Moment-angle complexes, monomial ideals, and Massey products

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio

On the homotopy Lie algebra of an arrangement

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincar\'e polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras.Comment: 20 pages; accepted for publication by the Michigan Math. Journa

Predictive coding in auditory perception: challenges and unresolved questions.

Predictive coding is arguably the currently dominant theoretical framework for the study of perception. It has been employed to explain important auditory perceptual phenomena, and it has inspired theoretical, experimental and computational modelling efforts aimed at describing how the auditory system parses the complex sound input into meaningful units (auditory scene analysis). These efforts have uncovered some vital questions, addressing which could help to further specify predictive coding and clarify some of its basic assumptions. The goal of the current review is to motivate these questions and show how unresolved issues in explaining some auditory phenomena lead to general questions of the theoretical framework. We focus on experimental and computational modelling issues related to sequential grouping in auditory scene analysis (auditory pattern detection and bistable perception), as we believe that this is the research topic where predictive coding has the highest potential for advancing our understanding. In addition to specific questions, our analysis led us to identify three more general questions that require further clarification: (1) What exactly is meant by prediction in predictive coding? (2) What governs which generative models make the predictions? and (3) What (if it exists) is the correlate of perceptual experience within the predictive coding framework

Critical points and resonance of hyperplane arrangements

If F is a master function corresponding to a hyperplane arrangement A and a collection of weights y, we investigate the relationship between the critical set of F, the variety defined by the vanishing of the one-form w = d log F, and the resonance of y. For arrangements satisfying certain conditions, we show that if y is resonant in dimension p, then the critical set of F has codimension at most p. These include all free arrangements and all rank 3 arrangements.Comment: revised version, Canadian Journal of Mathematics, to appea

Torsion in Milnor fiber homology

In a recent paper, Dimca and Nemethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each prime p, there is a polynomial with p-torsion in the homology of the Milnor fiber. The techniques make use of properties of characteristic varieties of hyperplane arrangements.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-16.abs.htm

Modelling the Emergence and Dynamics of Perceptual Organisation in Auditory Streaming

Many sound sources can only be recognised from the pattern of sounds they emit, and not from the individual sound events that make up their emission sequences. Auditory scene analysis addresses the difficult task of interpreting the sound world in terms of an unknown number of discrete sound sources (causes) with possibly overlapping signals, and therefore of associating each event with the appropriate source. There are potentially many different ways in which incoming events can be assigned to different causes, which means that the auditory system has to choose between them. This problem has been studied for many years using the auditory streaming paradigm, and recently it has become apparent that instead of making one fixed perceptual decision, given sufficient time, auditory perception switches back and forth between the alternatives—a phenomenon known as perceptual bi- or multi-stability. We propose a new model of auditory scene analysis at the core of which is a process that seeks to discover predictable patterns in the ongoing sound sequence. Representations of predictable fragments are created on the fly, and are maintained, strengthened or weakened on the basis of their predictive success, and conflict with other representations. Auditory perceptual organisation emerges spontaneously from the nature of the competition between these representations. We present detailed comparisons between the model simulations and data from an auditory streaming experiment, and show that the model accounts for many important findings, including: the emergence of, and switching between, alternative organisations; the influence of stimulus parameters on perceptual dominance, switching rate and perceptual phase durations; and the build-up of auditory streaming. The principal contribution of the model is to show that a two-stage process of pattern discovery and competition between incompatible patterns can account for both the contents (perceptual organisations) and the dynamics of human perception in auditory streaming

Auditory Multi-Stability: Idiosyncratic Perceptual Switching Patterns, Executive Functions and Personality Traits

Multi-stability refers to the phenomenon of perception stochastically switching between possible interpretations of an unchanging stimulus. Despite considerable variability, individuals show stable idiosyncratic patterns of switching between alternative perceptions in the auditory streaming paradigm. We explored correlates of the individual switching patterns with executive functions, personality traits, and creativity. The main dimensions on which individual switching patterns differed from each other were identified using multidimensional scaling. Individuals with high scores on the dimension explaining the largest portion of the inter-individual variance switched more often between the alternative perceptions than those with low scores. They also perceived the most unusual interpretation more often, and experienced all perceptual alternatives with a shorter delay from stimulus onset. The ego-resiliency personality trait, which reflects a tendency for adaptive flexibility and experience seeking, was significantly positively related to this dimension. Taking these results together we suggest that this dimension may reflect the individual?s tendency for exploring the auditory environment. Executive functions were significantly related to some of the variables describing global properties of the switching patterns, such as the average number of switches. Thus individual patterns of perceptual switching in the auditory streaming paradigm are related to some personality traits and executive functions.</p