The Hitchin map for one-nodal base curves of compact type

Studying moduli spaces of semistable Higgs bundles (E, \phi) of rank n on a smooth curve C, a key role is played by the spectral curve X (Hitchin), because an important result by Beauville-Narasimhan-Ramanan allows us to study isomorphism classes of such Higgs bundles in terms of isomorphism classes of rank-1 torsion-free sheaves on X. This way, the generic fibre of the Hitchin map, which associates to any semistable Higgs bundle the coefficients of the characteristic polynomial of \phi, is isomorphic to the Jacobian of X. Focusing on rank-2 Higgs data, this construction was extended by Barik to the case in which the curve C is reducible, one-nodal, having two smooth components. Such curve is called of compact type because its Picard group is compact. In this work, we describe and clarify the main points of the construction by Barik and we give examples, especially concerning generic fibres of the Hitchin map. Referring to Hausel-Pauly, we consider the case of SL(2,C)-Higgs bundles on a smooth base curve, which are such that the generic fibre of the Hitchin map is a subvariety of the Jacobian of X, the Prym variety. We recall the description of special loci, called endoscopic loci, such that the associated Prym variety is not connected. Then, letting G be an affine reductive group having underlying Lie algebra so(4,C), we consider G-Higgs bundles on a smooth base curve. Starting from the construction by Bradlow-Schaposnik, we discuss the associated endoscopic loci. By adapting these studies to a one-nodal base curve of compact type, we describe the fibre of the SL(2,C)-Hitchin map and of the G-Hitchin map, together with endoscopic loci. In the Appendix, we give an interpretation of generic spectral curves in terms of families of double covers.

Potential Analysis for Hypoelliptic Second Order PDEs with Nonnegative Characteristic Form

In this Thesis we consider a class of second order partial differential operators with non-negative characteristic form and with smooth coefficients. Main assumptions on the relevant operators are hypoellipticity and existence of a well-behaved global fundamental solution. We first make a deep analysis of the L-Green function for arbitrary open sets and of its applications to the Representation Theorems of Riesz-type for L-subharmonic and L-superharmonic functions. Then, we prove an Inverse Mean value Theorem characterizing the superlevel sets of the fundamental solution by means of L-harmonic functions. Furthermore, we establish a Lebesgue-type result showing the role of the mean-integal operator in solving the homogeneus Dirichlet problem related to L in the Perron-Wiener sense. Finally, we compare Perron-Wiener and weak variational solutions of the homogeneous Dirichlet problem, under specific hypothesis on the boundary datum.

Circadian rhythms, sleep and autonomic function in progressive supranuclear palsy: characteristic features and reciprocal interactions

Background: Progressive supranuclear palsy (PSP) is a rare neurodegenerative condition. The aims of this study were to evaluate the association between sleep, the circadian system and autonomic function in a cohort of PSP patients. Methods: Patients with PSP diagnosed according to consensus criteria were recruited prospectively and retrospectively and performed the following tests: body core temperature (BcT), sleep-wake cycle, systolic and diastolic blood pressure (SBP, DBP) continuous monitoring for 48 h under controlled environmental conditions; cardiovascular reflex tests (CRTs). The analysis of circadian rhythmicity was performed with the single cosinor method. For state-dependent analysis, the mean value of variables in each sleep stage was calculated as well as the difference to the value in wake. Results: PSP patients presented a reduced total duration of night sleep, with frequent and prolonged awakenings. During daytime, patients had very short naps, suggesting a state of profound sleep deprivation across the 24-h. REM sleep behaviour disorder was found in 15%, restless legs syndrome in 46%, periodic limb movements in 52% and obstructive sleep apnea in 54%. BcT presented the expected fall during night-time, however, compared to controls, mean values during day and night were higher. However BcT state-dependent modulation was maintained. Increased BcT could be attributed to an inability to properly reduce sympathetic activity favoured by the sleep deprivation. At CRTs, PSP presented mild cardiovascular adrenergic impairment and preserved cardiovagal function. 14% had non-neurogenic orthostatic hypotension. Only 2 PSP presented the expected BP dipping pattern, possibly as a consequence of sleep disruption. State-dependent analysis showed a partial loss of the state-dependent modulation for SBP. Discussion: This study showed that PSP presented abnormalities of sleep, circadian rhythms and cardiovascular autonomic function that are likely to be closely linked one to another.