  Semantic communications represent a significant breakthrough with respect to
the current communication paradigm, as they focus on recovering the meaning
behind the transmitted sequence of symbols, rather than the symbols themselves.
In semantic communications, the scope of the destination is not to recover a
list of symbols symbolically identical to the transmitted ones, but rather to
recover a message that is semantically equivalent to the semantic message
emitted by the source. This paradigm shift introduces many degrees of freedom
to the encoding and decoding rules that can be exploited to make the design of
communication systems much more efficient. In this paper, we present an
approach to semantic communication building on three fundamental ideas: 1)
represent data over a topological space as a formal way to capture semantics,
as expressed through relations; 2) use the information bottleneck principle as
a way to identify relevant information and adapt the information bottleneck
online, as a function of the wireless channel state, in order to strike an
optimal trade-off between transmit power, reconstruction accuracy and delay; 3)
exploit probabilistic generative models as a general tool to adapt the
transmission rate to the wireless channel state and make possible the
regeneration of the transmitted images or run classification tasks at the
receiver side.
