Understanding the physical and evolutionary properties of Hot Stellar Systems (HSS) is a major challenge in astronomy. We studied the dataset on 13 456 HSS of Misgeld & Hilker (2011, MNRAS, 414, 3 699) that includes 12 763 candidate globular clusters using stellar mass ( $M_s$ ), effective radius ( $R_e$ ) and mass-to-luminosity ratio ( $M_s/L_\nu$ ), and found multi-layered homogeneous grouping among these stellar systems. Our methods elicited eight homogeneous ellipsoidal groups at the finest sub-group level. Some of these groups have high overlap and were merged through a multi-phased syncytial algorithm motivated from Almodóvar-Rivera & Maitra (2020, JMLR, 21, 1). Five groups were merged in the first phase, resulting in three complex-structured groups. Our algorithm determined further complex structure and permitted another merging phase, revealing two complex-structured groups at the highest level. A nonparametric bootstrap procedure was also used to estimate the confidence of each of our group assignments. These assignments generally had high confidence in classification, indicating great degree of certainty of the HSS assignments into our complex-structured groups. The physical and kinematic properties of the two groups were assessed in terms of $M_s$ , $R_e$ , surface density and $M_s/L_\nu$ . The first group consisted of older, smaller and less bright HSS while the second group consisted of brighter and younger HSS. Our analysis provides novel insight into the physical and evolutionary properties of HSS and also helps understand physical and evolutionary properties of candidate globular clusters. Further, the candidate globular clusters (GCs) are seen to have very high chance of really being GCs rather than dwarfs or dwarf ellipticals that are also indicated to be quite distinct from each other.

We cannot presume to summarize all of the science we've discussed in the talks, posters, and informal discussions. Here, we discuss a few of the themes that emerged, concentrating on the theoretical basis that Wojtek Dziembowiski and his colleagues have developed and explored over the past 40+ years. We connect those with observational results – especially those from recent ground-based surveys and space-based missions that have revolutionized the study of stellar variability.

Division V on Variable Stars consists of Commission 27, also called Variable Stars, and Commission 42, Close Binary Stars. The former deals with stars whose variations are intrinsic, whereas in the latter the variations are caused by the interactions between the components in the binary or multiple star system. There may be cases where the assignment of an object to one of the two Commissions may be in doubt. For example, the observation of pulsating stars in eclipsing binaries within nearby galaxies, or the relation between some types of oscillation modes and membership to binary systems, continue to be widely discussed.

The meeting started at 14h00. The president welcomed the participants to the business meeting of C27, and acknowledged the members of the Organizing Committee. Special thanks were expressed to the outgoing members: Conny Aerts, Peter Martinez, and Seetha Somasundaram.

The Organizing Committee of Commission 27 has decided to again provide a somewhat abbreviated bibliography as part of this triennial report, as astronomy-centered search engines and on-line publications continue to blossom. We focus on selected highlights in variable star research over the past three years. Further results can be found in numerous proceedings of conferences held in the time frame covered by this report.

Division V deals with all aspects of stellar variability, either intrinsic or due to eclipses by its companion in a binary system. In the case of intrinsic stellar variability the analysis of pulsating stars, surface inhomogeneities, stellar activity and oscillations are considered. For close binaries, classical detached eclipsing binaries are studied as well as more interacting systems, like contact and semi-detached binaries, or those with compact components, like cataclysmic variables and X-ray binaries, including the physics of accretion processes.

The meeting started at 16h00. The president welcomed the 24 participants to the business meeting of Commission 27. After the approval of the agenda, she gave an overview of the activities of Commission 27 of the past three years.

White dwarfs rotate. The angular momentum in single white dwarfs must originate early in the life of the star, but also must be modified (and perhaps severely modified) during the many stages of evolution between birth as a main–sequence star and final appearance as a white dwarf. Observational constraints on the rotation of single white dwarf stars come from traditional spectroscopy and from asteroseismology, with the latter providing hints of angular velocity with depth. Results of these observational determinations, that white dwarfs rotate with periods ranging from hours to days (or longer), tells us that the processes by which angular momentum is deposited and/or drained from the cores of AGB stars are complex. Still, one can place strong limits on these processes by considering relatively simple limiting cases for angular momentum evolution in prior stages, and on subsequent angular momentum evolution in the white dwarfs. These limiting-case constraints will be reviewed in the context of the available observations.

The interplay between theory and observation has been particularly close in asteroseismology, ranging from white dwarfs through roAp stars. This trend should continue as the field expands to include new classes such as solar-type stars. This synergistic relationship relies on communication between theorists and observers, but also on healthy skepticism in both directions. Here, I outline a (somewhat personal) view of the theoretical side - what kinds of observations are most helpful for optimizing utility of theoretical research efforts ? Where might theory be of closer guidance as observing programs are planned ?

