NCMW - Real Algebraic Geometry (2026)

Venue: IIT, Bombay

Dates: 23 Nov 2026 to 28 Nov 2026

The workshop is a focused introduction to real algebraic geometry. It starts from real algebra (ordered and real closed fields, the real spectrum), which is the natural entry point for students trained in commutative algebra, moves on to the geometry and topology of semi-algebraic sets,and then to the central theme of positivity: Hilbert’s 17th problem, Positivstellensätze and their effective versions, sums of squares, and their matrix counterparts. The last strand shows how symmetry and convexity interact with these questions, and leads to hyperbolic polynomials and spectrahedra, where current research begins. Compared with the first version of this plan, the number of strands has been reduced from six to three, so that each strand receives at least six hours of lectures and is developed with proofs and worked examples rather than as an overview. An opening lecture assuming only undergraduate algebra establishes the vocabulary and the guiding questions of the week. Research-level topics that were part of the first plan (random and metric real algebraic geometry, advanced convex algebraic geometry) are deferred to a proposed advanced follow-up workshop (see Organizational notes).