Michele Grassi | Università di Pisa (original) (raw)

Papers by Michele Grassi

Coastal Structures 2011, 2012

Journal of Geometry and Physics, Feb 1, 2009

We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of ... more We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of rank 3. We describe in detail the associated Lie super-algebra L 3,C of global sections. We show that L 3,C is a product of sl(4, C) with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of so(3, R). We give an explicit description of a geometrically natural real form of L 3,C. This real form is made up of so(3, R)-invariant operators which preserve the Poincaré pairing on the bundle of forms.

arXiv (Cornell University), Feb 13, 2008

We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whos... more We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s + 1, s + 1), su(s+1, s+1) and sl(2s+2, R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-Kähler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves.

Journal of Geometry and Physics, 2009

We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of ... more We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of rank 3. We describe in detail the associated Lie super-algebra L 3,C of global sections. We show that L 3,C is a product of sl(4, C) with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of so(3, R). We give an explicit description of a geometrically natural real form of L 3,C. This real form is made up of so(3, R)-invariant operators which preserve the Poincaré pairing on the bundle of forms.

Abstract. Given an orientable weakly self-dual manifold X of rank two, we build a geometric reali... more Abstract. Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6, C) as a naturally defined algebra LC of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of LC. This construction gives a bundle on X which is related to the search for a natural Gauge theory on X. We consider this paper as a first step in the study of a rich and interesting algebraic structure. 1.

We determine the structure of the Lie superalgebra generated by a set of natural operators of an ... more We determine the structure of the Lie superalgebra generated by a set of natural operators of an orientable WSD manifold of rank three. This Lie superalgebra is formed by global sections of a natural Lie superalgebra bundle, and turns out to be a product of mathbfsl(4,C)\mathbf{sl}(4,\C)mathbfsl(4,C) with the full special linear superalgebras of some graded vector spaces isotypical with respect

Annales de la faculté des sciences de Toulouse Mathématiques, 2010

L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques »... more L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

Rendiconti del Seminario Matematico della Università di Padova, 2009

Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of ... more Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6; C) as a naturally defined algebra v C of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of v C. This construction gives a bundle on X which is related to the search for a natural Gauge theory on X. We consider this paper as a first step in the study of a rich and interesting algebraic structure.

Journal of Algebra, 1996

I first define Koszul modules, which are a generalization to arbitrary rank of complete intersect... more I first define Koszul modules, which are a generalization to arbitrary rank of complete intersections. After a study of some of their properties, it is proved that Gorenstein algebras of codimension one or two over a local or graded CM ring are Koszul modules, thus generalizing a well known statement for rank one modules. The general techniques used to describe Koszul modules are then used to obtain a structure theorem for Gorenstein algebras in codimension one and two, over a local or graded CM ring.

Duke Mathematical Journal, 2000

Notation. The notation that we use is fairly standard. For concepts on differential geometry, see... more Notation. The notation that we use is fairly standard. For concepts on differential geometry, see, for example, [BCGGG]. On commutative algebra, see [G] or [BH]. On homological algebra, see, for example, [HS]. Basic notions of Hopf algebra theory can LOCAL VANISHING OF CHARACTERISTIC COHOMOLOGY 309 be found in any reference text, for example, [Mo] or [Sw]. We used the terms lemma and theorem for our results (unless otherwise noted), while we used proposition and remark for results that are already known (for which sometimes we couldn't find a reference) or that are straightforward applications of well-known facts. As there are many module and algebra structures involved, to avoid confusion we indicate the category of the operation. This is especially true of the tensor product. Often we use the notation M ⊗ A N or M ⊗ f N when the tensor product is done with respect to the ring A or with respect to the B-module structure induced by the ring morphism f : B → A, if M and N are A-modules. We also use this notation for the multiplication by ideals, as in I • A M or I • f M. We use a, b, c A to denote the submodule generated by the elements a, b, and c over the ring A. We explain the less standard notation and terminology in the first section.

[

Arxiv preprint arXiv:0802.1822, 2008

Abstract. We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manif... more Abstract. We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s + 1, s + 1), su(s+1, s+1) and sl(2s+2, R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have ...

Arxiv preprint math/0202016, 2002

Starting with the paper [SYZ] by Strominger, Yau and Zaslow, the search for a geometric counterpa... more Starting with the paper [SYZ] by Strominger, Yau and Zaslow, the search for a geometric counterpart to mirror symmetry has beed directed mainly at special la-grangian fibrations of Calabi-Yau manifolds. In recent years however this approach has come under some criticism, ...

Arxiv preprint math/0006154, 2000

In this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE&... more In this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE's. This notion provides a generaliza-tion of symplectic manifolds which is very well suited for the geomet-ric study of PDE's with values in a smooth manifold. Some of the standard ...

