155 research outputs found
On the Falk invariant of hyperplane arrangements attached to gain graphs
The fundamental group of the complement of a hyperplane arrangement in a
complex vector space is an important topological invariant. The third rank of
successive quotients in the lower central series of the fundamental group was
called Falk invariant of the arrangement since Falk gave the first formula and
asked to give a combinatorial interpretation. In this article, we give a
combinatorial formula for the Falk invariant of hyperplane arrangements
attached to certain gain graphs.Comment: To appear in the Australasian Journal of Combinatorics. arXiv admin
note: text overlap with arXiv:1703.0940
On the monodromy action on Milnor fibers of graphic arrangements
We analyze the monodromy action, over the rationals, on the first homology
group of the Milnor fiber, for arbitrary subarrangements of Coxeter
arrangements. We propose a combinatorial formula for the monodromy action,
involving Aomoto complexes in positive characteristic. We verify the formula,
in cases A, B and D.Comment: 23 pages; updated reference
The maximum likelihood degree of a very affine variety
We show that the maximum likelihood degree of a smooth very affine variety is
equal to the signed topological Euler characteristic. This generalizes Orlik
and Terao's solution to Varchenko's conjecture on complements of hyperplane
arrangements to smooth very affine varieties. For very affine varieties
satisfying a genericity condition at infinity, the result is further
strengthened to relate the variety of critical points to the
Chern-Schwartz-MacPherson class. The strengthened version recovers the
geometric deletion-restriction formula of Denham et al. for arrangement
complements, and generalizes Kouchnirenko's theorem on the Newton polytope for
nondegenerate hypersurfaces.Comment: Improved readability. Final version, to appear in Compositio
Mathematic
Of matroid polytopes, chow rings and character polynomials
Matroids are combinatorial structures that capture various notions of independence. Recently there has been great interest in studying various matroid invariants. In this thesis, we study two such invariants: Volume of matroid base polytopes and the Tutte polynomial. We gave an approach to computing volume of matroid base polytopes using cyclic flats and apply it to the case of sparse paving matroids. For the Tutte polynomial, we recover (some of) its coefficients as degrees of certain forms in the Chow ring of underlying matroid. Lastly, we study the stability of characters of the symmetric group via character polynomials. We show a combinatorial identity in the ring of class functions that implies stability results for certain class of Kronecker coefficients
Valuative invariants for large classes of matroids
We study an operation in matroid theory that allows one to transition a given
matroid into another with more bases via relaxing a \emph{stressed subset}.
This framework provides a new combinatorial characterization of the class of
split matroids. Moreover, it permits to describe an explicit matroid
subdivision of a hypersimplex, which in turn can be used to write down concrete
formulas for the evaluations of any valuative invariant on these matroids. This
shows that evaluations on split matroids depend solely on the behavior of the
invariant on tractable subclass of Schubert matroids that we call
\emph{cuspidal matroids}. We address systematically the consequences of our
approach for several invariants. They include the volume and Ehrhart polynomial
of base polytopes, the Tutte polynomial, Kazhdan--Lusztig polynomials, the
Whitney numbers of the first and second kind, spectrum polynomials and a
generalization of these by Denham, chain polynomials and Speyer's
-polynomials, as well as Chow rings of matroids and their
Hilbert--Poincar\'e series. The flexibility of this setting allows us to give a
unified explanation for several recent results regarding the listed invariants;
furthermore, we emphasize it as a powerful computational tool to produce
explicit data and concrete examples.Comment: 69 pages, 10 figures. This is a substantially revised and condensed
version. The structure and numbering of the results and definitions differs
from v1 and v
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
This memoir constitutes the author's PhD thesis at Cornell University. It
serves both as an expository work and as a description of new research. At the
heart of the memoir, we introduce and study a poset for each
finite Coxeter group and for each positive integer . When , our
definition coincides with the generalized noncrossing partitions introduced by
Brady-Watt and Bessis. When is the symmetric group, we obtain the poset of
classical -divisible noncrossing partitions, first studied by Edelman.
Along the way, we include a comprehensive introduction to related background
material. Before defining our generalization , we develop from
scratch the theory of algebraic noncrossing partitions . This involves
studying a finite Coxeter group with respect to its generating set of
{\em all} reflections, instead of the usual Coxeter generating set . This is
the first time that this material has appeared in one place.
Finally, it turns out that our poset shares many enumerative
features in common with the ``generalized nonnesting partitions'' of
Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In
particular, there is a generalized ``Fuss-Catalan number'', with a nice closed
formula in terms of the invariant degrees of , that plays an important role
in each case. We give a basic introduction to these topics, and we describe
several conjectures relating these three families of ``Fuss-Catalan objects''.Comment: Final version -- to appear in Memoirs of the American Mathematical
Society. Many small improvements in exposition, especially in Sections 2.2,
4.1 and 5.2.1. Section 5.1.5 deleted. New references to recent wor
Computation and Physics in Algebraic Geometry
Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra.
First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case.
Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature.
Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry
Tools and Algorithms for the Construction and Analysis of Systems
This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems
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