Why is geometric quantization (esp. Berezin-Toeplitz quantization) interesting for a symplectic geometer/topologist?

Question

I know that many symplectic geometers are interested in quantization as well. From what I understood, quantization isn't expected to be used as a tool to answer symplectic questions (as in translating the problem to the quantum world, solve it there, and somehow go back), unlike the relation, for example between complex geometry and tropical geometry. Rather, it is supposed to be interesting and have appeal for reasons intrinsic to quantization.

As a person whose main interests are symplectic geometry, topology and algebraic geometry, why study quantization? What aspects or problems there may I find appealing?

For those who approach mathematics from the physics realm, I guess the answer is straightforward, as those ways of quantizing symplectic manifolds are attemps at formalization of some physical theories and beliefs.

I am looking for a more mathematically oriented answer, since I personally derive my interest and motivation from the intrinsic beauty of mathematics.

Thanks

This post imported from StackExchange MathOverflow at 2016-02-25 15:17 (UTC), posted by SE-user Blade

Comments

This is very broad to answer in a MO post. For orientation I suggest you read the 4-page intro to Kirillov's Lectures on the Orbit Method. (Berezin-Toeplitz is somewhat outside of the mainstream he describes.)

This post imported from StackExchange MathOverflow at 2016-02-25 15:17 (UTC), posted by SE-user Francois Ziegler