On the dual graph of Cohen-Macaulay algebras

Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result: 1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinski's theorem for simplicial polytopes.) 2) If X is a canonically embedded arrangement of lines no three of which meet in the same point, then the diameter of the graph G(X) is not larger than the codimension of X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.)Comment: Minor changes throughout, Remark 4.1 expanded, to appear in IMR

Fine gradings on the simple Lie algebras of type EE

Some fine gradings on the exceptional Lie algebras $\mathfrak{e}_6$, $\mathfrak{e}_7$ and $\mathfrak{e}_8$ are described. This list tries to be exhaustive.Comment: Note di Matematica, 201

Singular Radon transforms and maximal functions under convexity assumptions

We prove variable coefficient versions of L^p boundedness results on Hilbert transforms and maximal functions along convex curves in the plane.Comment: 19 pages, to appear in Revista Matematica Iberoamerican

Higgs bundles and local systems on Riemann surfaces

Lecture notes from the Third International School on Geometry and Physics at the Centre de Recerca Matematica in Barcelona, March 26--30, 2012.Comment: Final version. To appear in the collection CRM Advanced Courses in Mathematic

Io e la matematica. Un’indagine sul rapporto dei ragazzi con la matematica

Si presentano i risultati di un’indagine - svolta tramite questionario - sulla visione della matematica e sull’impatto del progetto “La matematica dei ragazzi: scambi di esperienze tra coetanei” tra gli allievi di scuola primaria e secondaria che vi hanno partecipato nel 2010. L’analisi dei questionari raccolti (in totale 167) può dare indicazioni utili per la progettazione di attività didattiche laboratoriali di matematica e di matematica integrata con le scienze