Perhaps multi-objective game theory would be an approach to setting up the model.

There is a difficult subject that Bob refers to which is planning scientific activities. By knowing what is impossible, the alchemists stopped trying to turn lead into gold. Once that and other fantasies were put on the back burner the pace of science and technology really picked up. There is more to this subject, knowing when the right time for a technology is, which is a much more complex topic. Edison learned about this with his voting machines, which were great but nobody wanted them. Stephen Wolfram has lectured on planning for science and technology (my only reference here though is from a talk at the summer school. There is probably a blog.) and he has a lot to say on the topic.

My point here is that maybe Bob, if I may restrict you to the list, that you would want a breakthrough in fundamental physics. Right now, traditional physics doesn't allow antigravity. Maybe what is actually true for the fundamental theory allows it or not, but I think that before you'd have antigravity you would need a theoretical "OK, this is possible."

I am a big fan of positive thinking, and taking chances, but I think we have to take into account that in the past wishful thinking has led to much wasted effort.

I'd like to see a mathematical theory of human behavior, such as described by Isaac Asimov in his Foundation novels.

'Psychohistory is a fictional science in Isaac Asimov's Foundation universe which combines history, sociology, and mathematical statistics to make general predictions about the future behavior of very large groups of people".

Given the curated data that Mathematica can access, it might be the language of choice to develop it.

I see good possibilities in two areas, but as usual, the "breakthroughs" are the culmination of long study:

AI:

This was a big deal 30 years ago, but most thinking then was around "expert systems" which were in fact coded in the beginning with all the knowledge they would ever have. So not really intelligent. Watson (the IBM AI computer) demonstrates an entirely different capability. It filters possibilities from a wide network of associations to produce and then score its decisions. But most significantly, it has a method for modifying its future scoring algorithm based on the success or failure of its past decisions. It learns!

Picture a supercomputer in control of the traffic lights of a major city. Two are built. As built they are children, and they play a game with each other. Each simulates responses for the others traffic control decisions. Smooth, fast flow is good; Collisions are bad. Their first few weeks are filled with gridlock and fatalities, but it gets better. After six months they perform far better than the systems we have today. But the fact is, not even their builders understand the strategies they employ.

This could of course go wrong. Terminator? I recall a short story from long ago. They built a super computer and asked it, "Is there a God?" It's response: "There is now!"

Bionic engineering

We see this happening more and more rapidly. Amputees are give electro-mechanical arms or legs. They are connected to the severed ends of the original nerve fibers, both motor and sensory. They don't need to be exactly the right ones because the brain adapts through practice. There is no conscious intellectual component to the learning -- eventually the new arm and hand moves like the old one did, with exactly the same feeling.

New arms, hands, legs, and feet. Auditory signals from electronic devices into the nerve fiber produce sound. Visual signals coupled to the optic nerve produce sight.

And of course this could also go wrong: Eventually, someone will want to house a brain in a life support system and couple it to the world through bionic connections. Like Harlan Ellison's "I have no mouth but I must scream."

Transportation revolution, i.e. modification of the local gravitational field. Aka mass repulsion, gravitational shielding, antigravity, whatever we want to call it. One of the biggest blind-spots in science is that this is "impossible", or too far off to begin working on. My attitude is... self-driving cars, rocket-ships, etc are expressing the right sentiment, but are ultimately a misallocation of resources. Going to mars is thinking way too small, and has a small payoff. While gravity is a difficult problem to solve, I don't have to list out the potential benefits. What's better... a picture of pluto that took years and billions$, or a 3d printable craft that can travel faster than light and visit 10 different solar systems in 10 days. Almost nobody is working on it. Some places to start looking? Ultra-strong, ultra-high-frequency rotating magnetic fields (can be accomplished with stationary coils), plasma centrifuge or gyroscope, etc. Novel phenomenon in novel configurations under extreme conditions. I think the two most important steps are:

1. Cracking the code for room temperature superconductivity

2. Building a high-resolution, large volume (~10 x 10 meter) 3d printer that can print in multiple metals, plastic support materials, high-permeability magnetic core material, and most importantly, a superconducting material. Open it up for design submission, pick the best 20, and boom... we discover unexpected gravitational effects within the year, and fully master gravity within five years.

Once we build an AI, and we ask it for a way to circumvent the limitations of the speed of light, is it really going to say "i have absolutely nothing to contribute on the matter."? No, it will probably giggle and print out the instructions for 10 different methods. Then again, since it would be so darn easy for AI, maybe we should just hold off on gravity research until we have AI, then let it do all the heavy lifting :)

"There is no such thing as science-fiction technology... only science to which we have not yet sufficiently applied ourselves." -Bob