Please tell me what are some advanced core exercises?

Two of them: 2 of the best core exercises are "toes to bar", where you hang from a pull up bar and lift your feet to the bar where you hands are holding on to the bar. The second excellent core exercise is front-elevated planks. Think of a normal plank, but instead of your forearms resting on the ground, they are elevated in a suspension device, such as the TRX. You don't have to elevate very high, maybe 10".

Consider the: Many variants of the "plank". One of the best core exercises. Really small adjustments can be made to stress various "areas" of the core or make the exercise easier or harder depending on your "skill" level. See below for more info: http://greatist.com/move/plank-variations-for-core-strength