Abstract:
This paper studies the regularity of the minimum time function, $T(\cdot)$, for a control system with a general closed target, taking the state equation in the form of a differential inclusion. Our first result is a sensitivity relation which guarantees the propagation of the proximal subdifferential of $T$ along any optimal trajectory. Then, we obtain the local $C^2$ regularity of the minimum time function along optimal trajectories by using such a relation to exclude the presence of conjugate times.

Abstract:
In optimal control,sensitivity relations are usually understood as inclusions that identify the pair formed by the dual arc and the Hamiltonian, evaluated along the associated minimizing trajectory, as a suitable generalized gradient of the value function. In this paper, sensitivity relations are obtained for the Mayer problem associated with the differential inclusion $\dot x\in F(x)$ and applied to derive optimality conditions. Our first application concerns the maximum principle and consists in showing that a dual arc can be constructed for every element of the superdifferential of the final cost. As our second application, with every nonzero limiting gradient of the value function at some point $(t,x)$ we associate a family of optimal trajectories at $(t,x)$ with the property that families corresponding to distinct limiting gradients have empty intersection.

Abstract:
This paper investigates the value function, $V$, of a Mayer optimal control problem with the state equation given by a differential inclusion. First, we obtain an invariance property for the proximal and Fr\'echet subdifferentials of $V$ along optimal trajectories. Then, we extend the analysis to the sub/superjets of $V$, obtaining new sensitivity relations of second order. By applying sensitivity analysis to exclude the presence of conjugate points, we deduce that the value function is twice differentiable along any optimal trajectory starting at a point at which $V$ is proximally subdifferentiable. We also provide sufficient conditions for the local $C^2$ regularity of $V$ on tubular neighborhoods of optimal trajectories.

Abstract:
Climate change and potential global warming has become important in many forums both nationally and internationally. Though there has previously been some opposition to the existence of human caused climate change and resulting global warming as a threat to the earth’s population and survival, scientific evidence has more recently been found to be compelling. One of the key industries that may be affected by global warming and climate change is the tourism industry. This is becoming a growing concern in Florida, U.S.A. where potential rising sea levels may have a profound effect. This paper discusses the development, importance and implications of climate change, its relationship to the tourism and hospitality industry and provides a case study of the Florida lodging industry regarding mechanisms and responses that the lodging and resort sector of the industry has been taking in addressing climate change factors.

Abstract:
We introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are derived from the parametrisation of the moduli spaces by the bundle of measured geodesic laminations over Teichm\"uller space of S and can be viewed as analytic continuations of the shear coordinates on Teichm\"uller space. In terms of these coordinates the gravitational symplectic structure takes a particularly simple form, which resembles the Weil-Petersson symplectic structure in shear coordinates, and is closely related to the cotangent bundle of Teichm\"uller space. We then consider the mapping class group action on the moduli spaces and show that it preserves the gravitational symplectic structure. This defines three distinct mapping class group actions on the cotangent bundle of Teichm\"uller space, corresponding to different values of the curvature.

Abstract:
We describe what can be called the "universal" phase space of AdS3 gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact spatial sections, as well as the moduli spaces of multi-black-hole spacetimes are realized as submanifolds. The universal phase space is parametrized by two copies of the Universal Teichm\"uller space T(1) and is obtained from the correspondence between maximal surfaces in AdS3 and quasisymmetric homeomorphisms of the unit circle. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we obtain a relation between the generators of quasiconformal deformations in each T(1) sector and the chiral Brown-Henneaux vector fields. We also relate the charges arising in the holographic description (such as the mass and angular momentum of an AdS3 spacetime) to the periods of the quadratic differentials arising via the Bers embedding of T(1)xT(1). Our construction also yields a symplectic map from T*T(1) to T(1)xT(1) generalizing the well-known Mess map in the compact spatial surface setting.

Abstract:
Hodge's formula represents the gravitational MHV amplitude as the determinant of a minor of a certain matrix. When expanded, this determinant becomes a sum over weighted trees, which is the form of the MHV formula first obtained by Bern, Dixon, Perelstein, Rozowsky and rediscovered by Nguyen, Spradlin, Volovich and Wen. The gravity MHV amplitude satisfies the Britto, Cachazo, Feng and Witten recursion relation. The main building block of the MHV amplitude, the so-called half-soft function, satisfies a different, Berends-Giele-type recursion relation. We show that all these facts are illustrations to a more general story. We consider a weighted Laplacian for a complete graph of n vertices. The matrix tree theorem states that its diagonal minor determinants are all equal and given by a sum over spanning trees. We show that, for any choice of a cocycle on the graph, the minor determinants satisfy a Berends-Giele as well as Britto-Cachazo-Feng-Witten type recursion relation. Our proofs are purely combinatorial.

Abstract:
We give a description of gravitons in terms of an SL(2,C) connection field. The gauge-theoretic Lagrangian for gravitons is simpler than the metric one. Moreover, all components of the connection field have the same sign in front of their kinetic term, unlike what happens in the metric formalism. The gauge-theoretic description is also more economic than the standard one because the Lagrangian only depends on 8 components of the field per spacetime point as compared to 10 in the Einstein-Hilbert case. Particular care is paid to the treatment of the reality conditions that guarantee that one is dealing with a system with a hermitian Hamiltonian. We give general arguments explaining why the connection cannot be taken to be real, and then describe a reality condition that relates the hermitian conjugate of the connection to its (second) derivative. This is quite analogous to the treatment of fermions where one describes them by a second-order in derivatives Klein-Gordon Lagrangian, with an additional first-order reality condition (Dirac equation) imposed. We find many other parallels with fermions, e.g. the fact that the action of parity on the connection is related to the hermitian conjugation. Our main result is the mode decomposition of the connection field, which is to be used in forthcoming works for computations of graviton scattering amplitudes.