Mirror symmetry and toric geometry.
Modules of splines on polyhedral complexes.
Nonrational, nonsimple convex polytopes in symplectic geometry.
Normal polytopes, triangulations, and Koszul algebras.
On a Hadwiger-Wills inequality concerning lattice points.
On Lattice Points in Polyhedral Cross-Sections.
On projective toric varieties whose defining ideals have minimal generators of the highest degree
It is known that generators of ideals defining projective toric varieties of dimension embedded by global sections of normally generated line bundles have degree at most . We characterize projective toric varieties of dimension whose defining ideals must have elements of degree as generators.
On quantum limits on flat tori.
On the computation of Clebsch-Gordan coefficients and the dilation effect.
On the graph labellings arising from phylogenetics
We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.
On the Graver complexity of codimension 2 matrices.
On the Number of Convex Lattice Polytopes.
On the refinements of a polyhedral subdivision.
Let pi: P --> Q be an affine projection map between two polytopes P and Q. Billera and Sturmfels introduced in 1992 the concept of polyhedral subdivisions of Q induced by pi (or pi-induced) and the fiber polytope of the projection: a polytope Sygma(P,pi) of dimension dim(P)-dim(Q) whose faces are in correspondence with the coherent pi-induced subdivisions (or pi-coherent subdivisions). In this paper we investigate the structure of the poset of pi-induced refinements of a pi-induced subdivision....
On the volume of a certain polytope.
Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator
Let be a family of rational polytopes parametrized by inequations. It is known that the volume of is a locally polynomial function of the parameters. Similarly, the number of integral points in is a locally quasi-polynomial function of the parameters. Paul-Émile Paradan proved a jump formula for this function, when crossing a wall. In this article, we give an algebraic proof of this formula. Furthermore, we give a residue formula for the jump, which enables us to compute it.
Perfect Delaunay polytopes in low dimensions.
Points entiers dans les polytopes convexes
Polytopal linear algebra.
Rademacher-Carlitz polynomials
We introduce and study the Rademacher-Carlitz polynomial where , s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum , which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz...
Rapport sur la géométrie par morceaux