Duality, Strings and Supergravity: a Status Report

We will report on recent advances in the understanding of non-perturbative interconnections between different string dualities. Weak-strong coupling duality (S-duality) and T-duality (symmetry under compactification on dual tori) allows one to compare and explore the strong coupling regime of seemingly unrelated theories. These theories naturally merge in a quantum version of supergravity called M-theory. The dynamical role of `branes' of different nature and the new dynamical tool of (M)atrix formulation of M-theory will be briefly mentioned.Comment: 12 pages, LaTeX; Plenary Talk given at the XII International Congress of Mathematical Physics, 13-19 July 1997, Brisbane, Australi

Holonomy and parallel transport for Abelian gerbes

In this paper we establish a one-to-one correspondence between $S^1$-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group $S^1$ on a simply connected manifold $M$ is a group morphism from the thin second homotopy group to $S^1$, satisfying a smoothness condition, where a homotopy between maps from $[0,1]^2$ to $M$ is thin when its derivative is of rank $\leq 2$. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids.Comment: Final version. Improved readibility (hopefully). LaTeX, 60 pages, 14 figures. To appear in Advances in Mathematic

Sharp weighted bounds for multilinear maximal functions and Calderón-Zygmund operators

In this paper we prove some sharp weighted norm inequalities for the multi(sub)linear maximal function M introduced in [18] A.K. Lerner, S. Ombrosi, C. Pérez, R.H. Torres and R. Trujillo-Gonz´alez, New maximal functions and multiple weights for the multilinear Caldern-Zygmund theory, Advances in Math. 220, 1222-1264 (2009). and for multilinear Calderón-Zygmund operators. In particular we obtain a sharp mixed “Ap − A∞” bound for M, some partial results related to a Buckley-type estimate for M, and a sufficient condition for the boundedness of M between weighted Lp spaces with different weights taking into account the precise bounds. Next we get a bound for multilinear Calderón-Zygmund operators in terms of dyadic positive multilinear operators in the spirit of the recent work [16] A.K. Lerner, On an estimate of Calderón-Zygmund operators by dyadic positive operators, J. Anal. Math. Then we obtain a multilinear version of the “A2 conjecture”. Several open problems are posed

Sasakian quiver gauge theories and instantons on Calabi-Yau cones

We consider SU(2)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M\times S^3/\Gamma$, where $M$ is a smooth manifold and $S^3/\Gamma$ is a three-dimensional Sasaki-Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associated to $\Gamma$. We relate them to those arising through translationally-invariant dimensional reduction over the associated Calabi-Yau cones $C(S^3/\Gamma)$ which are based on McKay quivers and ADHM matrix models, and to those arising through SU(2)-equivariant dimensional reduction over the leaf spaces of the characteristic foliations of $S^3/\Gamma$ which are K\"ahler orbifolds of $\mathbb{C} P^1$ whose quiver bundles are based on the unextended Dynkin diagram corresponding to $\Gamma$. We use Nahm equations to describe the vacua of SU(2)-equivariant quiver gauge theories on the cones as moduli spaces of spherically symmetric instantons. We relate them to the Nakajima quiver varieties which can be realized as Higgs branches of the worldvolume quiver gauge theories on D$p$-branes probing D$(p+4)$-branes which wrap an ALE space, and to the moduli spaces of spherically symmetric solutions in putative non-abelian generalizations of two-dimensional affine Toda field theories.Comment: 41 pages; v2: minor corrections; Final version published in Advances in Theoretical and Mathematical Physic

Sparse multivariate polynomial interpolation in the basis of Schubert polynomials

Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in the study of cohomology rings of flag manifolds in 1980's. These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials. In 2003, Lenart and Sottile introduced skew Schubert polynomials, which generalize skew Schur polynomials, and expand in the Schubert basis with the generalized Littlewood-Richardson coefficients. In this paper we initiate the study of these two families of polynomials from the perspective of computational complexity theory. We first observe that skew Schubert polynomials, and therefore Schubert polynomials, are in \CountP (when evaluating on non-negative integral inputs) and \VNP. Our main result is a deterministic algorithm that computes the expansion of a polynomial $f$ of degree $d$ in $\Z[x_1, \dots, x_n]$ in the basis of Schubert polynomials, assuming an oracle computing Schubert polynomials. This algorithm runs in time polynomial in $n$, $d$, and the bit size of the expansion. This generalizes, and derandomizes, the sparse interpolation algorithm of symmetric polynomials in the Schur basis by Barvinok and Fomin (Advances in Applied Mathematics, 18(3):271--285). In fact, our interpolation algorithm is general enough to accommodate any linear basis satisfying certain natural properties. Applications of the above results include a new algorithm that computes the generalized Littlewood-Richardson coefficients.Comment: 20 pages; some typos correcte

