Mathematicians are using the geometry of moving objects—the same mathematics that describes a double pendulum’s swing—to solve stubborn problems in the geometry of static shapes like lines and circles. Algebraic geometry studies shapes defined by equations, but many questions about counting solutions or describing complicated surfaces remain out of reach with classical tools. Symplectic geometry, rooted in physics and the equations of motion, offers a fresh perspective. A 1990s insight from string theory, called mirror symmetry, showed that problems in one field can be translated into the other. This project reverses that flow: instead of using algebraic geometry to solve symplectic problems, it applies symplectic intuition back into algebraic geometry. This is fundamental mathematics, not applied research with an immediate practical target. There is no direct impact on infrastructure or daily life today. But the same kind of deep structural insight that mirror symmetry provides has historically underpinned advances in cryptography, coding theory, and the algorithms that run navigation systems and data networks. A richer understanding of how these two geometries connect could, over decades, reshape the mathematical tools that quietly support modern technology.
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Algebraic geometry is the mathematical study of shapes defined by algebraic equations, such as lines and circles. Classical algebraic geometry answers questions like "How many intersection points are there between two lines?" which can be rephrased as finding the number of solutions to a set of algebraic equations. While techniques that we now use in the field are modern and sophisticated, much of our inspiration and methodology evolved from these classical ideas. Symplectic geometry, on the other hand, finds its roots in physics. As a mathematical field, it is relatively young compared to algebraic geometry, with its modern treatment in mathematics beginning in the 1970s. The objects studied in symplectic geometry are solutions to the equations of motion. One needs to look no further than a double pendulum to see that the geometry of moving objects is more fluid and flexible than the equations that govern algebraic geometry. In the 1990s, a remarkable prediction arose out of string theory: that studying algebraic geometry in one setting is equivalent to studying symplectic geometry in a "mirror" setting. This equivalence, called mirror symmetry, has provided beautiful insights into mathematics since Candelas, Ossa, Green, and Park employed it to make a collection of bold predictions in symplectic geometry. By leveraging our knowledge of classical algebraic geometry and applying mirror symmetry principles, previously unattainable questions in symplectic geometry were now in reach. My research focuses on applying this mirror equivalence in the other direction. In the last decade, our understanding of the mirror dictionary has become robust enough that we can finally pass our intuition from symplectic geometry through the mirror to provide us with new tools in algebraic geometry. As these methods come from different areas of mathematics, they are a fresh perspective on a classical area of study. My goal is to attack problems related to enumerative geometry (counting of solutions to equations) and resolutions (describing shapes as solutions to equations) in algebraic geometry via their symplectic analogs.
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