“Math is not just a game and it’s not as shallow as it looks. There is a scientific theory that can connect different things in profound ways and it’s very beautiful actually, and I just like that,” said Northeastern Math Professor Ana-Maria Castravet, upon sharing her excitement about algebra and geometry.
The National Science Foundation has just approved Castravet’s grant proposal for her project on the moduli spaces of rational curves. This topic is one of many in the field of Algebraic Geometry, which has been Castravet’s main focus since her undergraduate work. Algebraic Geometry, as she puts it, provides a way to do algebra motivated by geometry. The goal is to determine the geometry of shapes defined by the zeroes of polynomial equations in several variables. This is something that is done at a more basic level to determine the geometry of lines and circles, but can be applied to higher dimensional objects as well.
For her upcoming project, Castravet will be studying specific geometric shapes called moduli spaces. Moduli spaces capture the variation of shapes as they degenerate, as a reflection of slightly changing their equations. For example, a circle may degenerate to just a union of two lines, based on the equations. Castravet’s moduli space appears simple, capturing trees of lines with points on them that can vary and further break into more lines. She is looking to understand the shape’s geometry and if it’s truly as simple as it looks.
“It turns out we actually don’t know many things about this particular space, so I’m fascinated by this object,” Castravet said. “For example, we can prove new combinatorial identities about integer numbers, as a result of complicated theorems about the geometry of the space.”
Her work will focus in a few different directions. Moduli spaces are closely tied to theoretical physics, and one aspect of the project will focus on this connection. Additionally, the project will focus on connections with number theory and how these can be used to further study the geometry of the moduli space.
Castravet has some challenging work ahead of her. She hopes to gain inspirations from her colleagues and from the many conferences she attends. “Sometimes you wake up and it’s a new day, and something new comes to you. In the end, I hope that even hard work that fails will have a purpose,” she said.
And her project’s results will have implications for other areas in mathematics. Castravet noted, “If I succeed in doing what I wanted to do, it will open up possibilities for other people to solve other open problems.”