70 research outputs found
Tropical bounds for eigenvalues of matrices
We show that for all k = 1,...,n the absolute value of the product of the k
largest eigenvalues of an n-by-n matrix A is bounded from above by the product
of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute
value), up to a combinatorial constant depending only on k and on the pattern
of the matrix. This generalizes an inequality by Friedland (1986),
corresponding to the special case k = 1.Comment: 17 pages, 1 figur
Amoebas of algebraic varieties and tropical geometry
This survey consists of two parts. Part 1 is devoted to amoebas. These are
images of algebraic subvarieties in the complex torus under the logarithmic
moment map. The amoebas have essentially piecewise-linear shape if viewed at
large. Furthermore, they degenerate to certain piecewise-linear objects called
tropical varieties whose behavior is governed by algebraic geometry over the
so-called tropical semifield. Geometric aspects of tropical algebraic geometry
are the content of Part 2. We pay special attention to tropical curves. Both
parts also include relevant applications of the theories. Part 1 of this survey
is a revised and updated version of an earlier prepreint of 2001.Comment: 40 pages, 15 figures, a survey for the volume "Different faces in
Geometry
Phases of Five-dimensional Theories, Monopole Walls, and Melting Crystals
Moduli spaces of doubly periodic monopoles, also called monopole walls or
monowalls, are hyperk\"ahler; thus, when four-dimensional, they are self-dual
gravitational instantons. We find all monowalls with lowest number of moduli.
Their moduli spaces can be identified, on the one hand, with Coulomb branches
of five-dimensional supersymmetric quantum field theories on
and, on the other hand, with moduli spaces of local
Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore
the asymptotic metric of these moduli spaces and compare our results with
Seiberg's low energy description of the five-dimensional quantum theories. We
also give a natural description of the phase structure of general monowall
moduli spaces in terms of triangulations of Newton polygons, secondary
polyhedra, and associahedral projections of secondary fans.Comment: 45 pages, 11 figure
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