Professor Amir Ali Ahmadi will talk about; 'Disjunctive Sum of Squares'

We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach, where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares approach certifies nonnegativity using multiple algebraic identities. Our main result is a disjunctive Positivstellensatz showing that the degree of each algebraic identity can be kept as low as the degree of the polynomial whose nonnegativity is in question. Based on this result, we construct a semidefinite programming–based converging hierarchy of lower bounds for the problem of minimizing a polynomial over a compact basic semialgebraic set, in which the size of the largest semidefinite constraint remains fixed throughout the hierarchy. We further prove a second disjunctive Positivstellensatz, which leads to an optimization-free hierarchy for polynomial optimization. We specialize this result to the problem of proving copositivity of matrices. Finally, we describe how the disjunctive sum of squares approach can be combined with a branch-and-bound algorithm, and we present numerical experiments on polynomial, copositive, and combinatorial optimization problems. The talk is self-contained and assumes no prior background in sum of squares optimization.

TBA

I will present a generalisation of Mirzakhani’s recursion for the volumes of moduli spaces of bordered Klein surfaces, including non-orientable surfaces. On these moduli spaces, the top form introduced by Norbury diverges as the lengths of one-sided geodesics approach zero. However, integrating this form over Gendulphe’s regularised moduli space—where the systole of one-sided geodesics is bounded below by epsilon—yields a finite volume. Using Norbury’s extension of the Mirzakhani–McShane identities to the non-orientable setting, we derive an explicit formula for the volume of the moduli space of one-bordered Klein bottles, as well as a recursion for arbitrary topologies that fully captures the dependence on the geometric regularisation parameter epsilon. I will conclude with remarks on the relation to refined topological recursion, which leads us to a refinement of the Witten–Kontsevich recursion and of the Harer–Zagier formula for the orbifold Euler characteristic of the moduli space of curves of genus g with n marked points. Based on joint work with P. Gregori and K. Osuga; the final part reflects ongoing work with N. Chidambaram, A. Giacchetto, and K. Osuga.