Strichartz estimates are a manifestation of a dispersion phenomenon, exhibited by certain partial differential equations, which is detected by suitable Lebesgue space norms. In most cases the evolution propagator U (t) is a one parameter group of unitary operators. Motivated by the importance of decay estimates in group representation theory and ergodic theory, Strichartz-type estimates seem worth investigating when U (t) is replaced by a unitary representation of a non-compact Lie group, the group element playing the role of time. Since the Schrödinger group is a subgroup of the metaplectc group, the case of the metaplectic or oscillatory representation is of special interest in this connection. We prove uniform weak-type sharp estimates for matrix coefficients and Strichartz estimates for that representation. The crucial point is the choice of function spaces able to detect such a dispersive effect, which in general will depend on the given group action. The relevant function spaces here turn out to be the so-called modulation spaces from Time-frequency Analysis in Euclidean space, and Lebesgue spaces with respect to Haar measure on the metaplectic group. The proofs make use in an essential way of the covariance of the Wigner distribution with respect to the metaplectic representation.2010 Mathematics Subject Classification. 42B35, 22E45.

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We deal with an application of partial differential equations to the correct definition of a wine cellar. We present some historical details about this problem. We also discuss how to build or renew a wine cellar, creating ideal conditions for the aging process and improving the quality of wines. Our goal is to calculate the optimal depth z0 of a wine cellar in order to attenuate the periodic temperature fluctuations. What follows is a kind of survey of wine-related and optimization problems which have been solved by means of powerful math tools.

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The aim of this chapter is to discuss some applications of mathematics: in oenology and in food and wine pairing. We introduce and study some partial differential equations for the correct definition of a wine cellar and to the chemical processes involved in wine aging. Secondly, we present a mathematical method and some algorithmic issues for analyzing the process of food and wine pairing done by sommeliers.