Exponential formulas for models of complex reflection groups

In this paper we find some exponential formulas for the Betti numbers of the De Concini-Procesi minimal wonderful models Y_{G(r,p,n)} associated to the complex reflection groups G(r,p,n). Our formulas are different from the ones already known in the literature: they are obtained by a new combinatorial encoding of the elements of a basis of the cohomology by means of set partitions with weights and exponents. We also point out that a similar combinatorial encoding can be used to describe the faces of the real spherical wonderful models of type A_{n-1}=G(1,1,n), B_n=G(2,1,n) and D_n=G(2,2,n). This provides exponential formulas for the f-vectors of the associated nestohedra: the Stasheff's associahedra (in this case closed formulas are well known) and the graph associahedra of type D_n.Comment: with respect to v.1: misprint corrected in Example 3.

Incidence combinatorics of resolutions

We introduce notions of combinatorial blowups, building sets, and nested sets for arbitrary meet-semilattices. This gives a common abstract framework for the incidence combinatorics occurring in the context of De Concini-Procesi models of subspace arrangements and resolutions of singularities in toric varieties. Our main theorem states that a sequence of combinatorial blowups, prescribed by a building set in linear extension compatible order, gives the face poset of the corresponding simplicial complex of nested sets. As applications we trace the incidence combinatorics through every step of the De Concini-Procesi model construction, and we introduce the notions of building sets and nested sets to the context of toric varieties. There are several other instances, such as models of stratified manifolds and certain graded algebras associated with finite lattices, where our combinatorial framework has been put to work; we present an outline in the end of this paper.Comment: 20 pages; this is a revised version of our preprint dated Nov 2000 and May 2003; to appear in Selecta Mathematica (N.S.

Chow rings of toric varieties defined by atomic lattices

We study a graded algebra D=D(L,G) defined by a finite lattice L and a subset G in L, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi. Our main result is a representation of D, for an arbitrary atomic lattice L, as the Chow ring of a smooth toric variety that we construct from L and G. We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Groebner basis of the relation ideal of D and a monomial basis of D over Z.Comment: 23 pages, 7 figures, final revision with minor changes, to appear in Invent. Mat

Impedance matching and emission properties of optical antennas in a nanophotonic circuit

An experimentally realizable prototype nanophotonic circuit consisting of a receiving and an emitting nano antenna connected by a two-wire optical transmission line is studied using finite-difference time- and frequency-domain simulations. To optimize the coupling between nanophotonic circuit elements we apply impedance matching concepts in analogy to radio frequency technology. We show that the degree of impedance matching, and in particular the impedance of the transmitting nano antenna, can be inferred from the experimentally accessible standing wave pattern on the transmission line. We demonstrate the possibility of matching the nano antenna impedance to the transmission line characteristic impedance by variations of the antenna length and width realizable by modern microfabrication techniques. The radiation efficiency of the transmitting antenna also depends on its geometry but is independent of the degree of impedance matching. Our systems approach to nanophotonics provides the basis for realizing general nanophotonic circuits and a large variety of derived novel devices

Tropical Discriminants

Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any A-discriminant, without any smoothness assumption.Comment: Major revisions, including several improvements and the correction of Section 5. To appear: Journal of the American Mathematical Societ

Der Kampf um Rohstoffe : wie das Recht Kosten und Nutzen der Rohstoffausbeutung verteilt

Große Rohstoffvorräte lagern in den Entwicklungsländern, doch ihre Ausbeutung führt in diesen Ländern oft weder zu steigendem Wirtschaftswachstum noch zu verbesserten Lebensverhältnissen der Bevölkerung. Von der Milliarde der ärmsten Menschen lebt fast ein Drittel in den rohstoffreichen Ländern. Kann das transnationale Rohstoffrecht dazu beitragen, dass die Verteilung gerechter abläuft und nicht nur die Investoren und Konsumenten der Nordhemisphäre und der Schwellenländer von den Rohstoffen der Welt profitieren? Die Juniorprofessorin Isabel Feichtner untersucht die Verteilungsgerechtigkeit im Rohstoffrecht

Representation theory for the Kriz model

The natural action of the symmetric group on the configuration spaces F(X; n) induces an action on the Kriz model E(X; n). The represen- tation theory of this DGA is studied and a big acyclic subcomplex which is Sn-invariant is described.Comment: 25 page