Equivariant chain complexes, twisted homology and relative minimality of arrangements

We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).Comment: 22 page

Development, implementation, and evaluation of an online English placement test at college level: a case study.

The primary purpose of the present project was to research the case study of current English placement practices at Intercollege in view of incorporating change, improvement and efficiency, within the framework of current work based learning and applied linguistics (and more particularly English online language testing) research discipline. The review of work based learning and current theories and practices in applied linguistics research discipline helped establish the characteristics of an insider researcher and the research approach and research techniques that would best serve such a project. The review of current theories and practices in second language (L2) teaching and learning in general, and in L2 testing in particular revealed that there is an extensive range of practices: these range from testing discrete points to integrative tasks. Tests are also delivered both in pen-and-paper as well as in electronic form, the latter being either computer based testing (CBT) or computer adaptive testing (CAT). The review of current English placement practices at Intercollege indicated the need for a new English placement test, developed in a scientific way, informed by current theories and practices, based on current test design models and taking advantage of more efficient methods of delivery, and placement. This review also revealed the need for more efficiency in the mode of delivery, administration, marking, reporting and test duration. Finally, this study of the current English placement practices at Intercollege established the need for a placement test that would incorporate a mechanism of continuous testing of reliability and validity as well as improvement. The detailed study of the specific context, setting, particular language programme, resources, test-takers, instructors, etc. informed by current theories and practices in second language (L2) testing online, helped in the development of the New English Placement Test Online (NEPTON) test specifications, and as a consequence, the development of the proposed test itself. The study of test delivery modes and the consideration of the specific work based conditions and requirements. For example administration, delivery, time and money efficiency, urgent need of an improved and more efficient English placement test (EPT) resulted in the selection of computer based testing delivery, with many features of the computer adaptive testing delivery mode incorporated in it such as randomized selection of test items and fewer items. The test item writing and item modération process resulted in the formation of a substantial pool of varied items in different skills, text types, topics, settings, and covering a variety of lexical and grammatical points and communicative, authentic-like situations in ali six levels. The field test which was took place in May 2004 in pen-and-paper form by almost 1200 students in ali three Intercollege campuses helped check the content and the test trial which took place in the period of August-September in its electronic form helped come up with the test cutoff points, and the fine-tuning of the test. The item analysis ensured the appropriateness of ali items. Pre-test questionnaires established test-takers' biographical data and information about test-taker computer familiarity. The test face validity (stakeholders' attitudes and feelings about the NEPTON) was established through the use of pre and post-test questionnaires. Experts in the area Coming, from the three campuses, also studied the test specifications and the test itself (both in its electronic and pen-and-paper format) and completed a questionnaire, thus contributing to the establishment of the test content and construct validity. The test reliability was established through a split half reliability index process and a series of other aspects or processes such as the size of the item bank, the instructions, the moderation process, and the item analysis, which are explained in chapter 5 in more details. The research project consists of two components: (a) The report, which describes the way work based and applied linguistics research approaches were used to investigate the case study of English placement test at college level at Intercollege in Cyprus and to what extent this has broad change, improvement and effìciency to current practices; and (b) The evidence of such a research project, which is the New English Placement Test Online (NEPTON), in other words, the test itself, developed, implemented and evaluated in order to materialize this change, improvement and efficiency aimed at by this project

Non-abelian resonance: product and coproduct formulas

We investigate the resonance varieties attached to a commutative differential graded algebra and to a representation of a Lie algebra, with emphasis on how these varieties behave under finite products and coproducts.Comment: 12 page

The abelianization of the Johnson kernel

We prove that the first complex homology of the Johnson subgroup of the Torelli group Tg is a non-trivial, unipotent Tg-module for all g ≥ 4 and give an explicit presentation of it as a Sym H 1(Tg,C)-module when g ≥ 6. We do this by proving that, for a finitely generated group G satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of G. In this setup, we also obtain a precise nilpotence test. © European Mathematical Society 2014

Around the tangent cone theorem

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.Comment: 39 pages; to appear in the proceedings of the Configurations Spaces Conference (Cortona 2014), Springer INdAM serie

Geometric and homological finiteness in free abelian covers

We describe some of the connections between the Bieri-Neumann-Strebel-Renz invariants, the Dwyer-Fried invariants, and the cohomology support loci of a space X. Under suitable hypotheses, the geometric and homological finiteness properties of regular, free abelian covers of X can be expressed in terms of the resonance varieties, extracted from the cohomology ring of X. In general, though, translated components in the characteristic varieties affect the answer. We illustrate this theory in the setting of toric complexes, as well as smooth, complex projective and quasi-projective varieties, with special emphasis on configuration spaces of Riemann surfaces and complements of hyperplane arrangements.Comment: 30 pages; to appear in Configuration Spaces: Geometry, Combinatorics and Topology (Centro De Giorgi, 2010), Edizioni della Normale, Pisa, 201

Complements of hypersurfaces, variation maps and minimal models of arrangements

We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in $\mathbb C^n$ by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants

Chamber basis of the Orlik-Solomon algebra and Aomoto complex

We introduce a basis of the Orlik-Solomon algebra labeled by chambers, so called chamber basis. We consider structure constants of the Orlik-Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen's minimal complex from the Aomoto complex.Comment: 16 page

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more citations, to appear in Mathematische Zeitschrif

The Pure Virtual Braid Group Is Quadratic

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of such a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a (not necessarily homomorphic) universal finite type invariant.Comment: 53 pages, 15 figures. Some clarifications added and inaccuracies corrected, reflecting suggestions made by the referee of the published version of the pape