It is well known that solving Chess is practically impossible using brute force methods. I'm interested to know if there have been any serious attempts using alternate methods. What theory and mathematical tools have been developed to solve (in the weaker sense) Chess? What has been done in abstracting the rules of Chess into a workable mathematical framework?
I first started working on this question about 25 years ago when I tried programming chess AI onto a TI-81 calculator, which has a 2400 byte memory limit. I've been playing tournament chess for about 25 years as well.
the answer is yes there are ways to prove a position is winning or drawn. one example of a self-evident drawn position is the 'checkmate is impossible' example here:
here is a similar theoretical discussion of a draw at the chess stack exchange forum:
the most common 'proofs' I see regarding chess is when human experts can prove a position is drawn but computers still assign a winning evaluation to either player (a computer evaluation of +2.0 or more for example). a 'fortress' in chess is when one side is at a disadvantage but proves a draw. in fact such pathological draws sometimes can provide inspiration and extra insight, and motivate computer chess programmers to re-work their engines until the correct evaluation is reached.
once you establish proofs about draws, it's easy to disturb the balance and prove a win. one way to prove wins is by counting tempi: http://en.wikipedia.org/wiki/Tempo_(chess)
former world champion Botvinnik perhaps was the first to make serious progress in the area you are asking about:
there may be various theorems or proofs, but if the person who discovered them is a competitive player they would be disinclined to share results. I imagine that across all books, academic papers, and message boards there is a large theory that has been developed but has never been concentrated into one place.
also... human experts can look at a position and often determine if white is provably winning, provably drawing, or provably lost. an example would be white has a queenside majority vs black's crippled kingside pawn majority.. which tends to win for white since she can create a passed pawn while black cannot. another example would be arguments regarding move opposition and which king achieves opposition. a subtle example would one side having a light-square or dark-square color complex and the other side having no compensation. such discussions largely revolve around chess 'initiative' and 'compensation'. this is all assuming perfect play.
there is a general theory regarding about if chess is solved, will it be a win for white, draw for white, or loss for white. It is largely believed to not be a loss for white. IM John Watson discusses chess and information theory in his book 'Chess Strategy in Action'.