There is virtually no information or mentions of Hypatia's mother in any text. One can assume that she had a mother since she existed; it's unclear whether she was present throughout Hypatia's childhood, though. Her father (Theon) has been mentioned numerous times, but not her mother. It's probable that Hypatia was not raised by her birth mother, considering the absence of any mention regarding her. Note that it was common during her time for women to be disincluded in historical and religious texts; therefore, Hypatia's mother very well may have been present. There is no evidence to support such a claim, however.
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Hypatia of Alexandria went to school at Platonism or Platonist.
Hypatia's interest in mathematics likely stemmed from her upbringing in Alexandria, Egypt, which was a hub of learning and scholarship. Her father, Theon of Alexandria, was a mathematician and philosopher, and he is believed to have introduced her to mathematics at a young age. Hypatia went on to become a renowned mathematician and philosopher herself, contributing to the field through her teaching and writings.
Hypatia held a teaching chair in Philosophy. She was particularly known for teaching (what we now call) Neo-platonism. She taught mathematics as a discipline for the mind that would allow students to better comprehend philosophy. Two good books on Hypatia are Hypatia of Alexandria by Maria Dzielska (Harvard University Press, 1995) and Hypatia of Alexandria: Mathematician and Martyr by Michael A. B. Deakin (Prometheus Books, 2007.) I've reviewed both books on my blog and have a number of other posts on Hypatia.
Any discoveries made by Hypatia did not survive.
Unfortunately, none of Hypatia's works survived.