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Riemann Surfaces
This course serves as a follow-up of Math 506. The students are assumed to know complex analysis and have some basic knowledge in algebra and topology. The courses are divided into two parts: TheExpand
Canonical bases for cluster algebras
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonicalExpand
Mirror symmetry for log Calabi-Yau surfaces I
We give a canonical synthetic construction of the mirror family to pairs (Y,D) where Y is a smooth projective surface and D is an anti-canonical cycle of rational curves. This mirror family isExpand
Compact moduli of plane curves
We construct a compactication Md of the moduli space of plane curves of degreed. We regard a plane curveC P 2 as a surface-divisor pair (P 2 ;C) and dene Md as a moduli space of pairs (X;D) where XExpand
Smoothable del Pezzo surfaces with quotient singularities
Abstract We classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in theExpand
Birational geometry of cluster algebras
We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the LaurentExpand
IRAS Faint Source Catalogue, version 2.0.
Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces
AbstractWe give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification Expand
Moduli of surfaces with an anti-canonical cycle
Abstract We prove a global Torelli theorem for pairs $(Y,D)$ where $Y$ is a smooth projective rational surface and $D\in |-K_{Y}|$ is a cycle of rational curves, as conjectured by Friedman in 1984.Expand
The homology of tropical varieties
Given a closed subvarietyX of an algebraic torusT, the associated tropical variety is a polyhedral fan in the space of 1-parameter subgroups of the torus which describes the behaviour of theExpand