The LHC has been running more-or-less continually since last December, and at 7 TeV center-of-mass energy since mid-March. I think there was a web feed at the time.
I don't think anything we do there will relate to Fermat's last theorem, and the big bang is pretty much established. What it will do is probe higher energies than we've been able to until now; the Standard Model is the most successful theory in science (in my not completely unbiased opinion), but at 1 TeV the broken electroweak symmetry must be reconciledhow? The answer to this question will determine the direction of high-energy physics research for the next several decades. We're discovering the fundamental building blocks of the universe and the rules governing their interactions.
