So far, medical science has done essentially nothing whatsoever to stop ageing from killing us. Instead, current medicine stops us dying prematurely of other causes.
This is in no small part due to a moving of the goal-posts. Medicine has done quite a bit to address many problems with aging. People are able to live much longer lives despite the aging of the cardio-vascular system. As medicine has improved in these areas, the bits that they are good at have their own names and are removed from the 'aging' bucket. Now, with those items removed, 'aging' is only left with things that medicine hasn't yet figured out.
I see no reason at all to think we're just going 'solve' ageing overnight, as the professor seems to think.
Admittedly, the claim being made here is rather optimistic, but it isn't entirely without merit. There is an open question about how difficult the aging problem really is. Aging *could* be surprisingly simple, with just a few genes needing to be tweaked to stop chemical timers that kill cells and inhibit healing. We have many examples of creatures which effectively don't age or even reverse aging during certain events, so we may just need to find analogues in humans, turn them on, and bam, we stop aging. It could be that the only reason we haven't done this previously is we didn't have the right tools for analyzing and altering genes until the last decade or so.
Of course, we probably don't have enough information to know how difficult a problem aging is going to be. Even if this claim is accurate, it is likely that anything it creates will just uncover new problems which will, in turn, need addressing. On the other hand, we thought that gastric ulcers were a hard problem and when a researcher suggested that treatment for most could be as simple as taking a course of antibiotics, he was laughed out of the room.