Asian Journal of Mathematics, 2005

For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and ... more For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and use it to interpolate between the anticanonical family in CP m+1 and its mirror dual. The main tool is the notion of self-dual manifold.

Asian Journal of Mathematics, 2005

For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and ... more For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and use it to interpolate between the anticanonical family in CP m+1 and its mirror dual. The main tool is the notion of self-dual manifold.

A modular agriculture system for use underwater. The system comprises aplurality of node elements... more A modular agriculture system for use underwater. The system comprises aplurality of node elements (101) each including an agricultural unit and a plurality of connection elements (201) for interconnecting at least some of said nodes ot form a mesh-like structure with nodes at the mesh junctions.

There is provided an apparatus for absorbing or converting energy from a moving body of water. Th... more There is provided an apparatus for absorbing or converting energy from a moving body of water. The apparatus com - prises an energy capture element (3) which, in use, moves in response to movement of the body of water in which the energy capture element (3) si placed, and an elongate guide element (1) defining a guide path along which the energy capture element (3) can move. The energy capture element (3) si avolume. nI use, the energy capture element (3) and the guide element (1) are arranged os that the energy capture element (3) moves along the guide path ni a substantially horizontal plane in response ot differences ni water pressure along a length of the energy capture element (3) parallel ot the guide path and ni response ot movement of the body of water sur - rounding the energy capture element (3).

A wave energy converter apparatus comprising at least two members joined by a connector movable s... more A wave energy converter apparatus comprising at least two
members joined by a connector movable so as to allow dis
placement of the members relative to one another in response
to waves in water where the apparatus is located. The appa
ratus also includes an energy converter for converting motion
of the connector to electrical energy, and a device for storing and/or transporting elsewhere the energy produced. The
members, when in use and located in a body of water, are
non-floating and completely Submerged, the apparatus, as a
whole, being neutrally buoyant. A member is additionally
provided for maintaining each of the Submerged members
substantially at rest relative to the surrounding water with
which they are in contact, Such that the Submerged members
move under wave-induced action Substantially in the same
manner as a water particle would move under wave-induced
action if located in the same region as the respective Sub
merged member.

We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for aff... more We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for affine-Kähler manifolds and for elliptic curves. Their geometric properties, especially the link with special lagrangian fibrations and the existence of a transformation similar to the Fourier-Mukai functor, suggest that this approach may be able to explain mirror symmetry also in other situations.

Coastal Structures 2011, 2012

Journal of Geometry and Physics, Feb 1, 2009

We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of ... more We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of rank 3. We describe in detail the associated Lie super-algebra L 3,C of global sections. We show that L 3,C is a product of sl(4, C) with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of so(3, R). We give an explicit description of a geometrically natural real form of L 3,C. This real form is made up of so(3, R)-invariant operators which preserve the Poincaré pairing on the bundle of forms.

arXiv (Cornell University), Feb 13, 2008

We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whos... more We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s + 1, s + 1), su(s+1, s+1) and sl(2s+2, R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-Kähler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves.

Journal of Geometry and Physics, 2009

We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of ... more We prove the existence of a natural *-Lie super-algebra bundle on any orientable WSD manifold of rank 3. We describe in detail the associated Lie super-algebra L 3,C of global sections. We show that L 3,C is a product of sl(4, C) with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of so(3, R). We give an explicit description of a geometrically natural real form of L 3,C. This real form is made up of so(3, R)-invariant operators which preserve the Poincaré pairing on the bundle of forms.

Abstract. Given an orientable weakly self-dual manifold X of rank two, we build a geometric reali... more Abstract. Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6, C) as a naturally defined algebra LC of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of LC. This construction gives a bundle on X which is related to the search for a natural Gauge theory on X. We consider this paper as a first step in the study of a rich and interesting algebraic structure. 1.

We determine the structure of the Lie superalgebra generated by a set of natural operators of an ... more We determine the structure of the Lie superalgebra generated by a set of natural operators of an orientable WSD manifold of rank three. This Lie superalgebra is formed by global sections of a natural Lie superalgebra bundle, and turns out to be a product of mathbfsl(4,C)\mathbf{sl}(4,\C)mathbfsl(4,C) with the full special linear superalgebras of some graded vector spaces isotypical with respect

Annales de la faculté des sciences de Toulouse Mathématiques, 2010

L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques »... more L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

Rendiconti del Seminario Matematico della Università di Padova, 2009

Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of ... more Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6; C) as a naturally defined algebra v C of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of v C. This construction gives a bundle on X which is related to the search for a natural Gauge theory on X. We consider this paper as a first step in the study of a rich and interesting algebraic structure.

