Università degli Studi Roma Tor Vergata
Dipartimento di Matematica

Date: 24 June 2026

Schedule: h: 16:00 Rome Time

Where: Conference Room 1201 "R. Dal Passo"

Speaker: Francesca La Piana - University of Saarland (Oslo)

Title: " Point and line arrangements: moduli spaces and operator actions "

Abstract:
Quantum magic squares are non-commutative analogues of doubly stochastic matrices, where the entries are positive elements of a C*-algebra summing to the identity along each row and column. In the classical setting, the Birkhoff–von Neumann theorem characterizes bistochastic matrices as convex combinations of permutation matrices. In 2020, De Las Cuevas, Drescher and Netzer showed that the quantum analogue fails already for n=3. Motivated by questions on quantum symmetries of graphs, we introduce a graph-based variant of quantum magic squares and study the corresponding matrix convex sets. We show that for every k-regular graph Γ , the set M^Γ forms a free spectrahedron, given explicitly by a monic linear pencil. This provides a geometric framework to analyse Arveson extreme points and matrix convex hulls. For the cycle C_4 we also construct an explicit counterexample showing that the Birkhoff–von Neumann analogy fails also in the graph setting.

Note: This talk is part of the activity of the M.U.R. Department of Excellence Project MatMod@TOV 2023-2027 (CUP E83C23000330006)