Centered Stationary Gaussian Processes (SGPs) are real valued
continuous stochastic processes on R^d or Z^d whose marginals
are centered normal random variables. Gaussianity occurs when
a process is obtained as a sum of many infinitesimal independent
contributions, and Stationarity occurs when the phenomenon in
question is invariant under translations in time or in space.
This makes stationary Gaussian processes an excellent model for
stationary noise and random signals, placing them amongst the
most well studied stochastic processes.

Level crossings of SGPs have been extensively studied for several
reasons: firstly, as a point process related to particle systems
in various media. Secondly, as an instrument for understanding the
behaviour of the Gaussian process itself, and finally, due to the fact
that many physical processes are closely approximated by a stationary
Gaussian processes conditioned not to cross a certain level.

In this talk we will define SGPs and survey classical and recent
results concerning their level crossings, starting from early works
in the 1940s by Kac, Rice and Slepian, through works of Dembo and
Bryc in early 2000's and ending up with recent state of the art
developments obtained with several co-authors. Our journey shall
take us through the forming relations between the theory of SGPs and
convex geometry, Hilbert spaces and finally -- harmonic analysis.
Our focus will be the direction of progress, its interaction with
various subfields of analysis and the ultimate goals it pursues.

No prior knowledge of the subject will be assumed.

16:15- tea