By Robert S. Doran, Calvin C. Moore, Robert J. Zimmer

George Mackey was once a rare mathematician of significant energy and imaginative and prescient. His profound contributions to illustration thought, harmonic research, ergodic concept, and mathematical physics left a wealthy legacy for researchers that keeps this day. This e-book relies on lectures provided at an AMS distinctive consultation held in January 2007 in New Orleans devoted to his reminiscence. The papers, written particularly for this quantity by way of internationally-known mathematicians and mathematical physicists, variety from expository and old surveys to unique high-level examine articles. The impact of Mackey's basic rules is clear all through. The introductory article comprises memories from former scholars, acquaintances, colleagues, and relations in addition to a biography describing his uncommon profession as a mathematician at Harvard, the place he held the Landon D. Clay Professorship of arithmetic.

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**Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey**

George Mackey was once a rare mathematician of serious energy and imaginative and prescient. His profound contributions to illustration thought, harmonic research, ergodic thought, and mathematical physics left a wealthy legacy for researchers that maintains at the present time. This booklet is predicated on lectures provided at an AMS targeted consultation held in January 2007 in New Orleans devoted to his reminiscence.

**Extra info for Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey**

**Sample text**

Subtracting, we see that if Ε φ F , ( ψ , ψ ) must be zero. Likewise, the discrete eigenfunctions are orthogonal t o all the eigendifferentials, and the Ε Ε 38 GROUP T H E O R Y A N D ATOMIC SPECTRA eigendifferentials are orthogonal to one another, provided the regions t o which they belong do not overlap. More than one linearly independent eigenfunction m a y belong to one eigenvalue of, say, the discrete spectrum. " Every possible linear combination o f degenerate eigenfunctions is also an eigenfunction with the same "eigenvalue.

We have seen that every operator which can be applied to a finite-dimensional vector is equivalent to a matrix. The infinitedimensional operators also have a matrix form, but it is often strongly singular. 1) '. It transforms the vector ψ into the vector q ^ , with components QM i 2 -"Xf) x x = 2 = Σ Χχ-Xf tai^i - ι ΧχΧι χ * i ^¥2 δ - ν( Ί Χ χ/ ö " ' 'f) x Xf*t Χ Μ χ ι · ' · */)· This vector is exactly the function x y into which ψ is transformed by the operation "multiplication by x . " The matrix which corresponds to the operator "differentiation with respect to x " is denoted by (i'/^)Pi since (Hji)djdx corresponds to p x t x 1 \ / x — x\ 1 f x' x ·" χ/ Δ _ > x ϋ It transforms the vector ψ into 2 χ '»·χ/ χ Hm i {Δ ΐΑ, > ΧΙ+ ΧΙ ί^-ΐΔ,^') δ χχ 2 ô 2 Ψ( ν χ x x/ f X 2> " ' > x f) ^ = lim — Δ—0 Δ (ψ(χ χ + £Δ, χ, 2 · · · , x) f — ψ(χ χ — |Δ, # , · · · , 2 x )) f and this is precisely the derivative of ψ with respect to x .

IF (Φ, Φ) = 1, then Ψ is said t o be normalized. I f the integral oo is finite, then Φ can always be normalized b y multiplication b y a constant ^1/c in the case above, since ^— ,— J = i j . T w o FUNCTIONS are orthogonal if their scalar product is zero. The scalar product given in the Eq. 7) is constructed by considering the functions Φ(χ · · · X ), G(X · · · # / ) of X X , · · · , X as vectors, whose components are labeled by / continuous indices. The function vector Φ{χ · · · X ) is defined in an /-fold infinite-dimensional space.