Martín Sombra

Universitat de Barcelona (UB)

Experimental Sciences & Mathematics

Born in 1970 in Ezpeleta (Argentina), Martín Sombra studied Mathematics as an undergraduate at the University of La Plata. He did his PhD thesis on Computer Algebra at the University of Buenos Aires. He then did postdoctoral stays at the MSRI at Berkeley, the IAS at Princeton, and the IMJ at Paris. He became Maître de Conférences at the University of Lyon 1, then spent four years as a "Ramón y Cajal" Researcher at the University of Barcelona, and became afterwards Full Professor at the University of Bordeaux 1. He finally moved back to Barcelona, joining ICREA in 2009. He works on problems at the interface of Algebraic Geometry, Number Theory and Complexity Theory. He currently collaborates with research groups in Barcelona, Paris, Caen, Bordeaux and Buenos Aires.

Research interests

Polynomials appear in a wide variety of contexts in Mathematics, Engineering and Computer Science. Polynomials in those situations are not random but come up with a certain structure which is important to exploit. I am interested in systems of structured polynomial equations and particularly in questions like: how many solutions does a given system have? How complicated those solutions can be? Can we predict where they will accumulate? Can we efficiently solve systems of polynomial equations? These problems have conduced me to study combinatorial objects like polytopes and fans, geometrical objects like curves and surfaces, and arithmetic objects like height of points and Diophantine equations. This gives a rich interplay between Complexity Theory, Combinatorics, Algebraic Geometry and Number Theory, leading to interesting results and stimulating research directions.

Selected publications

– Burgos Gil JI, Moriwaki A, Philippon P & Sombra M 2016, ‘Arithmetic positivity on toric varieties’, Journal of Algebraic Geometry, vol. 25, pp. 201-272

– Burgos Gil JI, Philippon P & Sombra M 2016, ‘Height of varieties over finitely generated fields’, Kyoto Journal of Mathematics, vol. 56, pp. 13-32

– Basu S & Sombra M 2016, ‘Polynomial partitioning on varieties of codimension two and point-hypersurface incidences in four dimensions’, Discrete & Computational Geometry, vol. 55, pp 158-184.

Selected research activities

Invited talks (selected)

– Mini-course at the CIMPA-ICTP research school on toric methods in geometry, arithmetic and dynamics at U. Católica de Chile, Santiago de Chile, 11 – 22 January

– Meeting on Okounkov bodies at Télécom Paris, 16 – 18 March

– Conference on geometric and combinatorial methods in number theory at Iasi, Romania, 27 June – 1 July

– Conference on Arakelov geometry at U. Regensburg, Germany, 5 – 9 September

– Colloquium of the Department of Mathematics of the U. La Plata, Argentina, 15 December


– CIMPA-ICTP research school on toric methods in geometry, arithmetics and dynamics at U. Católica de Chile, Santiago de Chile, 11 – 22 January

– IMUB Colloquium

– Algebra and Geometry Meeting, UB, 30 November – 2 December

Outreach activity

– “El que diuen els nombres”. Talk for high-school students from three schools at El Prat de Llobregat, on aspects of applied probability and the nature of the work of a researcher in Mathematics

Supervision of PhD thesis

– Marta Narváez-Clauss, “Quantitative equidistribution of Galois orbits of points of small height on the algebraic torus”, UB, 21 June (joint with C. D’Andrea)