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A Boolean network is a discrete dynamical system operating on vectors of Boolean variables. The action of a Boolean network can be conveniently expressed as a system of Boolean update functions, computing the new values for each component of the Boolean vector as a function of the other components. Boolean networks are widely used in modeling biological systems that can be seen as consisting of entities which can be activated or deactivated, expressed or inhibited, on or off. P systems on the other hand are classically introduced as a model of hierarchical multiset rewriting. However, over the years the community has proposed a wide range of P system variants including diverse ingredients suited for various needs. In this work, we propose a new variant—Boolean P systems—specifically designed for reasoning about sequential controllability of Boolean networks, and use it to first establish a crisp formalization of the problem, and then to prove that the problem of sequential controllability is PSPACE-complete. We further claim that Boolean P systems are a demonstration of how P systems can be used to construct ad hoc formalisms, custom-tailored for reasoning about specific problems, and providing new advantageous points of view.

We derive a weak-strong uniqueness and stability principle for the Landau equation in the soft potentials case (including Coulomb interactions). The distance between two solutions is measured by their relative entropy, which to our knowledge was never used before in stability estimates. The logarithm of the strong solution is required to have polynomial growth while the weak solution can be any H-solution with sufficiently many moments at initial time. Since we require a substantial amount of regularity on the strong solution, we also provide an example of sufficient conditions on the initial data that ensure this regularity in the Coulomb (and very soft potentials) case

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove the integral Hodge conjecture for one-cycles on complex Jacobian varieties, and the real integral Hodge conjecture modulo torsion for real abelian threefolds. For the theory of real algebraic cycles, and for several other purposes in real algebraic geometry, it is useful to have moduli spaces of real varieties to our disposal. Insight in the topology of a real moduli space provides insight in the geometry of a real variety that defines a point in it, and the other way around. In the moduli space of real abelian varieties, as well as in the Torelli locus contained in it, we prove density of the set of moduli points attached to abelian varieties containing an abelian subvariety of fixed dimension. Moreover, we provide the moduli space of stable real binary quintics with a hyperbolic orbifold structure, compatible with the period map on the locus of smooth quintics. This structure identifies the moduli space of stable real binary quintics with a non-arithmetic ball quotient.