Meetings: 3106 MSB from 8:30 to 10:00 on Tuesdays.
Credit: The variable unit request form can be found here https://www.math.ucdavis.edu/grad/gr ad-student-handbook and should be filled out if you want course credit.

References:
[CLS] Toric Varieties by D Cox, J Little and H Schenck
[F] Introduction to Toric Varieties by W Fulton
[C] Recent developments in toric geometry by D Cox
[R1] Decompositions of toric morphisms by M Reid
[R2]A young persons guide to toric geometry by M Reid
[NS] Toric symplectic geometry and full spark frames by T Needham and C Shonkwiler
[Z] Kahler-Ricci flow on a toric manifold with positive first Chern class by X Zhu
[SY] On symplectic packing problems in higher dimensions by K Siegel and X Yao
[A] On the sphere and cylinder by Archimedes



Complex toric varieties are simply varieties which contain a complex torus as a dense subvariety but many also arise from projective space as a git quotient or symplectic reduction. Their key property is a strong connection to polyhedra. This connection is bijective giving a discrete but large indexing of toric varieties. Singularities and projectivity of the varieties are clear from the polyhedra. Smooth projective complex varieties inherit much from projective space including a Kahler structure which includes complex, symplectic and Riemannian structures along with a measure and two filtrations (a Hodge structure) on the cohomology. These structures as well as the possible divisors in, line bundles on and blow-ups of toric varieties have associated polyhedral structures which are mostly useful for studying structures on the varieties but Stanley [F] used to extract facts about polyhedra. Finally there is an actual map from the projective variety to the polyhedron which provides Morse functions and geometric flows as well as a remarkably close relationship between measures on the variety and polytope with connections to symplectic squeezing and packing and include Archemedes' observation that the volumes of slices of spherical and cylindrical shells are equal.

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Seminar Outline: This is a rough outline and (apart from exam dates) subject to change.
DaySpeakerTopic
October 15Timothy PaczynskiAffine TV from [CLS] Ch1.1Related Exercises from [CLS] 1.1: 1,4,5,6,10,12,15 and a computation by Qin.
October 22Timothy PaczynskiAffine TV from [CLS] Ch1.2
October 29Evan Ortiz Affine TV from [CLS] Ch1.3 Related Exercises from [CLS] 1.3: 3, 4, 6, 7 and 12.
November 5NO MEETING
November 12Evan Ortiz Projective TV from [CLS] Ch 2.
November 19NO MEETING
November 26Soyeon Kim
December 3