Courses

June 2025

In the week 23-27 June, I will teach a 10-hours course at SISSA, Triste. The title is A course on Nonnegative Polynomials, and we will explore the theory of nonnegative polynomials from both classical and modern perspectives, highlighting connections with convex and algebraic geometry, semidefinite programming, and current research directions.

Where: SISSA, Via Bonomea, Room 136

When: 2pm–4pm everyday

1. Monday: The convex cone of nonnegative polynomials
   • Motivation: Hilbert’s 17th problem and optimization
   • Basics on convex geometry
   • Nonnegative polynomials on the real projective line
2. Tuesday: The dual moment cone
   • Conic duality and Haviland’s theorem
   • Richter’s theorem on Dirac measures
   • Bombieri-Weil inner product, Carathéodory numbers and tensor decomposition
3. Wednesday: Nonnegativity on real algebraic curves
   • Faces of the nonnegative cone
   • Extreme rays of the nonnegative cone and the real Jacobian
   • Carathéodory numbers for elliptic normal curves
4. Thursday: Volumes of nonnegative polynomials and sums of squares
   • The gauge of the nonnegative cone
   • The gauge of the dual to sums of squares
   • Open questions
5. Friday: Polynomial optimization and sums of squares representations
   • Schmüdgen and Putinar's theorem
   • The sums of squares hierarchy
   • Convergence and effective aspects

2024-2025

Between mid-October 2024 and mid-February 2025, I am organizing a course in Real Algebraic Geometry. See also this webpage.

Where: University of Leipzig, Seminargebäude 214

When: Every Friday 15:00-17:00, starting October 2024

Exercise sessions: they will take place on Tuesdays, 11:00-13:00, at the University of Leipzig, room P701, in the following dates: 05.11.2024, (next dates to be discussed)

A tentative list of topics includes:

1. Basics on Real Closed Fields
   • Intermediate Value Theorem
   • Real Puiseux Series
   • Real Root Counting
2. Semialgebraic Geometry
   • Tarski-Seidenberg Theorem
   • Curve Selection Lemma
   • Boundedness and Closedness
   • Triangulations and Stratifications
   • Sard and Hardt Theorems
3. Topological Properties
   • Real Algebraic Curves and Harnack Theorem
   • Thom-Milnor Bound
4. Algebraic Models of Manifolds
   • Basics of Cobordism
   • Nash-Tognoli Theorem

Bibliography

• Bochnak, Coste, Roy, Real Algebraic Geometry
• Basu, Pollak, Roy, Algorithms in Real Algebraic Geometry
• Scheiderer, A Course in Real Algebraic Geometry
• Benedetti, Risler, Real Algebraic and Semi-algebraic Sets
• Lerario, Metric Algebraic Geometry
• Mangolte, Real Algebraic Varieties

2023

Between the 19th and 23rd of June 2023 I was lecturer for the compact course Semidefinite Programming for Algebra, Combinatorics and Geometry, together with Sebastian Debus, at the University of Magdeburg.