Columbia Algebraic Geometry Seminar

11 am: Jørgen Ellegaard Andersen (Centre for the Quantum Geometry of Moduli Spaces, Aarhus University)

The geometric construction of the Reshetikhin-Turaev mapping class group representations and asymptotics

We will review the geometric construction of the mapping class group representations which are part of the Reshetikhin-Turaev TQFT. Following this we will discuss asymptotics in the moduli space of curves and asymptotics in the quantum level of the theory. Finally, some applications of these limits will be covered.

2 pm: Seán Keel (University of Texas)

Mirror symmetry made easy

I will describe my conjecture (and theorem in dimension two), joint with Hacking and Gross, which gives the mirror family to an affine CY manifold (with maximal boundary) as the spectrum of an explicit ring: A vector space with a canonical basis, a natural Mori-theoretic generalisation of Thurston's boundary to Teichmüller space, and a multiplication rule (the coefficients for expanding the product of two basis elements as a linear combination of basis elements) in terms of counts of rational curves. The construction is very elementary -- if you know what is meant by the order of zero or pole of a rational function along a divisor, you know enough to understand the main idea -- yet has surprising implications, for example that any such variety has "theta functions," a canonical basis of the vector space of global functions. This gives canonical functions on some of the most basic objects in algebraic geometry, e.g. all cluster varieties, the SL(n) character variety of a punctured Riemann surface, or the complement to an anti-canonical cycle of rational curves on a smooth projective surface, like Cayley's affine cubic surface.

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