We prove the stability of Symp(X,w)∩Diff_0(X) for a one-point blow-up of irrational ruled surfaces and study their topological colimit. Non-trivial generators of π _0[Symp(X,w)∩Diff_0(X)] that differ from Lagrangian Dehn twists are detected.

We continue our previous work to prove that for any non-minimal ruled surface (M,ω), the stability under symplectic deformations of π_0, π_1 of Symp(M,ω) is guided by embedded J-holomorphic curves. Further, we prove that for any fixed sizes blowups, when the area ratio μ between the section and fiber goes to infinity, there is a topological colimit of Symp(M,ω_μ). Moreover, when the blowup sizes are all equal to half the area of the fiber class, we give a topological model of the colimit which induces non-trivial symplectic mapping classes in Symp(M,ω) ∩ Diff_0(M), where Diff_0(M) is the identity component of the diffeomorphism group. These mapping classes are not Dehn twists along Lagrangian spheres.

AbstractWe prove packing stability for rational symplectic manifolds. This will rely on a general symplectic embedding result for ellipsoids which assumes only that there is no volume obstruction and that the domain is sufficiently thin relative to the target. We also obtain easily computable bounds for the Embedded Contact Homology capacities which are sufficient to imply the existence of some symplectic volume filling embeddings in dimension 4.

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in \N$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the ECH capacities give sharp inequalities for embedding ellipsoids into polydisks.

Consider symplectic ruled surfaces M-lambda(g) = (Sigma(g) x S-2, lambda sigma(Sigma g) circle plus sigma S-2) such that Sigma(g) has area lambda and S-2 has area 1. We show that for k >= [g/2] the homotopy type of the symplectomorphism groups G g. of M g. is constant as. increases in the interval (k, k + 1], thus generalizing an existent result of Abreu-McDuff for the rational ruled surfaces with g = 0. We also investigate the changes in the groups p* G g. as. passes an integer k and show the existence of higher Samelson products in pi(4k+ 2g)G(lambda)(g). that exist only for lambda in the range (k, k + 1]. To prove these results we introduce a refinement of the negative inflation technique introduced by Li-Usher.

We study symplectic embeddings of ellipsoids into balls. In the main construction, we show that a given embedding of 2m-dimensional ellipsoids can be suspended to embeddings of ellipsoids in any higher dimension. In dimension 6,s if the ratio of the areas of any two axes is sufficiently large then the ellipsoid is flexible in the sense that it fully fills a ball. We also show that the same property holds in all dimensions for sufficiently thin ellipsoids E(1,..., a). A consequence of our study is that in arbitrary dimension a ball can be fully filled by any sufficiently large number of identical smaller balls, thus generalizing a result of Biran valid in dimension 4.

We provide a new way of understanding the multiplicative structure of the rational homotopy groups pi(*)(X-lambda) circle times Q for a family of topological spaces, once we know enough about their additive structure. This allows us to interpret the condition of realizing as an A(k) map a multiple of a map f : S-1 -> G between two topological groups in terms of the existence of a rational Whitehead product of order k. Our main example will be when the X-lambda are classifying spaces of symplectomorphism groups BSymp(Sigma(g) x S-2, omega(lambda)) where omega(lambda) is a symplectic deformation on the trivial ruled surface Sigma(g) x S-2. Our method of detecting nontriviality is based on computations of equivariant Gromov-Witten invariants. One application gives a homotopy-theoretic counterpart to a geometric result obtained by Karshon. Another application concerns the ring structure of H-*(BSymp(S(2)xS(2),omega(lambda))).

