Applications of symplectic geometry to algebra
Lead Research Organisation:
University of St Andrews
Department Name: Mathematics and Statistics
Abstract
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.
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.
