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

Institutions and reciprocity in the employment relationship

Homo oeconomicus has dominated mainstream Economics during the last century. One of the main assumptions of this model is that humans maximise their own utility functions. In other words, homo oeconomicus, before taking action, considers the consequences on their own future interests, which are generally assumed to be monetary. This thesis provides experimental results showing that human behaviour often differs from that of homo oeconomicus, particularly in environments where trust and reciprocity are salient concerns. To be precise, this dissertation analyses the employment relationship, focusing particularly on the importance of trust and the role of direct reciprocity in the relationship between managers and workers. Reciprocity is an important contract enforcement device in the presence of incomplete labour contracts. By reciprocity between employer and employee, what is meant is a predisposition, within the institutional context of defined employment tasks, to cooperate with the other party even at personal cost, and a willingness to punish the other party if they violate cooperative norms, even when punishment is costly to the individual. The original contribution of this thesis goes beyond this result and shows the impact of informal employment rules on reciprocity. In particular, it uses experimental methods to identify two distinct governance patterns for employment relationships: the rigid governance structure and the flexible governance structure. The former is characterised by task-centred rules and defines the boundaries of jobs in a much more specific way than the latter, which is characterised by function-centred rules, and gives rise to a more flexible and discretionary model of employment relationships. The most important original experimental result of this thesis is that rigid governance characterised by taskcentred rules and low reciprocity is better suited to one-shot transactions, whereas flexible governance characterised by function-centred rules and a high level of reciprocity is better suited to repeated transactions

A Multidisciplinary Approach to Capability in Age and Ageing

This open access book provides insight on how to interpret capability in ageing – one’s individual ability to perform actions in order to reach goals one has reason to value – from a multidisciplinary approach. With for the first time in history there being more people in the world aged 60 years and over than there are children below the age of 5, the book describes this demographic trends as well as the large global challenges and important societal implications this will have such as a worldwide increase in the number of persons affected with dementia, and in the ratio of retired persons to those still in the labor market. Through contributions from many different research areas, it discussed how capability depends on interactions between the individual (e.g. health, genetics, personality, intellectual capacity), environment (e.g. family, friends, home, work place), and society (e.g. political decisions, ageism, historical period). The final chapter summarizes the differences and similarities in these contributions. As such this book provides an interesting read for students, teachers and researchers at different levels and from different fields interested in capability and multidisciplinary research

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

A Multidisciplinary Approach to Capability in Age and Ageing

This open access book provides insight on how to interpret capability in ageing – one’s individual ability to perform actions in order to reach goals one has reason to value – from a multidisciplinary approach. With for the first time in history there being more people in the world aged 60 years and over than there are children below the age of 5, the book describes this demographic trends as well as the large global challenges and important societal implications this will have such as a worldwide increase in the number of persons affected with dementia, and in the ratio of retired persons to those still in the labor market. Through contributions from many different research areas, it discussed how capability depends on interactions between the individual (e.g. health, genetics, personality, intellectual capacity), environment (e.g. family, friends, home, work place), and society (e.g. political decisions, ageism, historical period). The final chapter summarizes the differences and similarities in these contributions. As such this book provides an interesting read for students, teachers and researchers at different levels and from different fields interested in capability and multidisciplinary research