Is there reincarnation?
Short answer: Reincarnation is a theory, and if viewed correctly, a helpful one. But it cannot actually be proven. It may be true, it may not.
Long answer: Does it really matter? Even if there is reincarnation, the next “you” will not remember who it was before. Even if someone claims to remember a past life, how is that relevant to enlightenment? Does knowing that you were so and so in a past incarnation help you remove ignorance of your limitless nature? Thinking you are a person subject to reincarnation is the root of ignorance itself. So in the context of Vedanta, the only helpful way to view reincarnation is as the continual identification with our thoughts. Every moment we are identifying with a thought that arises (is born) and then it goes away (dies). We are born to each thought and then we die to it. Then another thought arises and the process starts again. This is the constant cycle of birth and rebirth. Continuing to identify with it is samsara.