Bridging Frameworks via Mirror Symmetry

点击翻译按钮获取中文摘要

Stand in one place. Ask the question "What are the possible ways you could face while standing there?'' One answer is from zero degrees to 360 degrees, but that is not a fully-satisfying answer. The most intuitive answer is you can turn around in a circle. This answer is an example of a geometric classification of possible solutions, or a moduli space. Moduli spaces are ubiquitous in geometry. From conic sections to the range of motion of a robot, one is studying moduli spaces. In algebraic geometry, we study the geometry of the solutions of polynomials and associated geometric classification problems. When one has many variables and uses higher degrees, such questions become difficult. Such shapes formed by Typically there are three ways to study varieties: looking at other objects that sit inside them, finding ways that they sit inside other objects, and finding invariants that help classify them.In the last 25 years, string theory has giving intuitive frameworks for studying certain classical algebro-geometric objects, Calabi-Yau shapes. In string theory, Calabi-Yau shapes are added to the space-time continuum in order to get physical models for the universe. In mathematics, this led to a geometric duality called mirror symmetry which focuses on the duality between Type IIA and IIB string theory. This rich framework allows many connections between mathematical fields, typically symplectic geometry and algebraic geometry.Many of the connections made have to do with enumerative geometry, studying how many curves of a certain type sit inside higher dimensional objects. Mirror symmetry turned this problem in symplectic geometry into an algebro-geometric problem, making it easier to compute the answer. Some of the connections sit in number theory. Varieties have number-theoretic analogues where one can study them over a finite field, providing geometric analogues to the Riemann zeta function. The proposed research plan focuses on finding bridges amongst fields motivated by mirror symmetry. The proposal involves the following projects:1.) Providing a method to compute the FJRW-invariants in symplectic geometry by linking the invariants to an algebro-geometric setting then using tropical geometry. These invariants describe how many curves of a certain type sit in a generalized version of a Calabi-Yau shape, called a Landau-Ginzburg model.2.) Studying the number theoretic properties of Calabi-Yau shapes when viewed under mirror symmetry, harnessing properties of the zeta function associated to these shapes.3.) Classify a certain class of higher-dimensional analogues to polygons by using their correspondence to algebraic objects by using geometric quotients, consequently giving a classification of certain types of Calabi-Yau shapes.4.) Codify what mirror symmetry means for another type of string theory, heterotic mirror symmetry.The work presented here will provide more links amongst mathematical fields, creating a more cohesive mathematical community. Each project takes two fields and connects them in a way so that both fields can contribute to the understanding of Calabi-Yau shapes.