MATH 8102: Real Algebraic Geometry
Fall 2026, University of Missouri
Office and Office hours:
MSB-218. Open door policy. You can come by with questions whenever I am there, or just to say hello.
Appointments made by email are also welcome. Contact: tduff AT missouri DOT edu
Assessment:
4 homework assignments
1 week of scribed lectures which will be posted on this webpage.
Each enrolled student must scribe one full week of lectures using this LaTeX template.
Lectures: MWF 12:00 -- 12:50 PM in MSB 111
Topics:
The beginning of the course will focus on the first two topics. The remaining topics will be considered as time permits. I encourage students to let me know which topics sound interesting to them.
Real-closed fields, real root counting, and Tarski's theorem.
Nonnegativity and infeasibility certificates: Real Nullstellensatz and Theorems of Krivine, Stengle, Putinar, Schmudgen, Polya, and Handelman.
Polyhedra and spectrahedra, conic and polynomial optimization.
Semialgebraic geometry: dimension, smoothness, and connectivity.
Algebraic aspects: Pusieux series, enumerative geometry, real spectrum.
References:
Our primary references will be the two recent textbooks on real algebraic geometry by Scheiderer and Theobald. Additional resources will be posted here at a later date.
Weekly Schedule (tentative)
Week 1: Overview, Polyhedra, and Polya's Theorem (Theobald Sections 2.1 and 6.5)
Week 2: Theorems of Handelman and Putinar (Theobald Section 6.6)