Convex Polytopes, Algebraic Geometry, and Combinatorics

  title={Convex Polytopes, Algebraic Geometry, and Combinatorics},
  author={Laura Escobar and Kiumars Kaveh},
  journal={Notices of the American Mathematical Society},
In the last several decades, convex geometry methods have proven very useful in algebraic geometry specifically to understand discrete invariants of algebraic varieties. An approach to study algebraic varieties is to assign to a family of varieties a corresponding family of combinatorial objects which encode geometric information about the varieties. Often, the combinatorial objects that arise are convex polytopes, and convex geometry has been an essential tool for this strategy. The emergence… 
Euclidean Distance Degree and Mixed Volume
We initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface $$f=0$$ f = 0 defined by a polynomial f that is general


The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and
Convex Bodies Associated to Linear Series
In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study
Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any
Brunn–Minkowski inequality for multiplicities
This is a convex subset of the real vector space P⊗ZR. In fact, it is a convex polytope; see [Br], where this polytope is discussed from an algebraic point of view. It is known that the total mass of
Convex bodies associated to actions of reductive groups
We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of
Convex bodies and multiplicities of ideals
We associate convex regions in ℝn to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically
Crystal bases and Newton–Okounkov bodies
Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G coincides with a natural valuation
Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry
The notion of a Khovanskii basis for $(A, \mathfrak{v})$ is introduced which provides a framework for far extending Gr\"obner theory on polynomial algebras to general finitely generated algeBRas and construct an associated compactification of $Spec(A)$.
Introduction to toric varieties
The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio
Brick Manifolds and Toric Varieties of Brick Polytopes
  • Laura Escobar
  • Mathematics, Computer Science
    Electron. J. Comb.
  • 2016
It is proved that in some cases the general fiber, which the authors christen a brick manifold, is a toric variety, and a nice description of the toric varieties of the associahedron is given.