We review our most recent results on near-IR studies of human brain activity, which have been evolving in two directions: detection of neuronal signals and measurements of functional hemodynamics. We discuss results obtained so far, describing in detail the techniques we developed for detecting neuronal activity, and presenting results of a study that, as we believe, confirms the feasibility of neuronal signal detection. We review our results on near-IR measurements of cerebral hemodynamics, which are performed simultaneously with functional magnetic resonance imaging (MRI) These results confirm the cerebral origin of hemodynamic signals measured by optical techniques on the surface of the head. We also show how near-IR methods can be used to study the underlying physiology of functional MRI signals.

We found correlation between reduced scattering coefficient and total hemoglobin concentration measured on muscles by the frequency-domain spectroscopy during venous occlusion protocol. This can be a useful parameter, which can be employed in clinical studies.

In a series of recent papers ([19], [20], [22], [23]) we extended the notion of square integrability, for representations of nilpotent Lie groups, to that of stepwise square integrability. There we discussed a number of applications based on the fact that nilradicals of minimal parabolic subgroups of real reductive Lie groups are stepwise square integrable. In Part I we prove stepwise square integrability for nilradicals of arbitrary parabolic subgroups of real reductive Lie groups. This is technically more delicate than the case of minimal parabolics. We further discuss applications to Plancherel formulae and Fourier inversion formulae for maximal exponential solvable subgroups of parabolics and maximal amenable subgroups of real reductive Lie groups. Finally, in Part II, we extend a number of those results to (infinite dimensional) direct limit parabolics. These extensions involve an infinite dimensional version of the Peter-Weyl Theorem, construction of a direct limit Schwartz space, and realization of that Schwartz space as a dense subspace of the corresponding L2 space.

This is a semi-expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so many years, much of that Memoir is up to date, but of course there have been a number of refinements, advances and new developments, most of which have applied to smaller classes of real reductive Lie groups. Here we rewrite that AMS Memoir in in view of these advances and indicate the ties with some of the more recent (or at least less classical) approaches to geometric realization of unitary representations.

In earlier papers we studied direct limits $${(G,\,K) = \varinjlim\, (G_n,K_n)}$$ of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which $${G_n = N_n\rtimes K_n}$$ with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections $${\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}$$ for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit $${L^2(G/K) := \varinjlim L^2(G_n/K_n)}$$ is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces $${\mathcal{A}(G_n/K_n)}$$ of regular functions on G n /K n and injections $${\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}$$ for m ≧ n related to restriction by $${\nu_{m,n}(f)|_{G_n/K_n} = f}$$ . Thus the direct limit $${\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}$$ sits as a particular G-submodule of the much larger inverse limit $${\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}$$ . Further, we define a pre Hilbert space structure on $${\mathcal{A}(G/K)}$$ derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space $${\mathcal{A}(G/K)}$$ , and also defines a G-invariant inner product on $${\mathcal{A}(G/K)}$$ for which the left regular representation of G is multiplicity-free.

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author’s 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if Γ \ M is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold M, then Γ \ M is homogeneous if and only if each isometry γ∈ Γ is of constant displacement. The description of bounded isometries in this paper gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford’s use of quaternions in non-Euclidean geometry, and we sketch that in a historical note.