Given a symplectic surface (Σ, ω) of genus g ≥ 4, we show that the free group with two generators embeds into every asymptotic cone of (Ham(Σ, ω), dH), where dH is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds. * This paper was the outcome of the authors' work in the computational symplectic topology graduate student team-based research program held Hofer's geometryLet (M, ω) be a symplectic manifold. Given a smooth function H :where H t (p) := H(t, p). Let ϕ t H : M → M be the flow of the ODEẋ(t) = X t (x(t)), making sufficient assumptions to ensure that the flow is globally defined on the time interval [0, 1] (for example, we could take M to be compact). Inside the group Symp(M, ω) = {φ ∈ Diff(M ) : φ * ω = ω} of symplectomorphisms we have the subgroup of Hamiltonian diffeomorphisms Ham(M, ω), which consists of the time-one maps ϕ 1 H : M → M of flows as above. The group Ham(M, ω) is equipped with a geometrically meaningful bi-invariant metric introduced by Hofer. The resulting metric group is an important object of study in symplectic geometry. For φ ∈ Ham(M, ω), we define the Hofer norm φ H = inf H 1 0

We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin-Toeplitz quantization, Donaldson's dynamical system on the space of Hermitian inner products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the Q-operator on Kähler manifolds with constant scalar curvature, and yield exponential convergence of Donaldson's iterations to the fixed point. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.

We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin-Toeplitz quantization, Donaldson's dynamical system on the space of Hermitian products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results incidentally confirm a prediction of Donaldson for the spectrum of the Q-operator on Kähler manifolds with constant scalar curvature. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance. Contents 7 Two concepts of quantum noisea We thank S. Nonnenmacher for this explanation. b Note that after renormalization, there is a missing factor of 1/2 in front of the second term of the analogous formula in [30, (1.2)].

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