  We present the first results from a very deep Chandra X-ray observation of
the core of the Perseus cluster of galaxies. A pressure map reveals a clear
thick band of high pressure around the inner radio bubbles. The gas in the band
must be expanding outward and the sharp front to it is identified as a shock
front, yet we see no temperature jump across it; indeed there is more soft
emission behind the shock than in front of it. We conclude that in this inner
region either thermal conduction operates efficiently or the co-existing
relativistic plasma seen as the radio mini-halo is mediating the shock. If
common, isothermal shocks in cluster cores mean that we cannot diagnose the
expansion speed of radio bubbles from temperature measurements alone. They can
at times expand more rapidly than currently assumed without producing
significant regions of hot gas. Bubbles may also be significantly more
energetic. The pressure ripples found in earlier images are identified as
isothermal sound waves. A simple estimate based on their amplitude confirms
that they can be an effective distributed heat source able to balance radiative
cooling. We see multiphase gas with about 10^9 Msun at a temperature of about
0.5 keV. Much, but not all, of this cooler gas is spatially associated with the
optical filamentary nebula around the central galaxy, NGC 1275. A residual
cooling flow of about 50 Msun/yr may be taking place. A channel is found in the
pressure map along the path of the bubbles, with indications found of outer
bubbles. The channel connects in the S with a curious cold front.
