Title: Mirror Symmetry, Elliptic Curves, and Theta Functions
Abstract: Mirror symmetry predicts a duality between Calabi–Yau manifolds that exchanges complex geometry and symplectic geometry. Kontsevich’s mathematical formulation of this idea, known as homological mirror symmetry, proposes an equivalence between two categories of geometric objects: on one side, Lagrangians in the symplectic manifold, and on the other, coherent sheaves on the complex manifold. The Strominger–Yau–Zaslow conjecture gives a geometric picture of this correspondence in terms of torus fibrations and T-duality.
In this talk, I will discuss these ideas in the simplest and most concrete example, the elliptic curve. I will explain how line bundles and Lagrangians correspond in this setting, and how this leads to a canonical basis of sections of line bundles given by theta functions. I will then describe how the formulas for theta functions and their products can be understood from the symplectic side through Floer-theoretic triangle counts. If time permits, I will briefly indicate how the Gross–Siebert program generalizes this picture and produces canonical bases in much broader settings.
Note: The talk will start at 4:10 pm
Daniel Mallory
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