Schubert calculus and shifting of interval positroid varieties

Consider k x n matrices with rank conditions placed on intervals of columns. The ranks that are actually achievable correspond naturally to upper triangular partial permutation matrices, and we call the corresponding subvarieties of Gr(k,n) the _interval positroid varieties_, as this class lies within the class of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert and opposite Schubert varieties, and their intersections, and is Grassmann dual to the projection varieties of [Billey-Coskun]. Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain degenerations to positively compute the H^*-classes of Richardson varieties, each summand recorded as a (2+1)-dimensional "checker game". We use his same degenerations to positively compute the K_T-classes of interval positroid varieties, each summand recorded more succinctly as a 2-dimensional "K-IP pipe dream". In Vakil's restricted situation these IP pipe dreams biject very simply to the puzzles of [Knutson-Tao]. We relate Vakil's degenerations to Erd\H os-Ko-Rado shifting, and include results about computing "geometric shifts" of general T-invariant subvarieties of Grassmannians.Comment: 35 pp; this subsumes and obviates the unpublished http://arxiv.org/abs/1008.430

Behavioral Impact of Unisensory and Multisensory Audio-Tactile Events: Pros and Cons for Interlimb Coordination in Juggling

Recent behavioral neuroscience research revealed that elementary reactive behavior can be improved in the case of cross-modal sensory interactions thanks to underlying multisensory integration mechanisms. Can this benefit be generalized to an ongoing coordination of movements under severe physical constraints? We choose a juggling task to examine this question. A central issue well-known in juggling lies in establishing and maintaining a specific temporal coordination among balls, hands, eyes and posture. Here, we tested whether providing additional timing information about the balls and hands motions by using external sound and tactile periodic stimulations, the later presented at the wrists, improved the behavior of jugglers. One specific combination of auditory and tactile metronome led to a decrease of the spatiotemporal variability of the juggler's performance: a simple sound associated to left and right tactile cues presented antiphase to each other, which corresponded to the temporal pattern of hands movement in the juggling task. A contrario, no improvements were obtained in the case of other auditory and tactile combinations. We even found a degraded performance when tactile events were presented alone. The nervous system thus appears able to integrate in efficient way environmental information brought by different sensory modalities, but only if the information specified matches specific features of the coordination pattern. We discuss the possible implications of these results for the understanding of the neuronal integration process implied in audio-tactile interaction in the context of complex voluntary movement, and considering the well-known gating effect of movement on vibrotactile perception

COMBINATORIAL ASPECTS OF EXCEDANCES AND THE FROBENIUS COMPLEX

In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that is the order complex of a poset whose definition was motivated by the classical Frobenius problem. We determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. We end with an enumeration of Q-factorial posets. Open questions and directions for future research are located at the end of each chapter

Hyperbrain features of team mental models within a juggling paradigm: a proof of concept

Background Research on cooperative behavior and the social brain exists, but little research has focused on real-time motor cooperative behavior and its neural correlates. In this proof of concept study, we explored the conceptual notion of shared and complementary mental models through EEG mapping of two brains performing a real-world interactive motor task of increasing difficulty. We used the recently introduced participative “juggling paradigm,” and collected neuro-physiological and psycho-social data. We were interested in analyzing the between-brains coupling during a dyadic juggling task, and in exploring the relationship between the motor task execution, the jugglers’skill level and the task difficulty. We also investigated how this relationship could be mirrored in the coupled functional organization of the interacting brains. Methods To capture the neural schemas underlying the notion of shared and complementary mental models, we examined the functional connectivity patterns and hyperbrain features of a juggling dyad involved in cooperative motor tasks of increasing difficulty. Jugglers’ cortical activity was measured using two synchronized 32-channel EEG systems during dyadic juggling performed with 3, 4, 5 and 6 balls. Individual and hyperbrain functional connections were quantified through coherence maps calculated across all electrode pairs in the theta and alpha bands (4–8 and 8–12 Hz). Graph metrics were used to typify the global topology and efficiency of the functional networks for the four difficulty levels in the theta and alpha bands. Results Results indicated that, as task difficulty increased, the cortical functional organization of the more skilled juggler became progressively more segregated in both frequency bands, with a small-world organization in the theta band during easier tasks, indicative of a flow-like state in line with the neural efficiency hypothesis. Conversely, more integrated functional patterns were observed for the less skilled juggler in both frequency bands, possibly related to cognitive overload due to the difficulty of the task at hand (reinvestment hypothesis). At the hyperbrain level, a segregated functional organization involving areas of the visuo-attentional networks of both jugglers was observed in both frequency bands and for the easier task only. Discussion These results suggest that cooperative juggling is supported by integrated activity of specialized cortical areas from both brains only during easier tasks, whereas it relies on individual skills, mirrored in uncorrelated individual brain activations, during more difficult tasks. These findings suggest that task difficulty and jugglers’ personal skills may influence the features of the hyperbrain network in its shared/integrative and complementary/segregative tendencies