" are space-time with its metric). Since these high-energy degrees of freedom are, strictly speaking, unknown to us it is impossible to make a prediction about the behaviour of the theory in these very strong curvature regimes (the big bang is a curvature singularity). In particular is it not possible to assume that the notion of space-time continues to make sense.
