From Exploring the World of Lucid Dreaming:

I discovered in high school that I was a lucid dreamer when I learned that I could study complicated mathe-matical and geometry problems before going to bed and discovered that I was able to solve the problems when I awakened. This phenomenon followed me through college and medical school. When I was in medical school, I began to apply my sleep-solving abilities to medical problems, quickly running through the questions of the day and usu-ally finding useful solutions or useful additional ques-tions in the process (even today I will occasionally wake up at 3: 00 in the morning and call the hospital to order a special laboratory test on a problem patient, the pos-sible solution of which had occurred to me in a lucid dream). At this point, the greatest use to which I have been able to put this facility is in the practice of surgery. Each night before retiring I review my list of surgical cases and I actually practice these cases in my sleep. I have gained a reputation for being a rapid and skilled surgeeon with almost no major complications. This surgical “practice” has allowed me from the very beginning to constantly review the anatomy and to refine and polish technique by eliminating unnecessary motions. I am presently able to perform most major complex procedures < 35 percent to 40 percent of the time taken by most off my peers. (R. V., Aiken, South Carolina)

I recently pulled second place in a math competition. When I received a copy of the problems (five in all), I spent most of the day mulling over various approaches. When I went to sleep that night, I dreamed lucidly of looking through a particular math reference book I own. I don’t think I dreamed of reading anything in particular in the book, just the act of flipping through it. Subjec-tively, the dream was only a couple of seconds long. When I woke, I didn’t have an opportunity to look through the book until that evening. When 1 did, 1 discovered the trick I needed to solve one of the problems. (T. D., Clarksville, Tennessee)

A little over a year ago, I was in a linear algebra class that introduced me to vector spaces. I was having a lot of trouble understanding the topic on more than a superficial level. After about a week of serious studying, I had a lucid dream about an abstract vector space. I perceived directly a four-dimensional space. The dream did not have a visual component, but such abstract dreams are not uncommon for me. The best I can describe that dream is to say that I perceived four coordinate axes that were mutually perpendicular. Since that night, both math and dreaming have been more fun for me, and I’ve had relatively little trouble understanding vector space calculus. (T. D., Clarksville, Tennessee)

There's a lot of mostly-unexplored territory in lucid dreaming. For instance, people have reported growing more eyes and seeing through all eyes simultaneously, or even being able to see in all directions at once. I once opened a poetry book in a lucid dream and found a poem I hadn't ever seen before, and on waking the poem actually seemed pretty good.