We construct new classes of Parseval frames for a Hilbert space which allow signal reconstruction from the absolute value of the frame coefficients. As a consequence, signal reconstruction can be done without using noisy phase or its estimation. This verifies a longstanding conjecture of the speech processing community.

Partially supported by an NSF post-doctoral fellowship that the equivariant groups A G * (X) depend only on the stack [X/G] and not on its presentation as a quotient. If X is smooth, then A 1 G (X) is isomorphic to Mumford's Picard group of the stack, and the ring A * G (X) can naturally be identified as an integral Chow ring of [X/G] (Section 5.3).These results imply that equivariant Chow groups are a useful tool for computing Chow groups of quotient spaces and stacks. For example, Pandharipande ([Pa1], [Pa2]) has used equivariant methods to compute Chow rings of moduli spaces of maps of projective spaces as well as the Hilbert scheme of rational normal curves. In this paper, we compute the integral Chow rings of the stacks M 1,1 and M 1,1 of elliptic curves, and obtain a simple proof of Mumford's result ([Mu]) that P ic f un (M 1,1 ) = Z/12Z. In an appendix to this paper, Angelo Vistoli computes the Chow ring of M 2 , the moduli stack of smooth curves of genus 2.Equivariant Chow groups are also useful in proving results about intersection theory on quotients. It is easy to show that if X is smooth then there is an intersection product on A G * (X). The theorem on quotients therefore implies that there exists an intersection product on the rational Chow groups of a quotient of a smooth algebraic space by a proper action. The existence of such an intersection product was shown by Gillet and Vistoli, but only under the assumption that the stabilizers are reduced. This is automatic in characteristic 0, but typically fails in characteristic p. The equivariant approach does not require this assumption and therefore extends the work of Gillet and Vistoli to arbitrary characteristic. Furthermore, by avoiding the use of stacks, the proof becomes much simpler.Finally, equivariant Chow groups define invariants of quotient stacks which exist in arbitrary degree, and associate to a smooth quotient stack an integral intersection ring which when tensored with Q agrees with rings defined by Gillet and Vistoli. By analogy with quotient stacks, this suggests that there should be an integer intersection ring associated to an arbitrary smooth stack, which could be nonzero in degrees higher than the dimension of the stack.We remark that besides the properties mentioned above, the equivariant Chow groups we define are compatible with other equivariant theories such as cohomology and K-theory. For instance, if X is smooth then there is a cycle map from equivariant Chow theory to equivariant cohomology (Section 2.8). In addition, there is a map from equivariant K-theory to equivariant Chow groups, which is an isomorphism after completion; and there is a localization theorem for torus actions, which can be used to give an intersection theoretic proof of residue formulas of Bott and Kalkman. These topics will be treated elsewhere.

The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2-designs in finite-dimensional real or complex Hilbert spaces. Examples of such frames are two-uniform frames and mutually unbiased bases, which include discrete chirps. The number of operations required for reconstruction with these frames grows at most as the cubic power of the dimension of the Hilbert space. Moreover, we present a very efficient algorithm which gives reconstruction on the order of d operations for a d-dimensional Hilbert space.

Abstract. A complex frame is a collection of vectors that span C M and define measurements, called intensity measurements, on vectors in C M . In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from 4M − 4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.

A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of H 2 (X, G m ) is surjective.