NCMW - Real Algebraic Geometry (2026)
Speakers and Syllabus
| Speaker and affiliation | No. of Lectures | Detailed Syllabus |
| Cordian Riener CR UiT The Arctic University of Norway, Tromsø |
4 | A first tour of real algebraic geometry (opening lecture) and Symmetry, convexity, and hyperbolicity (3 lectures). Description: The opening lecture assumes only undergraduate algebra: real versus complex solution sets; nonnegative polynomials that are not sums of squares; Descartes’ rule of signs, Sturm sequences, and Hermite’s quadratic form; a map of the week. The main series: actions of finite groups, in particular symmetric groups, on polynomial rings and real varieties; degree and half-degree principles for symmetric polynomials; topology of symmetric semi-algebraic sets; sums-of-squares cones, moment cones, spectrahedra, and semidefinite relaxations; exploiting group actions to reduce semidefinite programs; hyperbolic polynomials, hyperbolicity cones, and determinantal representations, with the elementary symmetric polynomials as running example. |
| Manoj Keshari MK IIT Bombay |
3 | Real algebra: ordered fields, real closed fields, and the real spectrum (3 lectures). Description: Orderings and positive cones of fields; formally real fields and ArtinSchreier theory; real closed fields, their characterizations and the real closure; the real spectrum of a commutative ring; real radical ideals and the real Nullstellensatz; the Artin–Lang homomorphism theorem; Hilbert’s 17th problem and Artin’s theorem as an application. The series develops real algebraic geometry from the commutative algebra side, which is the background most participants will bring. |
|
Saugata Basu |
3 | Semi-algebraic geometry and quantitative topology (3 lectures). Description: Semi-algebraic sets and maps; the first-order language of ordered fields, quanti fier elimination, the Tarski–Seidenberg theorem and the Tarski transfer principle; cylindrical and cell decomposition, triangulation, semi-algebraic connected components, curve selection, and local conic structure; Betti numbers as measures of topological complexity, the Thom–Milnor bounds for real varieties, single-exponential bounds for semi-algebraic sets defined by s polynomials of degree at most d in k variables, bounds for projections and sign conditions, and equivariant extensions in the presence of group actions. |
|
Marie-Françoise Roy |
3 | Positivstellensätze: certificates, effectivity, and algorithms (3 lectures). Description: Preorderings and quadratic modules; the Krivine–Stengle Positivstellensatz and the real Nullstellensatz as certificates for emptiness,infeasibility, and positivity; Schmüdgen’s Positivstellensatz and the Archimedean condition; degree and size bounds in Positivstellensatz certificates and their dependence on the number of variables, constraints, and degrees; effective real Nullstellensatz; algorithms over real closed fields: sign conditions, real root isolation, cylindrical algebraic decomposition, critical point methods, and complexity-theoretic consequences. |
|
Apoorva Khare |
3 | Positivity in polynomials and matrices (3 lectures). Description: Putinar’s Positivstellensatz, Archimedean quadratic modules, and the moment–SOS hierarchy in polynomial optimization; positive semidefinite matrices, the Schur product theorem, and entrywise or functional preservers of positivity; total positivity and determi nantal inequalities; real-rootedness, stability, symmetric functions, and positivity-preserving transformations, with connections to hyperbolic polynomials. |
Tutor
| Tutor | Abbv. | Affiliation | Courses associated to |
| Niladri Sekhar Patra | T1 | IIT Bombay | MK + SB (foundations) |
| Charu Goel | T2 | IIM Mumbai | MFR + AK (positivity) |
| Daniel Bosch | T3 | UiT, Norway | CR 0 + CR 1–3 (opening lecture; symmetry, convexity, hyperbolicity) |
| Hugues Verdure | T4 | UiT, Norway | SB + CR; coordination of the open-problem session |
Time Table
| Date | 9.00–10.30 Lecture |
10.30 Tea |
11.00–12.30 Lecture |
12.30 Lunch |
14.30–16.00 Lecture |
16.00 Tea |
16.15–17.15 Tutorial / Session |
| Mon 23 Nov | CR 0 | MK 1 | MK 2 | Tutorial 1 | |||
| Tue 24 Nov | MK 3 | SB 1 | SB 2 | Tutorial 2 | |||
| Wed 25 Nov | SB 3 | MFR 1 | MFR 2 | Tutorial 3 | |||
| Thu 26 Nov | MFR 3 | AK 1 | CR 1 | Tutorial 4 / Excursion | |||
| Fri 27 Nov | CR 2 | CR 3 | AK 2 | Tutorial 5 / Posters | |||
| Sat 28 Nov | AK 3 | Open problems | Concluding discussion |