Perhaps it's a story where a character is protecting a child from someone who wrongly believes the child is theirs. In the Harry Potter universe, there could be dark wizards trying to take the child for some evil purpose, thinking it's their own for some convoluted reason. And the real parent or guardian has to use magic and their wits to keep the child safe.
Well, it could be about a character in the Harry Potter world who discovers that a child is theirs and not someone else's. Maybe it's a story of a wrong assumption of paternity, and the real parent has to fight to claim the child. For example, a wizard who was wrongly thought to be the father by another wizard, and through a series of magical adventures and revelations, the truth comes out.
Hermione could be a main character. She's smart and resourceful. If there's a child involved and a case of mistaken identity regarding the child's parentage, Hermione would surely be involved in sorting out the truth, using her knowledge of magic and logic.
A plot might be about Harry's child struggling with the pressure of living up to their father's legacy. At Hogwarts, they constantly feel compared to Harry. But then, they find their own niche in the wizarding world by being really good at potion - making. They enter a big potion - making competition and end up using their skills to save the day when a dark potion goes wrong and starts affecting students. They prove that they can be great in their own way, not just because they are Harry Potter's child.
In such a fanfiction, perhaps the plot could involve Harry seeing a child in a difficult situation, like an orphan in the wizarding world who has no family due to a dark wizard's attack. Harry, having been an orphan himself, feels a strong connection and decides to adopt the child. He then teaches the child about magic, love, and how to face the challenges in the wizarding world.
Perhaps the plot could be that the child of Harry Potter gets sent back in time accidentally. There, they meet a young Harry and his friends. The child has to be very careful not to disrupt the past too much while also trying to find a way back to their own time. This would be filled with lots of tricky situations, like when they almost reveal future events or when they have to avoid being seen by their own young parents.
Well, perhaps the Yautja is on a hunt in its own world but gets caught in a magical portal that leads it to Hogwarts. The students are both scared and intrigued. Hermione starts researching about it, trying to find a way to send it back. Meanwhile, the Yautja sees the Death Eaters as worthy prey and starts stalking them. Harry has to convince the Yautja not to go on a full - out killing spree which might endanger innocent lives as well.
A possible plot could be that a Smilodon escapes from a magical menagerie in the Forbidden Forest. Harry and his friends discover it and try to return it safely without getting hurt or getting in trouble with the Hogwarts authorities.
Perhaps Harry could be a wizard - geisha hybrid. He might use his magic to enhance the traditional geisha arts, like creating magical illusions during his performances. He could be in a Hogwarts - like school but also trained in the ways of the geisha, with spells and potions being used in the context of geisha traditions.
Well, one possible plot could be that Orochimaru somehow finds his way into the Harry Potter universe. He might be interested in the unique magic there, like the Elder Wand or the Philosopher's Stone. He could try to use his ninja skills and devious ways to steal these powerful magical items, and Harry and his friends would have to stop him using their own magic.
In this 'harry potter math fanfiction', it might also be that there is a new subject in Hogwarts called 'Magical Math'. The teachers could be magical creatures who are experts in both magic and math. The students would have to learn things like how to use equations to create spells or use geometric shapes to build magical structures. And there could be a big competition where the students use their math skills to win a great prize, like a magical artifact that can enhance their magic power.
A possible plot could be Morgoth's return in a weakened form in the Forbidden Forest. His presence starts to corrupt the magical creatures there. Harry and his friends sense the disturbance and go to investigate. They soon realize it's a much greater threat than they thought and must find a way to either banish Morgoth back to his realm or destroy his essence completely.