Dan Edidin

An algebraic characterization of injectivity in phase retrieval

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 4 M − 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.

Localization in equivariant intersection theory and the Bott residue formula

We prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections and Schubert varieties.

Characteristic classes and quadric bundles

In this paper we construct Stiefel-Whitney and Euler classes in Chow cohomology for algebraic vector bundles with nondegenerate quadratic form. These classes are not in the algebra generated by the Chern classes of such bundles and are new characteristic classes in algebraic geometry. On complex varieties, they correspond to classes with the same name pulled back from the cohomology of the classifying space BSO(N,C). The classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups ([T2], [EG]). We begin by using the geometry of quadric bundles to study Chern classes of maximal isotropic subbundles. If V → X is a vector bundle with quadratic form, and if E and F are maximal isotropic subbundles of V then we prove (Theorem 1) that ci(E) and ci(F ) are equal mod 2. Moreover, if the rank of V is 2n, then cn(E) = ±cn(F ), proving a conjecture of Fulton. We define Stiefel-Whitney and Euler classes as Chow cohomology classes which pull back to Chern classes of maximal isotropic subbundles of the pullback bundle. Using the above theorem we show (Theorem 2) that these classes exist and are unique, even though V need not have a maximal isotropic subbundle. These constructions also make it possible to give “Schubert” presentations,

Notes on the Construction of the Moduli Space of Curves

The purpose of these notes is to discuss the problem of moduli for curves of genus g ≥ 31 and outline the construction of the (coarse) moduli scheme of stable curves due to Gieseker. The notes are broken into four parts.

A new identity for Parseval frames

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.

Equivalence of Reconstruction From the Absolute Value of the Frame Coefficients to a Sparse Representation Problem

The purpose of this letter is to prove, for real frames, that signal reconstruction from the absolute value of the frame coefficients is equivalent to solution of a sparse signal optimization problem, namely a minimum lscrp (quasi)norm over a linear constraint. This linear constraint reflects the coefficients relationship within the range of the analysis operator

Frames for linear reconstruction without phase

The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communications, including wireless and fiber-optical transmissions. The algorithms discussed here rely on suitable rank-one operator valued frames defined on finite-dimensional real or complex Hilbert spaces. Examples of such operator-valued frames are the rank-one Hermitian operators associated with vectors from maximal sets of equiangular lines or maximal sets of mutually unbiased bases. A more general type of examples is obtained by a tensor product construction. We also study erasures and show that in addition to loss of phase, a maximal set of mutually unbiased bases can correct for erased frame coefficients as long as no more than one erasure occurs among the coefficients belonging to each basis, and at least one basis remains without erasures.

Fast algorithms for signal reconstruction without phase

We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important in signal processing, especially speech recognition technology, and has relevance for state tomography in quantum theory. We show that a generic frame gives reconstruction from the absolute value of the frame coefficients in polynomial time. An improved efficiency of reconstruction is obtained with a family of sparse frames or frames associated with complex projective 2-designs.

The Integral Chow Ring of the Stack of at Most 1-Nodal Rational Curves

We give a presentation for the stack of rational curves with at most 1 node as the quotient by GL3 of an open set in a 6-dimensional irreducible representation. We then use equivariant intersection theory to calculate the integral Chow ring of this stack.

Riemann-Roch for Quotients and Todd Classes of Simplicial Toric Varieties

Abstract In this paper we give an explicit formula for the Riemann-Roch map for singular schemes which are quotients of smooth schemes by diagonalizable groups. As an application we obtain a simple proof of a formula for the Todd class of a simplicial toric variety. An equivariant version of this formula was previously obtained for complete simplicial toric varieties by Brion and Vergne (Brion M. and Vergne M. ([1997]). An equivariant Riemann-Roch theorem for complete simplicial toric varieties. J. Reine. Agnew. Math.482:67–92) using different techniques. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.