Today, we are beginning to probe the interior of stars through the new science of stellar seismology. Certain stars, ranging from our own Sun to white dwarfs, undergo natural vibrations that can be detected with sensitive time-series photometry and/or spectroscopy. Since the signal we seek is an unbroken time-series to allow determination of the vibration frequencies, data from a single-site is usually incapable of uniquely identifying the pulsation modes, no matter how large the telescope being used. In many cases, the observational goals can be achieved using small-ish telescopes in well-coordinated global networks. Here, I briefly describe the work of one such international network of observatories and scientists known as the Whole Earth Telescope (WET). With the WET, we have sounded out the interiors of a large number of nonradially pulsating stars. Over the past 14 years, WET has observed dozens of stars in 20 separate observing campaigns. Our team has wide span of interests, and has observed several other classes of objects such as delta Scuti stars, CV stars, pulsating sdB stars, and rapidly oscillating Ap stars.

The development of helioseismology as a forefront area of solar and stellar astronomy has been fascinating to watch. Helioseismology takes full advantage of the benefits of lots of photons and a spatially resolved target. These allow measurement of tiny amplitude photospheric velocity and flux variations, and the precise identification of the oscillation modes corresponding to each observed frequency.

As we move into the next century of stellar astronomy, it is helpful to consider the modes by which stellar astronomy has advanced in the modern era. At this conference, our attention has been focussed on precision measurement of stellar fundamental properties such as mass, luminosity, and radius. This mode of inquiry is one of several that have borne fruit. In this review, I will discuss some of the topics that have arisen here with respect to how our understanding of stellar interiors progresses through interaction between theory and observation.

The first two pulsating planetary-nebula nuclei (PNNs), those of K 1-16 and Lo 4, were discovered by Grauer & Bond (1984) and Bond & Meakes (1990). They are nonradial multiperiodic g-mode pulsators, with typical periods near 25–31 min and low amplitudes (up to ∼ 0.05–0.1 mag). These PNNs have extremely high temperatures (Teff > 100,000 K), and are hydrogen-deficient with high abundances of C and O. Their spectra and pulsational properties are very similar to those of the pulsating GW Vir (PG 1159–035) white dwarfs.

During the final evolution of most stars, they shed their outer skin and expose their core of the hot ashes of nuclear burning. As these hot and very dense cores cool into white dwarf stars, they go through episodes of multiperiodic, nonradial g-mode pulsation. The tools of stellar seismology allow us to use the pulsation spectra as powerful probes into the deep interiors of these stars. Progress in white dwarf seismology has required significant international cooperation, since another consequence of the complex pulsations of these stars is decoding the true pulsation frequencies requires a coordinated global effort involving high-speed photometric observations. Through one such effort, the Whole Earth Telescope project, we have located subsurface composition changes, detected differential rotation and magnetic fields, and measured fundamental quantities such as stellar mass, luminosity, and distance to extraordinary accuracy.

The nonradially pulsating GW Vir (a.k.a. DOV, or pulsating PG1159) stars and the K1–16 type variable stars provide an unparalleled opportunity to probe the internal structure of planetary nebula central stars and their immediate descendants, hot pre-white dwarfs. Recent global photometry campaigns have provided data of sufficient quality to severely constrain evolutionary models of these stars.

The variable white dwarfs repeatedly force theory to conform to their observed properties so that further progress can be made in understanding the structure and evolution of all white dwarfs. We use the term “understanding” in a loose sense here because, as we will show, both observational constraints and interpretation of the observations vis-à-vis theory contribute to uncertainties in our understanding at this time. In any case, recent progress in this field (sometimes called white dwarf seismology) has provided some fascinating insights into the evolutionary and structural properties of white dwarfs and their progenitors. This short review is our attempt to describe recent progress made in the interaction of theory with observations.

Asymptotic analysis of the equations of nonradial adiabatic oscillation shows that there is a “characteristic period spacing” for g-modes with the same degree (l) and consecutive values of the radial wavenumber (n). If modes with the same l but different n are present in a pulsating star, then comparison of the characteristic period spacing in appropriate stellar models with period differences between the observed pulsation periods can provide mode identifications, and thereby constrain other physical properties of the star. A simple empirical model of the observed period spectrum of the hot white dwarf star PG1159-035 yields two statistically significant mean period intervals of 21.0±0.3 or 8.8±0.1s. We also evaluate the period interval for models of hot white dwarfs that are representative of PG1159 stars. The observed intervals correspond very closely to those derived from the theoretical models. This analysis indicates that the mass of PG1159-035 is 0.60M⊙ and that it is either a dipole (l=1) or l=3 pulsator (or both!), pulsating in high radial overtone g-modes.