Milena Hering

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.

SYZYGIES, MULTIGRADED REGULARITY AND TORIC VARIETIES

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1, . . . , Bl on X and m1, . . . , ml ∈ N, consider the line bundle L := B m 1 1 ⊗ � � � ⊗ B ml l . We give conditions on the mi which guarantee that the ideal of X in P(H 0 (X, L) � ) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.

Finitely many smooth d-polytopes with N lattice points

We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙn that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.

The membrane inclusions curvature equations

We examine a system of equations arising in biophysics whose solutions are believed to represent the stable positions of N conical proteins embedded in a cell membrane. Symmetry considerations motivate two equivalent refomulations of the system which allow the complete classification of solutions for small N<13. The occurrence of regular geometric patterns in these solutions suggests considering a simpler system, which leads to the detection of solutions for larger N up to 280. We use the most recent techniques of Grobner bases computation for solving polynomial systems of equations.

The T-Graph of a Multigraded Hilbert Scheme

The T-graph of a multigraded Hilbert scheme records the zero- and one-dimensional orbits of the T=(K*) n action on the Hilbert scheme induced from the T-action on . It has vertices the T-fixed points, and edges the one-dimensional T-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the T-graph depends on the ground field, resolving a question of Altmann and Sturmfels.