The homotopy type of the contactomorphism groups of tight contact 33-manifolds, part I

We prove that that the homotopy type of the path connected component of the identity in the contactomorphism group is characterized by the homotopy type of the diffeomorphism group plus some data provided by the topology of the formal contactomorphism space. As a consequence, we show that every connected component of the space of Legendrian long knots in $\R^3$ has the homotopy type of the corresponding smooth long knot space. This implies that any connected component of the space of Legendrian embeddings in \NS^3 is homotopy equivalent to the space K(G,1)\times\U(2), with $G$ computed by A. Hatcher and R. Budney. Similar statements are proven for Legendrian embeddings in $\R^3$ and for transverse embeddings in \NS^3. Finally, we compute the homotopy type of the contactomorphisms of several tight $3$-folds: \NS^1 \times \NS^2, Legendrian fibrations over compact orientable surfaces and finite quotients of the standard $3$-sphere. In fact, the computations show that the method works whenever we have knowledge of the topology of the diffeomorphism group. We prove several statements on the way that have interest by themselves: the computation of the homotopy groups of the space of non-parametrized Legendrians, a multiparametric convex surface theory, a characterization of formal Legendrian simplicity in terms of the space of tight contact structures on the complement of a Legendrian, the existence of common trivializations for multi-parametric families of tight $\R^3$, etc.Comment: 77 pages. Comments are welcome

Diverse Large HIV-1 Non-subtype B Clusters Are Spreading Among Men Who Have Sex With Men in Spain

In Western Europe, the HIV-1 epidemic among men who have sex with men (MSM) is dominated by subtype B. However, recently, other genetic forms have been reported to circulate in this population, as evidenced by their grouping in clusters predominantly comprising European individuals. Here we describe four large HIV-1 non-subtype B clusters spreading among MSM in Spain. Samples were collected in 9 regions. A pol fragment was amplified from plasma RNA or blood-extracted DNA. Phylogenetic analyses were performed via maximum likelihood, including database sequences of the same genetic forms as the identified clusters. Times and locations of the most recent common ancestors (MRCA) of clusters were estimated with a Bayesian method. Five large non-subtype B clusters associated with MSM were identified. The largest one, of F1 subtype, was reported previously. The other four were of CRF02_AG (CRF02_1; n = 115) and subtypes A1 (A1_1; n = 66), F1 (F1_3; n = 36), and C (C_7; n = 17). Most individuals belonging to them had been diagnosed of HIV-1 infection in the last 10 years. Each cluster comprised viruses from 3 to 8 Spanish regions and also comprised or was related to viruses from other countries: CRF02_1 comprised a Japanese subcluster and viruses from 8 other countries from Western Europe, Asia, and South America; A1_1 comprised viruses from Portugal, United Kingom, and United States, and was related to the A1 strain circulating in Greece, Albania and Cyprus; F1_3 was related to viruses from Romania; and C_7 comprised viruses from Portugal and was related to a virus from Mozambique. A subcluster within CRF02_1 was associated with heterosexual transmission. Near full-length genomes of each cluster were of uniform genetic form. Times of MRCAs of CRF02_1, A1_1, F1_3, and C_7 were estimated around 1986, 1989, 2013, and 1983, respectively. MRCA locations for CRF02_1 and A1_1 were uncertain (however initial expansions in Spain in Madrid and Vigo, respectively, were estimated) and were most probable in Bilbao, Spain, for F1_3 and Portugal for C_7. These results show that the HIV-1 epidemic among MSM in Spain is becoming increasingly diverse through the expansion of diverse non-subtype B clusters, comprising or related to viruses circulating in other countries

Fundamental groups of formal Legendrian and horizontal embedding spaces

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of nontrivial loops of Legendrian embeddings are already nontrivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at [Pi]1 –level

Convex integration with avoidance and hyperbolic (4,6) distributions

This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3,5) and (3,6) distributions follows as a particular case. We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differential relation fails to be ample along some principal subspaces, we implement an "avoidance trick" to ensure that these are avoided during convex integration. Using this trick we provide the first example of a differential relation that is ample in coordinate directions but not in all directions, answering a question of Eliashberg and Mishachev. This so-called "avoidance trick" is part of a general avoidance framework, which is the main contribution of this article. Given any differential relation, the framework attempts to produce an associated object called an "avoidance template". If this process is successful, we say that the relation is "ample up to avoidance" and we prove that convex integration applies. The example of hyperbolic (4,6) distributions shows that our framework is capable of addressing differential relations beyond the applicability of classic convex integration

Convex integration with avoidance and hyperbolic (4,6) distributions

This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3,5) and (3,6) distributions follows as a particular case. We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differential relation fails to be ample along some principal subspaces, we implement an "avoidance trick" to ensure that these are avoided during convex integration. Using this trick we provide the first example of a differential relation that is ample in coordinate directions but not in all directions, answering a question of Eliashberg and Mishachev. This so-called "avoidance trick" is part of a general avoidance framework, which is the main contribution of this article. Given any differential relation, the framework attempts to produce an associated object called an "avoidance template". If this process is successful, we say that the relation is "ample up to avoidance" and we prove that convex integration applies. The example of hyperbolic (4,6) distributions shows that our framework is capable of addressing differential relations beyond the applicability of classic convex integration

Convex integration with avoidance and hyperbolic (4,6) distributions

This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3,5) and (3,6) distributions follows as a particular case. We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differential relation fails to be ample along some principal subspaces, we implement an "avoidance trick" to ensure that these are avoided during convex integration. Using this trick we provide the first example of a differential relation that is ample in coordinate directions but not in all directions, answering a question of Eliashberg and Mishachev. This so-called "avoidance trick" is part of a general avoidance framework, which is the main contribution of this article. Given any differential relation, the framework attempts to produce an associated object called an "avoidance template". If this process is successful, we say that the relation is "ample up to avoidance" and we prove that convex integration applies. The example of hyperbolic (4,6) distributions shows that our framework is capable of addressing differential relations beyond the applicability of classic convex integration

Visualización y ejemplificación a través de las TIC

Proyecto de innovación educativa en el que se han generado recursos digitales para su uso en el aula en varias asignaturas de Grado y Doble Grado impartidas en la Facultad de Matemáticas. El objetivo es implementar algunos ejemplos que en clase se hacen ordinaria y analógicamente en un formato digital que ayude a mejorar la visualización y comprensión de los conceptos matemáticos teóricos.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasFALSEsubmitte