Moderately Discontinuous Homotopy

We introduce a metric homotopy theory, which we call moderately discontinuous homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the moderately discontinuous homology theory recently developed by the authors and E. Sampaio. The $k$-th MD homotopy group is a group $MDH^b_{\bullet }$ for any $b\in [1,\infty ]$ together with homomorphisms $MD\pi ^b\to MD\pi ^{b^{\prime}}$ for any $b\geq b^{\prime}$. We develop all its basic properties including finite presentation of the groups, long homotopy sequences of pairs, metric homotopy invariance, Seifert van Kampen Theorem, and the Hurewicz Isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For $b=1$, it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for $b=\infty $, the $MD$-homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-homology in the same case.

Inessential directed maps and directed homotopy equivalences

A directed space is a topological space $X$ together with a subspace $\vec {P}(X)\subset X^I$ of directed paths on $X$ . A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated spaces $\vec {P}(X)_-^+$ of directed paths between a source ( $-$ ) and a target ( $+$ )—up to homotopy. If it is, moreover, homotopic to the identity map—in a directed sense—such a symmetry will be called an inessential d-map, and the paper explores the algebra and topology of inessential d-maps. Comparing two d-spaces $X$ and $Y$ ‘up to symmetry’ yields the notion of a directed homotopy equivalence between them. Under appropriate conditions, all directed homotopy equivalences are shown to satisfy a 2-out-of-3 property. Our notion of directed homotopy equivalence does not agree completely with the one defined in Goubault (2017, arxiv:1709:05702v2) and Goubault, Farber and Sagnier (2020, J. Appl. Comput. Topol. 4, 11–27); the deviation is motivated by examples. Nevertheless, directed topological complexity, introduced in Goubault, Farber and Sagnier (2020) is shown to be invariant under our notion of directed homotopy equivalence. Finally, we show that directed homotopy equivalences result in isomorphisms on the pair component categories of directed spaces introduced in Goubault, Farber and Sagnier (2020).

Differential operators on almost-Hermitian manifolds and harmonic forms

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.

Some Singular Vector-Valued Jack and Macdonald Polynomials

For each partition τ of N, there are irreducible modules of the symmetric groups S N and of the corresponding Hecke algebra H N t whose bases consist of the reverse standard Young tableaux of shape τ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family G n , p , N of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups S N and the Hecke algebra H N t , there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by κ and q , t , respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is x 1 m ⊗ S , where S is an arbitrary reverse standard Young tableau of shape τ . The singular values depend on the properties of the edge of the Ferrers diagram of τ .

Feminismos latinoamericanos: una mirada panorámica

Resumen: Este trabajo recoge la charla de MaríaLuisa Femenías, transcrita y re-elaborada por la autora,presentada por videoconferencia el 22 de marzo de 2011,en un evento pensado como preparación para la cohorteen Estudios de Género del Doctorado de Humanidadesde la Universidad del Valle. Femenías inicia su charlapostulando tres niveles de feminismos, (a) un nivel teórico, (b) un feminismo militante y, por último, (c) el feminismoespontáneo del movimiento de mujeres. En el nivel teórico,la autora reconoce la necesidad de situar nuestros propiossaberes y poder dinamizar las categorías que nos llegande afuera, en función de nuestros propios problemas yexperiencias, nuestras situaciones particulares históricosocialesy geográficas. La autora reconoce que la historiade nuestros movimientos muestra, no calcos de lo ajeno,sino respuestas a acontecimientos locales “previos”, quehan dado lugar a movimientos de mujeres autónomos.En ese sentido, Femenías sostiene que el pensamientofeminista latinoamericano es original y es originario en lamedida en que parte de su propia situación y localización,y tiene características propias como la pluralidad étnicocultural,lo cual hace aún más imperativo que se tome encuenta la intersección género, clase y etnia. La autoraretoma el concepto de subalternidad, pues las mujeresconstituimos una “minoría”, no numéricamente, sino enrelación con nuestro escaso usufructo de los espacios depoder. Se examinan los modos en los cuales el problemade la violencia doméstica se estructura como un eje queinvisibiliza otras violencias, como por ejemplo la violenciade la exclusión por naturalización que sostiene la divisiónsexual del trabajo y de los espacios asociados. Finalmente,se admite que la mayoría de nuestras luchas han sido porla legalización de los derechos de las mujeres, lo cual nosvincula con el feminismo denominado “de la igualdad”,pero que este reclamo ha insuficiente , lo cual, de un modoo de otro, ha conducido a la revisión situada de esos conceptos,en relación con lo que se ha llamado la “culturafemenina” que remite a la preocupación “en red” de quetodos y todas puedan vivir una vida que merezca ser vivida.Palabras clave: feminismos, pensamiento feminista latinoamericano, subalternidad, género, clase, etniaLatin American Feminisms: An OverviewAbstract: This paper corresponds to a talk given byMaría Luisa Femenías, transcribed and reworked by theauthor, which was a videoconference given on March 22,2011, in an event preparing for the Doctorate of Humanities,emphasis on Gender Studies. Femenías postulatesthree levels of feminism, (a) a theoretical level, (b) feministmilitancy, and (c) spontaneous women’s movements. Onthe theoretical level, the author recognizes the need to placeour knowledge and retool foreign categories with regardto our own problems and experiences, our historical andsocial situations and geography. The author recognizesthat the history of our movements has mostly respondedto local events, giving rise to autonomous women’s movements.In this sense Latin American feminist thought isoriginal, for it obeys our own characteristics, such as ourethnic and cultural plurality, which makes it imperative totake into account the intersections among gender, class andethnicity. Femenías invokes the concept of subalternity, for women constitute a minority, not numerically, but in the sense of our low levels of access to power. The paper alludes to ways in which domestic violence becomes a screen making other forms of violence invisible, among them the sexual division of labor and its associated spaces. Finally, it is admitted that most of our struggles have been for legalization of women’s rights, which ties us to “equality feminism”, but this has been insufficient, which has led to the revision of concepts in relation to what has been called “feminine culture”, related in turn to the purpose of making it possible for everyone to lead a life worth living.Key Words: feminisms, Latina American feministthought, subalternity, gender, class, ethnicity