Armando Capasso

Short Bio. I am Armando, a Math Ph.D. student (from November 1st 2020) and Assistant Professor at Università degli Studi Roma Tre (Italy). I receveid my Bachelor Degree in Math at Università degli Studi di Napoli "Federico II" (Italy) in December 2011 under the supervision of Professor Clorinda De Vivo, my Master Science Degree at Università degli Studi di Trieste (Italy) in October 2014 under the supervision of Professor Ugo Bruzzo (SISSA, Trieste). After a sabbatical period, during which I went deep into the complex (algebraic) geometry and started my personal research on positivity conditions for Higgs bundles, I taught Math and Physics in some schools as Substituting Teacher (2017-2019) and Linear Algebra and Geometry at University as Professor on contract (2019-2020).

1-Numerically flatness for Higgs bundles over compact Kähler manifolds

Monday 14th February 2022, 11:00 -- 12:00, Room 211 Pal. C

In 1998 De Cataldo introduced new (semi)positive notions for vector bundles over compact Kähler manifolds: the t-nefness conditions. These was new also in the smooth complex projective setting.
Since these notions involves on particular order relations between the Hermitian forms on vector bundles, in 2007 Bruzzo and Graña-Otero introduced the Hitchin-Simpson connection on Higgs bundles and the relevant curvature form which is a Hermitian form of course; so they introduced the t-H-nefness conditions in this new setting. They extended all results of De Cataldo, and in particular classified the 1-H-nflat Higgs bundles.
Moreover, in 2019 Bruzzo and myself proved that 1-H-nflatness is equivalent to H-nflatness (introduced in 2007 by Bruzzo and Graña-Otero) in the smooth complex projective setting.
In this seminar I will introduce all the opportune notions in order to understand these objects and I will expose some geometric consequence of all these stuffs.

Algebraic hypersurfaces with vanishing Hessian and CW-complexes

Monday 17th February 2022, 11:00 -- 12:00, Room 211 Pal. C

In 1851 and 1859 O. Hesse conjuctured that the complex projective hypersurfaces having vanishing Hessian determinant at any points (hypersurfaces with indeterminated Hessian, for short) are cones; this is true until to dimenion 3 or in degree 2 as proved by Gordan and Noether in 1876. In the other cases, also Perazzo in 1900 and Permutti in 1957 provided examples of hypersurfaces with indeterminated Hessian which are not cone in any possible case.
Moreover, Gordan and Noether proved an algebraic criterion so that a complex projective hypersurface is either a cone or having indeterminated Hessian. In the modern algebra language, this criterion implies that some graded Artinian quotient algebras satisfying the Poincaré duality (which are all and the only Standard Graded Artinian Gorestein Algebras, SGAG Algebras for short) does not satisfy the Strong Lefschetz Property (SLP, fo short).
One tries to study these SGAG Algebras which does not satisfies the Weak Lefschetz Property (WLP, for short), condition more easy to disprove than SLP and implies by this last one. Indeed, if a SGAG Algebra has the WLP then the vector of dimenion of its graded piece (the so-called Hilbert vector) is symmetric and unimodal. But Gondim in 2017 proved that this last condition is not sufficient for the WLP.
In this seminar I explain all these (one directionial) implications and expose how to construct a CW-compex which permits to compute the Hilbert vector of a SGAG algebra.

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