Oscillatory regulatory networks have been discovered in many regulatory pathways. Due to their enormous complexity, it is necessary to study their dynamics by means of highly simplified models. These models have received particular value because artificial regulatory networks can be engineered experimentally. In this paper, we study dynamical properties of an artificial regulatory oscillator called repressilator. We have shown that oscillations arise from the existence of an absorbing toruslike region in the phase space of the model. This geometric structure requires monotonic repression at all promoters and the absence of any regulatory connections apart from a cyclic repression loop. We show that oscillations collapse as only weak extra connections are introduced if there is imbalance between the attended concentrations and those sufficient for saturation of the promoters. We found that a pair of diffusively coupled repressilators displays synchronization properties similar to those of relaxation oscillators if the regulatory connections in the cyclic repression loop are strong. Thus, the role of strengthening these connections can be viewed as introducing time scale separation among variables. This may explain controversial synchronization properties reported for repressilators in earlier studies.

The Repressilator is a genetic regulatory network used to model oscillatory behavior of more complex regulatory networks like the circadian clock. We prove that the Repressilator equations undergo a supercritical Hopf bifurcation as the maximal rate of protein synthesis increases, and find a large range of parameters for which there is a cycle.

Consider any symplectic ruled surface $(M^g_{\lambda},\omega_{\lambda})$ given by $(\Sigma_g \times S^2, \lambda \sigma_{\Sigma_g} \oplus \sigma_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EGW_{g,0}(M^g_{\lambda};H_k, A-kF)$ for all hamiltonian circle actions $H_k$ on $M^g_{\lambda}$, where $A=[\Sigma_g \times pt]$ and $F= [pt \times S^2]$. We use these invariants to show the nontriviality of certain higher order Whitehead products that live in the homotopy groups of the symplectomorphism groups $G_{\lambda}^g$, $g \geq 0$. Our results are sharper when $g=0,1$ and enable us to answer a question posed by D.McDuff in the case $g=1$ and provide a new interpretation of the multiplicative structure in the ring $H^*(BG^0_{\lambda} ;\Q)$ found by Abreu-McDuff.

Symplectic manifolds are the natural domains of the modern mathematical formulation of classical mechanics, and they play a prominent role in many areas of mathematics. Since the introduction of the theory J–holomorphic curves by Gromov in [5], a tremendous amount of progress has been made in the study of these manifolds and the maps which preserve their symplectic structures. This progress has been particularly dramatic in the case of symplectic manifolds of dimension 4. For example, for S×S, the symplectic forms are classified (see [5], [14]), their symplectomorphism groups are well understood (see [5], [3]), and their Lagrangian spheres are all known to be symplectically equivalent (see [6]). Ruled symplectic 4–manifolds have also been completely classified (see [11]). The proofs of these results all have the same starting point; the existence of foliations by J–holomorphic spheres for all tamed almost complex structures. There have been several attempts to establish the existence of such foliations in more general settings. It was recently shown in [7] that such existence statements do not hold for foliations by J–holomorphic spheres of manifolds of dimension greater than 4. For index reasons, one can also not expect to find foliations by J–holomorphic curves of higher genus. To overcome this latter limitation, in the setting of symplectizations of a contact 3–manifolds, H. Hofer proposed in [8] to replace the standard J–holomorphic map equation with a parameterized version where the parameter takes values in the space of harmonic 1–forms on the domain. More precisely, they introduce the notion of an H–holomorphic map which is a map u from a Riemann surface (Σ, j) to the symplectization (R×Z, J) of a contact manifold Z such that ∂̄Ju takes values inH = H0,1(u∗C)), the space of harmonic (0, 1)–forms on Σ with values in the trivial bundle u∗(C), where C ⊂ T (R×Z) is the trivial complex vector bundle generated by the R–factor. For H–holomorphic maps many new analytic difficulties arise. For example, local intersections of such maps need not be positive, and the space of these maps is in general not compact (see [15]). Despite these difficulties,H–holomorphic maps have been used to obtain foliations of contact 3–manifolds. In particular, in both [1] and [16], it is shown that every contact structure on a 3–manifold admits a contact form and an almost complex structure which support an open book decomposition whose pages are embedded H–holomorphic maps. The use of parameterized versions of the J–holomorphic map equation is not new. For example, they were used to find non–trivial elements in symplectomorphism groups by O. Buse in [4]. As well, the parameter space introduced in [10] was recently used to compute the Gromov-Witten invariants of Kähler surfaces in [12].