Buddhism after death

From what I understand the aggregates aren't self

Correct (per SN 22.59).

but now when a person dies

From the viewpoint of higher truth (that the aggregates are not self) there is no "person". The idea of a "person" is the same as the idea the aggregates are "self":

The Blessed One said, "And which is the burden? 'The five clung-to-aggregates,' it should be said. Which five? Form as a clung-to-aggregate, feeling as a clung-to-aggregate, perception as a clung-to-aggregate, fabrications as a clung-to-aggregate, consciousness as a clung-to-aggregate. This, monks, is called the burden.

And which is the carrier of the burden? 'The person,' it should be said. This venerable one with such a name, such a clan-name. This is called the carrier of the burden.

SN 22.22

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There are these five clung-to-aggregates, friend Visakha: form as a clung-to-aggregate, feeling as a clung-to-aggregate, perception as a clung-to-aggregate, fabrications as a clung-to-aggregate, consciousness as a clung-to-aggregate. These five clung-to-aggregates are the self-identification described by the Blessed One.

MN 44

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Why now do you assume 'a being'? Mara, have you grasped a view? This is a heap of sheer constructions: Here no being is found.

Just as, with an assemblage of parts, The word 'chariot' is used, So, when the aggregates are present, There's the convention 'a being.'

SN 5.10

the material aggregates of his body dissolve what remains who gets reincarnated ?

This question is unrelated to higher Buddhism. In MN 117, it is taught there are two sorts of right view. The view of 'not-self' is transcendent right view and the view of 'self reaping results of kamma' is worldly right view. The Buddha did not mix up the two sorts of right view.

An analogy is kindergarten for children vs university for adults. Calculus is not taught to children in kindergarten and basic arithmetic is not taught to adults at university. Similarly, not-self is not taught to children and reincarnation is not taught to adults.

When the new material aggregates of the new body are born then they are different from the ones of the person who died .What defines that THIS person is in THAT body?

In higher truth, the idea "a person" or "self" dies is called "annihilationism" and the idea "a person" or "self" is reincarnated is called "eternalism", as follows:

There are, bhikkhus, some recluses and brahmins who are eternalists, and who on four grounds proclaim the self and the world to be eternal.

There are, bhikkhus, some recluses and brahmins who are annihilationists and who on seven grounds proclaim the annihilation, destruction, and extermination of an existent being. And owing to what, with reference to what, do these honorable recluses and brahmins proclaim their views? Herein, bhikkhus, a certain recluse or a brahmin asserts the following doctrine and view: 'The self, good sir, has material form; it is composed of the four primary elements and originates from father and mother. Since this self, good sir, is annihilated and destroyed with the breakup of the body and does not exist after death, at this point the self is completely annihilated.' In this way some proclaim the annihilation, destruction, and extermination of an existent being.

DN 1

In cultural Buddhism, because most faith follower Buddhists are not enlightened, they are taught and believe in "eternalism"; which is why they cling so tenaciously to metaphysical ideas of "rebirth".

In the history of Buddhism, there were heretic teachers, such as Buddhaghosa, who taught "not-self phenomena are reincarnated". This idea is heretical for at least two reasons:

1. To believe not-self is reincarnated destroys the incentive to do good personal kamma.

2. To believe not-self is reincarnated destroys the view of impermanence & dispassion required to realise Nibbana.

In the Pali suttas, it is clearly taught the same "person" inherits the results of "his" or "her" actions. That good & bad results follow from good & bad actions is what the scholars loosely translate as "rebirth" or "reappearance".

According to SN 12.17, Eternalist have the view ‘The one who acts is the one who experiences’ and annihilationist have the view ‘The one who acts is someone other than the one who experiences.

"'The one who acts is the one who experiences [the result of the act]' amounts to the eternalist statement, 'Existing from the very beginning, stress is self-made.' 'The one who acts is someone other than the one who experiences'[2] amounts to the annihilationist statement, 'For one existing harassed by feeling, stress is other-made.' Avoiding these two extremes, the Tathagata teaches the Dhamma via the middle:

The Buddha teaches via the middle. So our view or the “Buddhist view” would be,

‘From ignorance as a requisite condition come fabrications. From fabrications as a requisite condition comes consciousness. From consciousness as a requisite condition comes name-&-form. From name-&-form as a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving. From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth. From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. Such is the origination of this entire mass of stress & suffering.’

People always ask what is reborn and the mechanism of rebirth, as if that mattered, but that's really not the point of the dhama. THe only thing you need to know about rebirth is that rebirth does not stop dukkha, what is the condition for rebirth and what is the condition of cessation of rebirth.

asking about the precise working of rebirth is like asking what happens to your old car that you trash, after you buy a new car. The answer is that the old car goes to so some junk yard, where it is dismantled. BUt for the people who ask what happens after death, they want to know more, they say it matters what is done with the thousands little parts of the car and where they go. That's really none of your business.

In practice, nobody cares much about that, what matters is that you have taken up a new car, the new car is what you focus on, and that having a new car is bad, so you must trigger the condition to stop having a new car. The main worry about rebirth is that there is the taking up of a new body, worrying about what happens with the old body is not important for anybody who wants to stop dukkha.