Cluster algebras, quiver representations and triangulated categories

This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria)and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.

An irreducibility criterion for group representations, with arithmetic applications

We prove a criterion for the irreducibility of an integral group representation p over the fraction field of a noetherian domain R in terms of suitably defined reductions of p at prime ideals of R. As applications, we give irreducibility results for universal deformations of residual representations, with a special attention to universal deformations of residual Galois representations associated with modular forms of weight at least 2.

Intelligent task-level grasp mapping for robot control

In the future, robots will enter our everyday lives to help us with various tasks.For a complete integration and cooperation with humans, these robots needto be able to acquire new skills. Sensor capabilities for navigation in real humanenvironments and intelligent interaction with humans are some of the keychallenges.Learning by demonstration systems focus on the problem of human robotinteraction, and let the human teach the robot by demonstrating the task usinghis own hands. In this thesis, we present a solution to a subproblem within thelearning by demonstration field, namely human-robot grasp mapping. Robotgrasping of objects in a home or office environment is challenging problem.Programming by demonstration systems, can give important skills for aidingthe robot in the grasping task.The thesis presents two techniques for human-robot grasp mapping, directrobot imitation from human demonstrator and intelligent grasp imitation. Inintelligent grasp mapping, the robot takes the size and shape of the object intoconsideration, while for direct mapping, only the pose of the human hand isavailable.These are evaluated in a simulated environment on several robot platforms.The results show that knowing the object shape and size for a grasping taskimproves the robot precision and performance

Supporting the reuse of effective CSCL learning designs through social structure representations

Distance and blended collaborative learning settings are usually characterized by different social structures defined in terms of groups' number, dimension, and composition; these structures are variable and can change within the same activity. This variability poses additional complexity to instructional designers, when they are trying to develop successful experiences from existing designs. This complexity is greatly associated with the fact that learning designs do not render explicit how social structures influenced the decisions of the original designer, and thus whether the social structures of the new setting could preclude the effectiveness of the reused design. This article proposes the usage of new representations (social structure representations, SSRs) able to support unskilled designers in reusing existing learning designs, through the explicit characterization of the social structures and constraints embedded either by the original designers or the reusing teachers, according to well-known principles of good collaborative learning practice. The article also describes an evaluation process that involved university professors, as well as the main findings derived from it. This process supported the initial assumptions about the effectiveness of SSRs, with significant evidence from both qualitative and qualitative data.

Computing congruences of modular forms and Galois representations modulo prime powers

This article starts a computational study of congruences of modular forms and modular Galoisrepresentations modulo prime powers. Algorithms are described that compute the maximum integermodulo which two monic coprime integral polynomials have a root in common in a sensethat is defined. These techniques are applied to the study of congruences of modular forms andmodular Galois representations modulo prime powers. Finally, some computational results withimplications on the (non-)liftability of modular forms modulo prime powers and possible generalisationsof level raising are presented.

Galois representations, embedding problems and modular forms

To an odd irreducible 2-dimensional complex linear representation of the absolute Galois group of the field Q of rational numbers, a modular form of weight 1 is associated (modulo Artin's conjecture on the L-series of the representation in the icosahedral case). In addition, linear liftings of 2-dimensional projective Galois representations are related to solutions of certain Galois embedding problems. In this paper we present some recent results on the existence of liftings of projective representations and on the explicit resolution of embedding problems associated to orthogonal Galois representations, and explain how these results can be used to construct modular forms.

Exact results for Wilson loops in arbitrary representations

We compute the exact vacuum expectation value of 1/2 BPS circular Wilson loops of TeX = 4 U(N) super Yang-Mills in arbitrary irreducible representations. By localization arguments, the computation reduces to evaluating certain integrals in a Gaussian matrix model, which we do using the method of orthogonal polynomials. Our results are particularly simple for Wilson loops in antisymmetric representations; in this case, we observe that the final answers admit an expansion where the coefficients are positive integers, and can be written in terms of sums over skew Young diagrams. As an application of our results, we use them to discuss the exact Bremsstrahlung functions associated to the corresponding heavy probes.

Action Dominates Valence in Anticipatory Representations in the Human Striatum and Dopaminergic Midbrain

The acquisition of reward and the avoidance of punishment could logically be contingent on either emitting or withholding particular actions. However,the separate pathways inthe striatumfor go and no-go appearto violatethis independence, instead coupling affect and effect. Respect for this interdependence has biased many studies of reward and punishment, so potential action- outcome valence interactions during anticipatory phases remain unexplored. In a functional magnetic resonance imaging study with healthy human volunteers, we manipulated subjects" requirement to emit or withhold an action independent from subsequent receipt of reward or avoidance of punishment. During anticipation, in the striatum and a lateral region within the substantia nigra/ventral tegmental area (SN/VTA), action representations dominated over valence representations. Moreover, we did not observe any representation associated with different state values through accumulation of outcomes, challenging a conventional and dominant association between these areas and state value representations. In contrast, a more medial sector of the SN/VTA responded preferentially to valence, with opposite signs depending on whether action was anticipatedto be emitted or withheld. This dominant influence of action requires an enriched notion of opponency between reward and punishment.

