Research School in Real Algebraic Geometry II

Abstract:
I will focus in these lectures on the enumeration of rational curves in geometrically rational algebraic surfaces, mainly from the real perspective. The main goal will be to describe how Welschinger invariants of different real forms on the same rational complex surface are related. These are the so-called real Abramovich-Bertram Formula. If time permits, I will explain how this shed some light on the recently defined quadratic Gromov-Witten invariants, which generalize Gromov-Witten and Weslchinger invariants to curve counting over other fields than $\mathbb C$ or $\mathbb R$. 

A tentative contents for the 4 lectures is as follows:

1. enumeration of complex and real rational curves in the projective plane (ie Gromov-Witten and Welschinger invariants or $P^2$)

2. generalisation to geometrically rational surfaces (including real-rational surfaces, but not only)

3. Weslchinger invariants and surgeries of real geometrically rational surfaces

4. generalization to quadratic Gromov-Witten invariants

Abstract: This lecture series presents some topics in the overlap of combinatorics and real algebraic geometry. 

The first focus will be on the topology of real algebraic hypersurfaces and the construction method of Viro’s combinatorial patchworking. We will see that a tropical formulation of Viro’s patchworking leads to a better understanding of the construction and its limitations. 

The second part of the lectures will generalize patchworking to higher codimensions and to objects outside of algebraic geometr, such as oriented matroids. These are matroids with extra structure inspired by real vector spaces or directed graphs. The patchworking construction applied here provides a strong link between oriented matroids and real toric varieties. 

Abstract: 

In this workshop I’d like to introduce the group of birational maps of the real projective plane and explore a few properties with a mix of real algebraic geometry and group theory. If time permits, I will also talk about the mysterious group of biregulous transformations, which are the birational transformations of the real projective plane that extend continuously at every real point. 

Abstract:
In the 1980s Frederick J. Almgren and William Browder started a program aimed at studying the regularity of minimal surfaces with tools and techniques from differential topology and homotopy theory. This led them to announce the solution to basic questions in geometric measure theory and, in particular, in the Federer-Fleming theory of integral currents; however, the project was never finished and no proof of the announced results ever appeared. In this seminar I will describe a joint study with William Browder and Camillo De Lellis in which we complete the program started with Almgren, providing a full answer to the question of how closely one can approximate an integral current representing an integral homology class by a smooth submanifold.

Abstract: 

The Kadomtsev–Petviashvili (KP) equation is a central object in the theory of integrable systems, whose solutions reveal deep connections with algebraic geometry and combinatorics. KP solutions can be constructed from algebraic curves or from Grassmannians points.

In this talk, I will discuss how these constructions behave under degenerations. We focus on banana curves—reducible rational nodal curves that arise as degenerations of hyperelliptic curves. I will describe the geometry and combinatorics of the tropical theta divisor, showing how it canonically encodes the matroid and Grassmannian structures underlying the associated KP multi-soliton solutions.

This is joint work with Simonetta Abenda, Türku Özlum Çelik, and Yelena Mandelshtam. Our framework naturally specializes to real and totally positive settings.

Abstract:

The BKK Theorem states that the number of complex roots of a complex polynomial system can be counted with the help of the volume of the Newton polytope. But what if we want to count real roots of real systems? There is no “generic” number of solution in that case. One can then adopt a probabilistic approach and count the “average” number of solutions of “random” systems. I will show, in the Gaussian case, how this leads to a result similar to the classical BKK theorem but where the volume of the Newton polytope is  the Riemannian volume for a certain (explicit) metric. I will then explain the conscequences in terms of monotonicity of the number of solution and how this invalidates a local version of a conjecture by Bürgisser.

Abstract: The Bourgain slicing conjecture (now a theorem) has, over the last decades, motivated extensive research on slices of convex bodies and their volumes. This inspires a joint work with Jared Miller and Mauricio Velasco, where we aim to solve or approximate similar optimization problems for general starshaped or convex bodies. We introduce and study the class of polystar bodies, namely starshaped sets whose gauge or radial functions are polynomial, which we use as approximators. Beyond establishing theoretical results based on spherical harmonic decompositions and convolution operators, we also develop practical tools for the effective computation of polystar approximations, achieving asymptotically optimal approximation rates.