Helioseismology, the study of the global solar oscillations, constitutes a formidable tool for the investigation of the internal structure and dynamics of the Sun. The oscillatory motions with periods around 5 minutes detected at the surface are due to standing acoustic waves (p-modes) trapped in resonant cavities between the Sun's surface and an inner turning point, whose depth is determined by the local speed of sound and the frequency and horizontal wave number of the wave. Since each wave, characterized by a specific frequency and wave number, propagates through a different region of the Sun, it probes the physical properties of the crossed medium, like temperature and composition, and thus makes it possible to deduce the internal stratification and dynamics of the Sun from the spectrum of resonant modes. 

In my thesis I approached issues related to the solar modeling and the helioseismic inversions. The inversion codes I developed for determining the stratification of the Sun and the radial profile of its angular velocity (Paterno', Sofia & Di Mauro 1996), were originally based on the optimally localized averaging technique (OLA) (Backus & Gilbert 1970). The latest version of the inversion algorithms (Di Mauro, Dziembowski, Paterno' 1998) is based on the SOLA method (Pijpers & Thompson, 1992), and it is more flexible to probe different physical conditions.

These inversion codes have been applied to a large amount of observed frequencies, now available from a variety of heliosismology experiments on Earth (GONG, IRIS, BISON, LOWL) and on space (MDI and GOLF instruments flying on board of SOHO satellyte), which offer a prime opportunity to enhance our understanding of the structure and dynamics of the Sun.

The results so far have shown that the solar structure is remarkably close to the predictions of the standard solar models, consistent within 1 % (Di Mauro 2000). Despite such overall success, this discipline has not yet exhausted its resources, since helioseismic results clearly suggest further refinements of the solar models.

The detailed structure of the convective zone and of the near-surface region is still quite uncertain, since there remains substantial ambiguity associated with modelling the convective flux, taking into account the non-adiabatic effects, explaining the excitation and damping of the solar oscillations and defining an appropriate equation of state to describe the thermodynamic properties of the solar structure.

The attempts to restore the solar core conditions, up to now, have been contradictory too. In fact p modes (as opposed to gravity modes, g modes) are not very sensitive to the core of the Sun. This indicates the necessity of using more accurate low degree p-mode data and to continue to investigate for the presence of g modes.

In addition, there is still much work ahead in getting a detailed understanding of the Sun's rotation. Some rotational features like the temporal changes which occur near the base of the convective envelope have not been yet explained.
Ever more precise helioseismic observations from ground and space can help us to reconstruct the complete picture of the Sun and, at the end, to solve the most discussed open questions in solar physics such as the solar neutrino problem, the history of the Sun's angular momentum, and the solar cycle generation mechanism, through the interaction of the convective motions with the rotation inside the Sun.