There is something called The Friedmann-Lemaître-Robertson-Walker (FLRW) metric. It’s one out of many solutions of the Einstein field equations of general relativity, but it is the one that has been singled out over the years because it does the best job in explaining the observational evidence for the kind of expansion we see in the universe. This is one of the mathematical substructures you will have to wrestle with if you are to challenge the Big Bang theory.

As for the original singularity, Lemaître had this idea in 1931 of a primeval atom which was “a mass with a measurable size”. It kinda collapsed during the years to come though due to a combination of the gargantuan pressure of gravity and Einsteins theory of general relativity.