1,723 research outputs found
Stanley's Major Contributions to Ehrhart Theory
This expository paper features a few highlights of Richard Stanley's
extensive work in Ehrhart theory, the study of integer-point enumeration in
rational polyhedra. We include results from the recent literature building on
Stanley's work, as well as several open problems.Comment: 9 pages; to appear in the 70th-birthday volume honoring Richard
Stanle
Computing N\'eron-Severi groups and cycle class groups
Assuming the Tate conjecture and the computability of \'etale cohomology with
finite coefficients, we give an algorithm that computes the N\'eron-Severi
group of any smooth projective geometrically integral variety, and also the
rank of the group of numerical equivalence classes of codimension p cycles for
any p.Comment: 22 pages; to appear in Compositio Mat
Cluster algebras, quiver representations and triangulated categories
This is an introduction to some aspects of Fomin-Zelevinsky's cluster
algebras and their links with the representation theory of quivers and with
Calabi-Yau triangulated categories. It is based on lectures given by the author
at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to
by now classical material, we present the outline of a proof of the periodicity
conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and
recent results on the interpretation of mutations as derived equivalences.Comment: 53 pages, references update
Topological Noetherianity of polynomial functors
We prove that any finite-degree polynomial functor is topologically
Noetherian. This theorem is motivated by the recent resolution of Stillman's
conjecture and a recent Noetherianity proof for the space of cubics. Via work
by Erman-Sam-Snowden, our theorem implies Stillman's conjecture and indeed
boundedness of a wider class of invariants of ideals in polynomial rings with a
fixed number of generators of prescribed degrees.Comment: Final versio
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