Toward a Theory of Experience

Science Education, Vol. 98, No. 1, pp. 106–126Experience is one of the most used terms in (science) education, and it is recognized as being related to learning (education). Yet what experience is and how it is related to learning and change remains untheorized. In this paper, we mainly draw on the work of J. Dewey and L. S. Vygotsky but also on M. Bakhtin and more recent advances on the topic of experience from French philosophy to contribute to a theory of this important category. Accordingly, experience is not something that belongs to or is had by individuals but rather denotes transactions in and across space and time within irreducible person-in-setting units; and it is perfused with an affect that is not (only) the result of mental constructions. An episode from an Australian physics classroom is used to exemplify what such a theory and its method-related implications have to accomplish in the analysis of concrete science lessons

World database of Happiness: Continuous register of research on subjective appreciation of life

Published in: Glatzer, W., VonBelow, S. & Stoffregen, M. (eds.), ‘Challenges for quality of life in the contemporary world: Advances in quality-of-life studies, theory and research’, Kluwer Academic Publishers, Dordrecht The Netherlands, 2004, Social Indicators Research Series, vol. 24, ISBN 1-4020—2890-3 (e-book 1-4020- 2903-9)The World Database of Happiness is an ongoing register of scientific research on subjective appreciation of life. It brings together findings that are scattered throughout many studies and prepares for research synthesis. The database stores research findings and presents these in standardized abstracts. This system differs from bibliographies that store publications and data-archives that store investigations. The system prepares for synthetic analysis by capitalizing on conceptual selectiveness, comparability and completeness. As the method is new, there is no common word for it. It is called a finding-browser

On Grothendieck Sets

[EN] We call a subset M of an algebra of sets A a Grothendieck set for the Banach space ba (A) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence fmn g n =1 in ba(A) which is pointwise convergent on M is weakly convergent in ba(A), i. e., if there is m 2 ba (A) such that mn ( A) ! m ( A) for every A 2M then mn ! m weakly in ba(A). A subset M of an algebra of sets A is called a Nikodym set for ba(A) if each sequence fm n g n =1 in ba(A) which is pointwise bounded on M is bounded in ba (A). We prove that if S is a s-algebra of subsets of a set W which is covered by an increasing sequence fS n : n 2 Ng of subsets of S there exists p 2 N such that S p is a Grothendieck set for ba(A). This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a s-algebra S is covered by an increasing sequence fS n : n 2 Ng of subsets, there is p 2 N such that S p is a Nikodym set for ba (S). This also refines the Grothendieck result stating that for each s -algebra S the Banach space ` (S) is a Grothendieck space. Some applications to classic Banach space theory are given.This research was funded by grant PGC2018-094431-B-I00 of Ministry of Scence, Innovation and universities of Spain.Ferrando, JC.; López Alfonso, S.; López Pellicer, M. (2020). On Grothendieck Sets. Axioms. 9(1):1-7. https://doi.org/10.3390/axioms9010034S1791Valdivia, M. (1979). On certain barrelled normed spaces. Annales de l’institut Fourier, 29(3), 39-56. doi:10.5802/aif.752Ferrando, J. C., López-Alfonso, S., & López-Pellicer, M. (2019). On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia. Filomat, 33(8), 2409-2416. doi:10.2298/fil1908409fLópez-Alfonso, S. (2015). On Schachermayer and Valdivia results in algebras of Jordan measurable sets. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 110(2), 799-808. doi:10.1007/s13398-015-0267-xFerrando, J. C., & Ruiz, L. M. S. (2004). A Survey on Recent Advances on the Nikodým Boundedness Theorem and Spaces of Simple Functions. Rocky Mountain Journal of Mathematics, 34(1). doi:10.1216/rmjm/1181069896Valdivia, M. (2012). On Nikodym boundedness property. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 107(2), 355-372. doi:10.1007/s13398-012-0081-7Ferrando, J. C., Ka̧kol, J., & López-Pellicer, M. (2017). On spaces Cb(X) weakly K-analytic. Mathematische Nachrichten, 290(16), 2612-2618. doi:10.1002/mana.201600406Rainwater, J. (1963). Weak convergence of bounded sequences. Proceedings of the American Mathematical Society, 14(6), 999. doi:10.1090/s0002-9939-1963-0155171-9Simons, S. (1972). A convergence theorem with boundary. Pacific Journal of Mathematics, 40(3), 703-708. doi:10.2140/pjm.1972.40.703Drewnowski, L., Florencio, M., & Paúl, P. J. (1994). Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections. Glasgow Mathematical Journal, 36(1), 57-69. doi:10.1017/s0017089500030548Saxon, S. A. (1972). Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology. Mathematische Annalen, 197(2), 87-106. doi:10.1007/bf0141958

Resolution of the equatorial spread F problem: Revisited

An overview of recent advances made in understanding the phenomenon of equatorial spread F (ESF) is presented and a discussion of unresolved issues that need to be addressed. The focus is on research that has occurred in the last decade. The topics include satellite observations, theory, and modeling. The suggested areas that require further exploration are a unified theory of turbulence extending from 100 s m to 10 s cm, the impact of geomagnetic storms on the development of equatorial spread F, the need for accurate thermospheric wind measurements and models, and identifying the underlying physics of ESF in the post-midnight sector during solar minimum