73 research outputs found
Tropical Geometry: new directions
The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that
have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative
geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress
in more traditional areas of tropical research, such as tropical
moduli spaces, tropical homology and tropical correspondence theorems
Smoothed projections over manifolds in finite element exterior calculus
We develop commuting finite element projections over smooth Riemannian
manifolds. This extension of finite element exterior calculus establishes the
stability and convergence of finite element methods for the Hodge-Laplace
equation on manifolds. The commuting projections use localized mollification
operators, building upon a classical construction by de Rham. These projections
are uniformly bounded on Lebesgue spaces of differential forms and map onto
intrinsic finite element spaces defined with respect to an intrinsic smooth
triangulation of the manifold. We analyze the Galerkin approximation error.
Since practical computations use extrinsic finite element methods over
approximate computational manifolds, we also analyze the geometric error
incurred.Comment: Submitted. 31 page
Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
Tropical geometry and its applications
Abstract. From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewise-linear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the problems in classical (real and complex) geometry
Convex and Algebraic Geometry
The subjects of convex and algebraic geometry meet primarily in the theory of toric varieties. Toric geometry is the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Recent developments in toric geometry that were discussed during the workshop include applications to mirror symmetry, motivic integration and hypergeometric systems of PDE’s, as well as deformations of (unions of) toric varieties and relations to tropical geometry
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