Career: hybrid surfaces to control cell adhesion and function

Issued as final reportNational Science Foundation (U.S.

Lifting Hamiltonian loops to isotopies in fibrations

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $\Psi$ of $H$ determines a $G$-equivariant principal bundle ${\mathcal P}$ on $M$ endowed with a $G$-invariant connection. We consider subgroups ${\mathcal G}$ of the diffeomorphism group ${\rm Diff}(M)$, such that, each vector field $Z\in{\rm Lie}({\mathcal G})$ admits a lift to a preserving connection vector field on ${\mathcal P}$. We prove that #\,\pi_1({\mathcal G})\geq #\,\Psi(Z(G)). This relation is applicable to subgroups ${\mathcal G}$ of the Hamiltonian groups of the flag varieties of a semisimple group $G$. Let $M_{\Delta}$ be the toric manifold determined by the Delzant polytope $\Delta$. We put $\varphi_{\bf b}$ for the the loop in the Hamiltonian group of $M_{\Delta}$ defined by the lattice vector ${\bf b}$. We give a sufficient condition, in terms of the mass center of $\Delta$, for the loops $\varphi_{\bf b}$ and $\varphi_{\bf\tilde b}$ to be homotopically inequivalent.Comment: 23 pages, 1 figure. To be published in Int. J. Geom. Methods Mod. Physic

Characteristic number associated to mass linear pairs

Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\mathbf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\Delta,\,{\mathbf b})$. Under certain hypotheses, we prove the equivalence between the fact that $(\Delta,\,{\mathbf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions. I.} Int. Math. Res. Not. IMRN 8 (2010) 1506-1574.) and the vanishing of a characteristic number of $E$. Denoting by ${\rm Ham}(M_{\Delta})$ the Hamiltonian group of the symplectic manifold defined by $\Delta$, we determine loops in ${\rm Ham}(M_{\Delta})$ that define infinite cyclic subgroups in $\pi_1({\rm Ham}(M_{\Delta}))$, when $\Delta$ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch surface, (ii) it is a $\Delta_p$ bundle over $\Delta_1$, (iii) $\Delta$ is the truncated simplex associated with the one point blow up of ${\mathbb C}P^n$.Comment: Revised version which will appear in ISRN Geometr