Related papers: On maximal rational poylhedral fans
We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program.…
For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral piecewise polynomial functions on P. We use the…
On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of…
Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups {\Gamma}, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases.…
We describe the structure of simplicial locally convex fans associated to even-dimensional complete toric varieties with signature 0. They belong to the set of such toric varieties whose even degree Betti numbers yield a top gamma vector…
We study the category of KM fans - a "stacky" generalization of the category of fans considered in toric geometry - and its various realization functors to "geometric" categories. The "purest" such realization takes the form of a functor…
In two previous papers we used the geometry of complete polyhedral fans to construct a subring $\TTT (l)$ of the modular forms on $\Gamma_{1} (l)$, and showed that for weight two the cuspidal part of $\TTT (l)$ coincides with the space of…
\noindent The simultaneous partition problems are classical problems of the combinatorial geometry which have the natural flavor of the equivariant topology. The $k$-fan partition problems have attracted a lot of attention \cite{Aki2000},…
This is an elementary review of our recent work on the classification of the spectra of those two-dimensional rational conformal field theories (RCFTs) whose (maximal) chiral algebras are current algebras. We classified all possible…
We study quotients of projective and affine spaces by various actions of the icosahedral group. Basically we concentrate on the rationality questions.
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
We investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope $P$. We show that the associated space of polytopes, called the inscribed cone of $P$, is closed under Minkowski…
OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number…
Graph associahedra are natural generalizations of the classical associahedra. They provide polytopal realizations of the nested complex of a graph $G$, defined as the simplicial complex whose vertices are the tubes (i.e. connected induced…
We provide formulas for projectors onto a polyhedral set, i.e. the intersection of a finite number of halfspaces. To this aim we formulate the problem of finding the projection as a convex optimization problem and we solve explicitly…
In the previous paper, we describe the intersection complexes of a toric variety as a finite complex of graded exterior modules on the associated fan. In this second part, we rewrite it explicitly by the barycentric subdivision of the fan.…
We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation…
The $g$-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete $g$-fans of rank 2. More explicitly, our first main result asserts that every complete…
Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical…
We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity…