And about new discoveries, some math problems were long considered impossible to solve. At the bottom of this Wikipedia page is a list of problems that were eventually and unexpectedly solved, and these are only fairly recent examples. And that's with "an exact science."
Isn't a maths problem a known problem? The fact it hasn't been solved is irrelevant to this debate. That nothing but a photon can travel at the speed of light and nothing can travel faster than the speed of light is not a problem. It's a fact.
Cosmologically speaking, the entire universe is expanding at many times faster than the speed of light; any object in the cosmos father away from us than 13 billion light years is in fact traveling away from us at FTL velocities. Mathematically speaking, this is in accordance with relativistic physics and is perfectly allowable, ergo FTL travel is still possible under certain circumstances with a sufficient curvature of space.
Mathematically speaking, it is possible to exceed the speed of light by changing the conditions of your local reference frame via the Alcubierre Metric or various other solutions to the field equations.
CONTEXTUALLY speaking, I'll again remind you that the article in the OP pertained to time travel, not FTL travel.
It's not fair that I should have to give you the key to understanding how mainstream astrophysics is full of beans without someone's taking up the challenge of my triangle question above, the answer to which is just a Google away and nicely illustrates thinking out of the box.
Okay, expanding universe? Big Bang? These silly conclusions were based on calculations using observed redshift of distant celestial objects, obviously. So what did they do wrong? They accounted for the known distribution of atomic hydrogen in space, which reined in their resulting speed of expansion a little. That form of hydrogen is easily detected with radioastronomy, But when two atoms of singlet hydrogen (H1) meet, what to they do? They mate for life and live happily ever after, or as long as can they avoid oxygen, etc., as H2. But since the electric field and spin of an H2 molecules' electrons are completely canceled, this unusual, happily married molecule is very difficult for astronomers to detect in space. We can do so now only if it's hot, and we have to develop technology to detect cold H2 in space. Hot H2 has been detected by the European Space Agency's Infrared Space Observatory. E.A. Valentijn and P.P. van der Werf detected huge amounts of molecular hydrogen (H2) in NGC 891, an edge-on galaxy 30 million light-years away in Andromeda (Valentijn and van der Werf 1999). In their report published in September 1999, they stated that their result "matches well the mass required to solve the problem of the missing mass of spiral galaxies." They concluded that the galaxy contained 5 to 15 times more molecular than atomic hydrogen. So there is perhaps ten times more molecular hydrogen in space than atomic hydrogen (that ratio is not known, since we don't have the proper instruments to measure it). But to deny it exists is like saying that hydrogen atoms don't pair off and become happy couples. Yet the above-mentioned calculations don't account for any H2 at all (let alone 5-15 times as much H2 as H1), and folks spin these wild yarns about expanding universe, Big Bang, and lots of other things that don't make sense but do address the problem of attracting students to fields of study they might otherwise find boring. So it does have that upside and continues unabated even though, as I mentioned, the key was detected over a decade ago, and anyone who brings it up is labeled a spoilsport. I've been called worse. But in the perhaps distant future, people will joke about this and compare it to the situation before the idea of meteorites as remains of meteors was accepted, in which mainstream scientists said, "Rocks don't fall from the sky!" And such thinking pretty much prevailed until that generation died off and was replaced by a generation who accepted the existence of meteorites as common knowledge. So it takes a while for things to sink in.
And achieving FTL is not about curvature of space. It's about using the unseen material that comprises the fabric of space as your fuel and creating a local vacuum in that fabric, into which every particle of matter comprising your vessel will be individually pushed forward by the ZPE (gravitons) comprising the fabric of space. You won't feel any g force when changing speed, since you are being pushed along by the same stuff that is sometimes called gravitons. Special relativity, as I have already explained upthread, applies if you use conventional rocketry, but not in this scenario. To even get started designing an FTL system you have to think of gravitons as not reaching across tens of thousands of light-years from the galactic core to pull on our solar system to keep it and our Orion Arm in orbit but being everywhere not occupied by so-called "real" subatomic particles and pushing. Think of a ball of water floating around in the Space Shuttle. Atmospheric pressure holds it together. So the acceleration you get with the type of FTL I described pushes every particle of matter individually in the direction of travel, a situation not covered by special relativity. This is a hot area of research today, but it's still in its infancy. And, as I mentioned, the terminology is all over the place, with many names for the same thing.
If the exponentially-increasing energy required for fast motion is due to drag from virtual particles (in some highly abstracted sense), then perhaps we can generate some kind of field which encourages virtual particles to form less frequently in a given area.
Drag is exactly right. It's the mechanism for Special Relativlty. It's like wind resistance except that this wind is made of not air molecules but stuff that affects every particle of you and your vessel's structure. But it's not so much a matter of stimulating virtual particles to make them form real particles less frequently as it is to stimulate them to take a form you can harvest and use as fuel as you clear them out of the way and form a pocket of graviton vacuum in your path, not an easy task but the key to FTL without "packing a lunch" (carrying fuel tanks).