Online-Workshop Computational Algebra 2021
Location & registration
The workshop took place on 21 May 2021 starting at 13:00 CEST. Note that for data protection reasons, joining required an access code. To get it, please register for the workshop by sending an email to caworkshop with subject “Workshop Computational Algebra 2021” and including your name and affiliation.
- 13:00: Words of welcome
- 13:05-13:55: Simon Brandhorst (Saarbrücken)
- 14:05-14:55: Jean Philippe Labbé (Berlin)
- 15:05-15:55: Timo de Wolff (Braunschweig)
- 16:05-16:55: Jonathan Kliem (Berlin)
Simon Brandhorst (Saarbrücken)
Equations for the K3-Lehmer map
The dynamical complexity of an automorphism of a complex surface is measured by its topological entropy. The entropy is the logarithm of a Salem number, that is, a real algebraic integer lambda greater than 1 which is conjugate to 1/lambda and all whose other conjugates lie on the unit circle. Conjecturally the smallest Salem number is Lehmer’s number lambda_10. Lehmer’s conjecture is true for entropies: log(lambda_10) is the minimum among entropies of automorphisms of complex surfaces.
In a series of papers McMullen proved the existence of such an automorphism on a complex projective K3 surface. His strategy combines ideas from integer programming with the theory of lattices, number fields and reflection groups. The final step of the proof relies on the Torelli-type Theorem for K3 surfaces which is non constructive. In this talk I present equations for this automorphism. To find it we used Kneser’s neighbor method, elliptic fibrations and their linear systems, finite non-symplectic automorphisms and p-adic lifting.
This is joint work with Noam D. Elkies.
Jonathan Kliem (Berlin)
A new face iterator for polyhedra and other lattices
We will discuss a depth-first algorithm to iterate over the faces of polyhedra (joint work with Christian Stump). It is very memory efficient and the implementation in SageMath is much faster than implementations of other algorithms. The algorithm can be applied to other situations such as simplicial complexes, finite polyhedral complexes, subdivisions of manifolds, extended tight spans and closed sets of matroids.
Jean Philippe Labbé (Berlin)
Lineup polytopes and applications in quantum physics
The set of all possible spectra of 1-reduced density operators for systems of N particles on a d-dimensional Hilbert space is a polytope called hypersimplex. If the spectrum of the original density operators is fixed, the set of spectra (ordered decreasingly) of 1-reduced density operators is also a polytope. A theoretical description of this polytope using inequalities was provided by Klyachko in the early 2000’s.
Adapting and enhancing tools from discrete geometry and combinatorics (symmetric polytopes, sweep polytopes, and the Gale order), we obtained such necessary inequalities explicitly, that are furthermore valid for arbitrarily large N and d.
These may therefore be interpreted as generalizations of Pauli’s exclusion principle for fermions. In particular, this approach leads to a new class of polytopes called lineup polytopes.
This is joint work with physicists Julia Liebert, Christian Schilling and mathematicians Eva Philippe, Federico Castillo and Arnau Padrol.
Timo de Wolff (Braunschweig)
An Introduction to Nonnegativity and Polynomial Optimization
In science and engineering, we regularly face hard, nonlinear polynomial optimization problems. Solving these problems is essentially equivalent to computing certificates of nonnegativity of multivariate, real polynomials — a key problem in real algebraic geometry. Since the 19th century, sums of squares (SOS) are a standard certificates for nonnegativity. These can be detected via semidefinite programming (SDP) in practice.
In 2014, Iliman and I introduced a new certificate of nonnegativity based on sums of nonnegative circuit polynomials (SONC), which I have developed further since then both in theory and practice joint with different coauthors. In particular, these certificates are independent of sums of squares and can be computed via convex optimization programs called relative entropy programs.
In the first part of this talk, I will give an overview about certificates of nonnegativity and its relation to polynomial optimization. In the second part, I discuss some highlights of recent work involving SONC certificates and their application.