What is the white ash on my soap?

This is sodium carbonate. It is formed during the soap making process when free sodium from the sodium hydroxide (lye) comes in contact with the carbon dioxide from the air. It is water soluble and harmless. It will rinse off of your bar after the first use.

Why does this circle appear in my soap?

When oils and lye are mixed the process is called saponification. This chemical reaction is exothermic, which means that heat is released. Soap makers refer to this as the gel stage. The reaction begins in the center of the soap and moves toward the edges. Some soap makers choose to add additional heat so that this reaction fully reaches the edges of the bar, while others choose to cool their bars so that the 'gel stage' does not occur. We choose to allow the reaction to occur, unassisted in any way. The only time we cool our bars is when there is the possibility of the reaction becoming too hot because of ingredients added. After the 'gel stage' the bar cools and hardens. It is safe to use and this will not change its performance.