Good stuff! I wouldn't mind betting that when paper-thin body-armour comes along, it will probably use carbon nanotubes (which are much stronger and lighter than even steel.) I'm sure that they will eventually outdo ceramic armour too.

The problem with thin is that even if the bullet doesn't go through, the bullet will still do tremendous damage as a blunt impact causing external and internal bruising, therefore this has to be accompanied by other thicker layers which can be of light material, but by itself it is still dangerous.

Anything that reduces the load on dismounts is great but there are a few issues that need to be addressed. In addition to stopping penetration, body armour and clothing needs also to address the blunt trauma injury caused by absorbing the energy from the strike. There is also performance against blast injury from IED to be considered. The rigidity of the armour will dictate how it is worn, either like the current plates or more like the steel armour of old, profiled to the body. There is likely to be the need for some energy absorbing material behind and this needs to be reconciled by heat management when the armour is worn in hot climates. Overall the soldier should end up with something much lighter and with as good or better protection but I hope developments will also make it more comfortable and less restricting and give dismounted troops back some of the mobility (together with the inherant protection that rapid movement provides) that they have lost in recent years.

From the picture it looks like the glass 'bullet' tore a gaping hole, right through the armor.
Am I looking at it, wrong?

What everyone else says: The paper-thin armor might stop penetration, but it won't stop broken ribs or the blunt trauma. Some sort of lightweight ceramic plate armor in conjunction with this polymer sheet might help absorb and dissipate the impact.

In regard to stopping blunt object inflicted injuries, all that would be required would be some means of keeping the armour a little distance from the skin- eg selective padding, rubber cones, etc.
Additionally, having the armour held away from the skin could mean that some form of simple ventilation system could be employed to keep the soldier cool- I hate to think how much those guys must sweat in the heat of Afghanistan or Iraq.

Why not just use good old empirical data?
Fabricate a few different materials and fire some bullets into them. That way the material gets out into the marketplace where it can be utilized and the over focused PhD's can figure out the "why it works" details later.
I wonder how far behind we'd be if we just discovered iron and it was up to the guys at MIT to determine why it's able to cut through wood before we made a saw.

“inability to reliably test such materials against projectile impacts” What? Give me a Barrett 50, an M-16, AK-47, and maybe a few exotics like the Steyr AUG and FN-SCAR…and limitless access to military and civilian ammunition… and a few “sheets” of this stuff and I will test it for you… has to be a lot simpler than setting up a pulse laser to propel microscopic glass beads for crying out loud… I get that they want to look at this material from the most scientific and molecular level but for crying out loud… analysis paralysis anyone… does it stop a bullet or not… how big… with what kinds of “tips”…what backing does it require to not kill the wearer due to the energy transfer on the backside… Throw some lead at it and Get’er done!

The better solution is robotic war where soldiers are no where near any harm. I know that robotic air craft are now common enough and the Navy will soon have very potent unmanned ships. The foot soldier remains the largest challenge for robotics. Supposed critics make noise about innocents lost in conflict. The normal slaughter of innocents is about 20-25% of all killed in war. So far robotic- drones have a far better gentleness to uninvolved parties.

RE: Blunt trauma: Wrong. Merely adding a puncture-proof layer of "paper thin" material over, say, a standard BDU, provides plenty of room for even the minute resistance of air trapped between the material and the body to slow the projectile (and the material itself) down to acceptable levels. The average bullet, whether pistol or rifle, has less kinetic energy than a decent fastball. The difference is it's really light, really small, and going wickedly fast, so you the timescales of interaction change from "oh, i need to catch that" to "oh, i was shot... and now i'm bleeding to death..."
So, extending the duration of the impact event from microseconds to milliseconds, turns a Gigawatt interaction into a longer, but lower power megawatt or tens-of-kilowatts interaction. Nobody's getting liquefied.
Things do change with a .50 BMG... that's more or less the kinetic energy of a full size pickup being dropped on your chest from about ten-twelve feet up. *Splat*