As stars of all masses evolve off the main sequence, they develop cool, extended envelopes that reprocess most of the emitted starlight into infrared radiation observable with TPF-I. Low- and intermediate-mass stars (M = 0.8 - 8 M) make up > 90 per cent of all the stars which have died in the Universe up to the present time. At the end of their main sequence life-time, they enter the high-luminosity asymptotic giant branch (AGB) phase. During the short so-called superwind phase, the stars eject their hydrogen-rich outer layers to reveal the chemically enriched deeper layers of the star. These stars obtain the highest luminosity and the largest diameter in their existence with the size or nearly an AU. AGB stars are expected to either vaporize, or swallow their planetary systems. Emission from silicates, SiC, PAHs, and some ices will provide powerful probes of physical and chemical evolution of these objects. In the closest AGB stars, TPF-I will resolve the photospheres; in more distant objects, it will probe their winds and dying planetary systems in great details.

After this final burst of activity these stars evolve into hot, compact white dwarfs with masses in the range 0.6 - 1.4 M_o. The expanding ejecta surrounding the star becomes ionized and forms a planetary nebula before dispersing into the interstellar medium (ISM).

Recently, the Spitzer Space Telescope had detected infrared excess emission from about 15 to 20% of old white dwarf stars near the Sun (Reach et al. 2006; Mullally et al. 2006). This emission indicates the presence of debris disks consisting of mostly large-solid particles that have resisted being dragged into the central white dwarf by the Poynting-Robertson drag for the age of the white dwarf. Such debris disks surrounding aging white dwarfs may trace the remnants of planetary systems that were destroyed during the post-main-sequence red-giant phase of their parent stars. There are hints that in-spiraling solids may be responsible for anomalous metal-abundances in white dwarf atmospheres. TPF-I / Darwin interferometric imaging may resolve these disks, determine their structure, and constrain their compositions. Such observations may shed independent light on the abundances of planetary systems and that have been destroyed millions to billions of years ago.

AGB mass-loss (e.g. Zijlstra et al. 2006) determines the mass distribution of stellar remnants, including the lower mass limit of type II supernovae progenitors. Stellar mass-loss also drives Galactic evolution through replenishment and chemical enrichment of the ISM. Such mass-loss contributes roughly half the total gas recycled by all stars (Maeder 1992), creates an amount of carbon roughly equal to that produced by supernovae and Wolf-Rayet stars (Dray et al. 2003; Gavilán, Buell, & Mollá 2005) and is the main source of carbonaceous interstellar dust (Dwek 1998; Edmunds 2001).

TPF-I / Darwin will resolve AGB stars at the distance of the Galactic center. These observations will directly measure the impacts of the high-pressure Galactic center environments, radiation, and outflows on the structure of AGB star envelopes. VLA and ground-based [NeII] observations have already revealed cometary tails around some evolved stars such as IRS7 (Yusef-Zadeh & Morris 1991). Interferometry will enable the study of such tails to be used as diagnostics of the environments.

There are about 200 moderately evolved AGB stars (Mira variables) known within 1 Kpc. Both oxygen-rich (M-type) and carbon-rich (C-type) AGB stars show spectral features around 9.7 micron and 11.3 micron due to silicate particles and SiC grain as well as spectral lines from other atomic and molecular species. IRAS detected more than 105 new AGB candidates. The improved spectral resolution of ISO and Spitzer allowed the study of the AGB population of the Magellanic Clouds and other Local Group galaxies at mid-infrared wavelengths.

Darwin/TPF will detect the circumstellar envelope (CSE) of AGB stars located well beyond the Local Group where these stars are being found photometrically as galaxy members (i.e. in the Sculptor group at 2.5 Mpc and in the M81 group at 4 Mpc). Furthermore, Darwin/TPF will provide detailed maps of the distribution dust and gas within the envelopes of AGB stars within the Galaxy. This will be essential information to tie basic stellar parameters to the properties of mass-loss. This is not only important for the usage of the properties of mass loss for addressing key astrophysical questions (see above), but also to advance theoretical models (e.g. Sandin & Hofner 2003) that cannot predict mass-loss rates from stellar parameters. The later is largely due to the complicated physics of the interplay of stellar pulsation, shock waves, dust formation and radiation pressure which combined all drive the mass-loss.