Sam Payne

Math 648: Tropical Scheme Theory

These are lecture notes for an ongoing seminar course led by Kalina Mincheva and Sam Payne at Yale University in Fall 2017. Seminar participants include Louis Brown, Daniel Corey, Netanel Friedenberg, David Jensen, Max Kutler, Shiyue Li, and Daping Weng.

We look forward to special guest lectures by Matt Baker, Noah Giansiracusa, and Felipe Rincón.

Notes for each lecture were taken by Friedenberg, and revised by Mincheva, Payne, and the speaker, with input from all participants.

• Lecture 1: Idempotent semi rings (Sam Payne)
• Lecture 2: Commutative algebra over idempotent semi rings (Kalina Mincheva)
• Lecture 3: Matroid theory (Max Kutler)
• Lecture 4: Matroids over partial hyperstructures (Matt Baker)
• Lecture 5: Commutative algebra over idempotent semi rings II (Kalina Mincheva)
• Lecture 6: Bend relations and affine tropical schemes (David Jensen)
• Lecture 7: Tropical subschemes of toric varieties (Kalina Mincheva)
• Lecture 8: Universal tropicalization (Noah Giansiracusa)
• Lecture 9: Valuated matroids and tropical cycles (Kalina Mincheva)
• Lecture 10: Tropical ideals (Felipe Rincón)
• Lecture 11: Tropical Nullstellensatz I (Sam Payne)
• Lecture 12: Tropical Nullstellensatz II (Kalina Mincheva)
• Lecture 13: The semiring of finitely generated ideals (Nati Friedenberg)