Journal of Algebra, 1996

I first define Koszul modules, which are a generalization to arbitrary rank of complete intersect... more I first define Koszul modules, which are a generalization to arbitrary rank of complete intersections. After a study of some of their properties, it is proved that Gorenstein algebras of codimension one or two over a local or graded CM ring are Koszul modules, thus generalizing a well known statement for rank one modules. The general techniques used to describe Koszul modules are then used to obtain a structure theorem for Gorenstein algebras in codimension one and two, over a local or graded CM ring.

Duke Mathematical Journal, 2000

Notation. The notation that we use is fairly standard. For concepts on differential geometry, see... more Notation. The notation that we use is fairly standard. For concepts on differential geometry, see, for example, [BCGGG]. On commutative algebra, see [G] or [BH]. On homological algebra, see, for example, [HS]. Basic notions of Hopf algebra theory can LOCAL VANISHING OF CHARACTERISTIC COHOMOLOGY 309 be found in any reference text, for example, [Mo] or [Sw]. We used the terms lemma and theorem for our results (unless otherwise noted), while we used proposition and remark for results that are already known (for which sometimes we couldn't find a reference) or that are straightforward applications of well-known facts. As there are many module and algebra structures involved, to avoid confusion we indicate the category of the operation. This is especially true of the tensor product. Often we use the notation M ⊗ A N or M ⊗ f N when the tensor product is done with respect to the ring A or with respect to the B-module structure induced by the ring morphism f : B → A, if M and N are A-modules. We also use this notation for the multiplication by ideals, as in I • A M or I • f M. We use a, b, c A to denote the submodule generated by the elements a, b, and c over the ring A. We explain the less standard notation and terminology in the first section.

[

Arxiv preprint arXiv:0802.1822, 2008

Abstract. We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manif... more Abstract. We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s + 1, s + 1), su(s+1, s+1) and sl(2s+2, R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have ...

Arxiv preprint math/0202016, 2002

Starting with the paper [SYZ] by Strominger, Yau and Zaslow, the search for a geometric counterpa... more Starting with the paper [SYZ] by Strominger, Yau and Zaslow, the search for a geometric counterpart to mirror symmetry has beed directed mainly at special la-grangian fibrations of Calabi-Yau manifolds. In recent years however this approach has come under some criticism, ...

Arxiv preprint math/0006154, 2000

In this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE&... more In this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE's. This notion provides a generaliza-tion of symplectic manifolds which is very well suited for the geomet-ric study of PDE's with values in a smooth manifold. Some of the standard ...

Asian Journal of Mathematics, 2005

For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and ... more For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and use it to interpolate between the anticanonical family in CP m+1 and its mirror dual. The main tool is the notion of self-dual manifold.

Asian Journal of Mathematics, 2005

For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and ... more For all m ≥ 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and use it to interpolate between the anticanonical family in CP m+1 and its mirror dual. The main tool is the notion of self-dual manifold.

A modular agriculture system for use underwater. The system comprises aplurality of node elements... more A modular agriculture system for use underwater. The system comprises aplurality of node elements (101) each including an agricultural unit and a plurality of connection elements (201) for interconnecting at least some of said nodes ot form a mesh-like structure with nodes at the mesh junctions.

There is provided an apparatus for absorbing or converting energy from a moving body of water. Th... more There is provided an apparatus for absorbing or converting energy from a moving body of water. The apparatus com - prises an energy capture element (3) which, in use, moves in response to movement of the body of water in which the energy capture element (3) si placed, and an elongate guide element (1) defining a guide path along which the energy capture element (3) can move. The energy capture element (3) si avolume. nI use, the energy capture element (3) and the guide element (1) are arranged os that the energy capture element (3) moves along the guide path ni a substantially horizontal plane in response ot differences ni water pressure along a length of the energy capture element (3) parallel ot the guide path and ni response ot movement of the body of water sur - rounding the energy capture element (3).

A wave energy converter apparatus comprising at least two members joined by a connector movable s... more A wave energy converter apparatus comprising at least two
members joined by a connector movable so as to allow dis
placement of the members relative to one another in response
to waves in water where the apparatus is located. The appa
ratus also includes an energy converter for converting motion
of the connector to electrical energy, and a device for storing and/or transporting elsewhere the energy produced. The
members, when in use and located in a body of water, are
non-floating and completely Submerged, the apparatus, as a
whole, being neutrally buoyant. A member is additionally
provided for maintaining each of the Submerged members
substantially at rest relative to the surrounding water with
which they are in contact, Such that the Submerged members
move under wave-induced action Substantially in the same
manner as a water particle would move under wave-induced
action if located in the same region as the respective Sub
merged member.

We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for aff... more We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for affine-Kähler manifolds and for elliptic curves. Their geometric properties, especially the link with special lagrangian fibrations and the existence of a transformation similar to the Fourier-Mukai functor, suggest that this approach may be able to explain mirror symmetry also in other situations.