First- and second-language phonological representations in the mental lexicon

Performance-based studies on the psychological nature of linguistic competence can conceal significant differences in the brain processes that underlie native versus nonnative knowledge of language. Here we report results from the brain activity of very proficient early bilinguals making a lexical decision task that illustrates this point. Two groups of SpanishCatalan early bilinguals (Spanish-dominant and Catalan-dominant) were asked to decide whether a given form was a Catalan word or not. The nonwords were based on real words, with one vowel changed. In the experimental stimuli, the vowel change involved a Catalan-specific contrast that previous research had shown to be difficult for Spanish natives to perceive. In the control stimuli, the vowel switch involved contrasts common to Spanish and Catalan. The results indicated that the groups of bilinguals did not differ in their behavioral and event-related brain potential measurements for the control stimuli; both groups made very few errors and showed a larger N400 component for control nonwords than for control words. However, significant differences were observed for the experimental stimuli across groups: Specifically, Spanish-dominant bilinguals showed great difficulty in rejecting experimental nonwords. Indeed, these participants not only showed very high error rates for these stimuli, but also did not show an error-related negativity effect in their erroneous nonword decisions. However, both groups of bilinguals showed a larger correctrelated negativity when making correct decisions about the experimental nonwords. The results suggest that although some aspects of a second language system may show a remarkable lack of plasticity (like the acquisition of some foreign contrasts), first-language representations seem to be more dynamic in their capacity of adapting and incorporating new information. &

On square permutations

Severini and Mansour introduced in [4]square polygons, as graphical representations of square permutations, that is, permutations such that all entries are records (left or right, minimum or maximum), and they obtained a nice formula for their number. In this paper we give a recursive construction for this class of permutations, that allows to simplify the derivation of their formula and to enumerate the subclass of square permutations with a simple record polygon. We also show that the generating function of these permutations with respect to the number of records of each type is algebraic, answering a question of Wilf in a particular case.

On the trace of an endofunctor of a small category

The trace of a square matrix can be defined by a universal property which, appropriately generalized yields the concept of "trace of an endofunctor of a small category". We review the basic definitions of this general concept and give a new construction, the "pretrace category", which allows us to obtain the trace of an endofunctor of a small category as the set of connected components of its pretrace. We show that this pretrace construction determines a finite-product preserving endofunctor of the category of small categories, and we deduce from this that the trace inherits any finite-product algebraic structure that the original category may have. We apply our results to several examples from Representation Theory obtaining a new (indirect) proof of the fact that two finite dimensional linear representations of a finite group are isomorphic if and only if they have the same character.

Pessebre virtual: disseny i exportació

Aquest projecte és una part d’un projecte més ampli consistent en estudiar un format gràfic que permeti exportar una escena modelada en Blender i importar aquesta mateixa escena en un entorn interactiu basat en Visual C++ amb OpenGL. D’aquesta forma, disposem de la capacitat de modelat de Blender i de la interacció i visualització de la llibreria OpenGL. Aquest format ha de representar geometria i textures imprescindiblement, i si és possible, d’altres factors importants com il·luminació, visualització i moviment. La part del projecte explicada en aquesta memòria consisteix en estudiar el format gràfic més adient per representar els diferents factors de realisme de l’escena (geometria, textura, etc.) havent triat el format OBJ per la seva capacitat de representació i fàcil edició. Per a provar el format, s’ha dissenyat un diorama de pessebre utilitzant les capacitats de modelatge de Blender. Pel que respecta les figures, aspecte important per a considerar l’escena com a pessebre, s’ha utilitzat un escàner 3D que ha obtingut representacions de malla 3D, a partir de figures reals de pessebre, que posteriorment han estat texturades. S’ha generat un vídeo del diorama de pessebre que permet veure’n tots els detalls navegant amb el punt de vista per l’escena. Aquest vídeo s’ha exposat en la mostra de pessebres de la Associació Pessebrista de Sabadell el Nadal del 2008.

La història es repeteix?: estudi comparatiu de la crisi del 29 amb l'actual

Treball de recerca realitzat per un alumne d'ensenyament secundari i guardonat amb un Premi CIRIT per fomentar l'esperit científic del Jovent l'any 2009. L’objectiu d’aquest treball ha estat donar pautes de reflexió sobre la crisi i el moment econòmic actual. Es volia valorar si era certa la idea que la situació econòmica dels nostres dies té certs paral•lelismes amb la crisi que va patir el món occidental el 1929. Per tal d’establir la veracitat de la hipòtesi plantejada es va dissenyar un model d’investigació basat en les entrevistes i en el recull i buidatge d’articles de premsa relacionats amb el tema, durant un període de temps no gaire extens, període que correspon als tres mesos a partir dels quals s’evidencia la situació de crisi. Al mateix temps era interessant copsar l’opinió de dos grans mestres en l’estudi de l’economia del nostre país, el Dr. Fabià Estapé i el Dr. Anton Costas, mestre i deixeble respectivament.. Sense esperar massa sorpreses, la crisi actual presenta molts punts de similitud amb el procés inciat l’any 1929. Ambdós esdeveniments econòmics tenen molts punts de connexió i les mesures proposades per solucionar o per pal•liar les conseqüències són també similars. En tots dos casos la figura de l’economista britànic John Maynard Keynes és cabdal per part dels diversos governs a l’hora de reconduir i adreçar la situació econòmica.

Modularity of the Consani-Scholten quintic

We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over Q, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livn´e method to induced four-dimensional Galois representations over Q. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by José Burgos Gil and the second author.