Asteroseismology has opened a window on the internal physics of thousands of stars, by relating pulsations of stars to their internal physics. Mode identification, namely the process of associating a measured oscillation frequency to the corresponding mode geometry and properties, is the preliminary step of the seismic analysis. In upper main-sequence stars, that often rotate rapidly, this identification is challenging and largely incomplete, as modes assume complex geometries and frequencies shift under the combined influence of the Coriolis force and centrifugal flattening.

In this contribution, I will describe the various classes of mode geometries that emerge in rapidly rotating stars and their differences with slow rotators. After discussing how their frequencies and periods relate with structural quantities, allowing us to derive constraints on the stellar evolution, I will discuss the approaches developed towards inversion methods.

Over the past decade, asteroseismic inversion techniques have emerged as crucial tools to help identify the missing physics in our understanding of stellar evolution. In this talk, I will provide a comprehensive overview of the recent progress in asteroseismology and showcase the major advancements in the field, with a focus on novel methods for probing stellar structure and evolution. I will present new inferences into various different types of pulsating stars and improvements in our ability to infer internal stellar dynamics. I will also share our latest research on non-linear inversion methods and the application of inversions to massive stars. Finally, I will discuss the future of asteroseismology, including the expected yield of several forthcoming missions. This talk aims to highlight the remarkable progress in asteroseismology and stimulate discussions on future avenues for continued advancement.

The eleven-year solar activity cycle is known to affect the acoustic p-modes; higher activity is correlated with increase in mode frequencies and decrease in their lifetimes. This is also seen in the autocorrelation function of the integrated light. Recently, the solar cycle is also observed in travel-time measurements of p-mode wavepackets for multiple skips [1]. In this work, we first construct a forward model to explain the variation in travel-time measurements with solar activity. A simplified model is constructed by considering axisymmetric averages of the magnetic activity associated with perturbations in the near-surface wave-speed. The perturbations are constructed by longitudinally averaging synoptic magnetograms from SDO/HMI and the SOHO/MDI. The maximum correlation between observed and modeled travel-time shifts is as high as $0.92$ for some skips, much less for others. Subsequently, we setup an inverse problem to invert for the latitudinal distribution of solar activity from travel-time observations. This work is a first step towards the goal of retrieving stellar butterfly diagrams from asteroseismic observables.

[1] V. Vasilyev, L. Gizon, 2023, submitted.

Solar inertial modes offer new possibilities to probe the solar interior down to the tachocline, and can be used to constrain such properties as the differential rotation or the spectrum of turbulent energy throughout the convection zone. Linear analysis enables us to compute the discrete eigenfrequencies of these modes [1,2]. However, because the inertial modes overlap in the frequency domain, especially for high azimuthal order $m$, this is not enough: it is necessary to model the power spectral density in the whole inertial frequency range, which can be done by modelling the stochastic source of excitation of the modes by turbulent vorticity.

In this presentation, I will show how this can be achieved in a 2D spherical setting, based on the formalism by [3]. I will then describe how this formalism can be used to relate changes in turbulent properties, with a focus on the turbulent viscosity, to their effects on the whole inertial range power spectral density (forward problem), as well as discuss the corresponding inverse problem.

[1] L. Gizon, D. Fournier, M. Albekioni. Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorial $\beta$ plane, Astron. Astrophys 642: A178, 2020.

[2] Y. Bekki, R.H. Cameron, L. Gizon. Theory of solar oscillations in the inertial frequency range: Linear modes of the convection zone. Astron. Astrophys 662: A16, 2022.

[3] J. Philidet, and L. Gizon. Interaction of solar inertial modes with turbulent convection, A 2D model for the excitation of linearly stable modes. Astron. Astrophys 673: A124, 2023.