A local to global argument on low dimensional manifolds

For an oriented manifold $M$ whose dimension is less than $4$, we use the contractibility of certain complexes associated to its submanifolds to cut $M$ into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation theory which says that the natural map between classifying spaces $\mathrm{B}\text{Homeo}^{\delta}(M)\to \mathrm{B}\text{Homeo}(M)$ induces a homology isomorphism where $\text{Homeo}^{\delta}(M)$ denotes the group of homeomorphisms of $M$ made discrete. Our proof shows that in low dimensions, Thurston's theorem can be proved without using foliation theory. Finally, we show that this technique gives a new perspective on the homotopy type of homeomorphism groups in low dimensions. In particular, we give a different proof of Hacher's theorem that the homeomorphism groups of Haken $3$-manifolds with boundary are homotopically discrete without using his disjunction techniques.Comment: Thoroughly revised. To appear in Transactions of the AM

Homological stability and stable moduli of flat manifold bundles

We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be detected on flat $(\#^g S^n \times S^n)$-bundles for $g$ large enough.Comment: Final version, to appear in Advances in Mathematic

Forecasting the economy with mathematical models: is it worth the effort?

Forecasting ; Mathematical models

Dynamical and cohomological obstructions to extending group actions

We study cohomological obstructions to extending group actions on the boundary $\partial M$ of a $3$-manifold to a $C^0$-action on $M$ when $\partial M$ is diffeomorphic to a torus or a sphere. In particular, we show that for a $3$-manifold $M$ with torus boundary which is not diffeomorphic to a solid torus, the torus action on the boundary does not extend to a $C^0$-action on $M$.Comment: Minor correction to statement of Theorem 1.

Experimental comparison between proportional and PWM-solenoid valves controlled servopneumatic positioning systems

The performance of the Dynamical Adaptive Backstepping-Sliding Mode Control (DAB-SMC) scheme for positioning of a pneumatic cylinder regulated by two types of PWM-solenoid valves is experimentally investigated. The goal is to study the compromise in controller’s performance as the system moves from using a proportional valve to employing the low-cost PWM-solenoid valves. Sinusoidal and multiple-step inputs are used as the reference position trajectories. Experimental results show that the DAB-SMC scheme works best with the proportional valve. The performance, however, deteriorates by more than twofold, once the system utilizes PWM- solenoid valves of 3/2-way or 2/2-way configurations. From this study, tradeoff between performances of different types of valves applied on a DAB-SMC scheme-controlled servo positioning system is successfully documented. This information helps to configure appropriate servopneumatic system for positioning applications