A question about the influence of The Mars Volta

I've heard that The Mars Volta happen to be an incredibly influential band in the modern prog scene. Would I be correct in saying that they were one of the pioneers of the kind of sound they have? With the strong influence of the post hardcore/alternative rock scene of their time and the generally chaotic, hyperactive nature of them?

Definitely influential, though many of the people here may not be familiar with some of the bands influenced by The Mars Volta. Thank You Scientist, to name but one, have a very strong MV vibe in their sound, and a very energetic, somewhat anarchic stage presence.

Thanks!  I must confess to being very behind in my listening, I know almost nothing of The Mars Volta (great band name BTW), if I were to listen to a single song/album, what would be their definitive work?  Best holiday wishes, Chuck in Chicago!  

The first one is the best one to start if you want something very catchy and modern. (de loused)
If you want to have something that is pretentious but also mind blowing, take Frances The Mute
The third one is similar to the second but more progressive less accessible and a bit less pretentious, but overlong, so maybe too hard to swallow for the first time listener. This one needed some time to grow on me. Now it is one of my favourites by them.
Number 4 lets some peoples ears bleed... so be warned. It takes all the complexities of the work right before and puts it into hyper speed hyper noisy relatively short songs and lets the sound appear to be relatively muddy and with little variation. Only if you want to be catapulted into another sphere at your first listen.

So these first 4 are all equally good in my opinion or are at least the essential ones.
I d recommend Frances The Mute as it is maybe the most representative in its mix and will definitely blow your mind completely nuts.

The Bedlam in Goliath is very uneven. There are some great songs, but others go on a bit too long, and come across as rather self-indulgent. On the other hand, Amputechture (their third) has become a fast favourite of mine. I'd even place it slightly above Frances the Mute. Octahedron is also a good bet for those who like shorter songs. Not too impressed with their final offering, Nocturniquet, though there are some very goods songs on it too.