Canada Research Chair in Equivariant Symplectic and Algebraic Geometry
Tier 2 - 2013-12-01
Using symplectic and algebraic geometry techniques to help visualize and analyze shapes beyond the second dimension.
This work will help connect several complex research areas, potentially leading to breakthroughs in various areas of applied mathematics such as robotics, kinematics, computational geometry, optimization theory, and biostatistics.
Visualizing the Shape of Space in Higher Dimensions
From ancient civilizations to modern times, human minds have consistently pondered the question: What is the shape of the universe in which we live?
An understanding of the physical universe requires us to be able to “see” shapes in higher dimension, but most of our day-to-day experiences are limited to two-dimensional representations of three-dimensional objects.
Mathematician Megumi Harada’s research helps people get around this problem. Her specialty is equivariant symplectic and algebraic geometry, a branch of geometry involving the study of symmetries of spaces with symplectic and/or algebraic structures.
She uses mathematical techniques in one- and two-dimensional situations, then generalizes those techniques, making it possible to analyze, in a concrete way, “shapes” in higher-dimensional spaces. Some examples of these techniques include gluing, direct products, fiber products, connect sum, and stereographic projection.
Symplectic geometry was originally developed to provide a mathematical framework for classical mechanics. In the 20th century, it developed into a mathematical theory, with applications to various aspects of physics, such as string theory and conformal field theory.
Algebraic geometry is the study of systems of polynomial equations and the geometry of their solutions. Algebraic geometry has many applications, including quantum computing, cryptography, and image and signal processing.
Harada works at the interface of symplectic geometry, algebraic geometry, geometric representation theory and algebraic combinatorics, and her research has potential applications outside of pure mathematics.
A particular focus of Harada’s current research is the “hot” new theory of Newton-Okounkov bodies. This theory builds upon the convex geometry of polytopes, which has applications in optimization theory. Harada intends to further develop this theory to further connect these research areas and potentially lead to new breakthroughs in applied mathematics.