Search

We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function |$f$| on a compact set |$K$|⁠ , the critical points of our approximants may be taken to lie in any given domain containing |$K$|⁠ , and all the critical values in any given neighborhood of the polynomially convex hull of |$f(K)$|⁠. [ABSTRACT FROM AUTHOR]

Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the special case of the conjecture where the variety has dominant projection to the domain of the exponential map, for abelian varieties and for algebraic tori. Furthermore, in the situation where the intersection is 0-dimensional, we exhibit structure in the intersection by parametrizing the sufficiently large points as the images of the period lattice under a (multivalued) analytic map. Our approach is complex geometric, in contrast to a real analytic proof given by Brownawell and Masser just for the case of algebraic tori. [ABSTRACT FROM AUTHOR]

We study the Picard group of the moduli space of semistable sheaves on a smooth quadric surface. We polarize the surface by an ample divisor close to the anticanonical class. We focus especially on moduli spaces of sheaves of small discriminant, where we observe new and interesting behavior. Our method relies on constructing certain resolutions for semistable sheaves and applying techniques of geometric invariant theory to the resulting families of sheaves. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an algorithm for computing |$p$| -adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of |$p$| -adic integration: Berkovich–Coleman integrals, which can be performed locally, and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve with basic wide-open sets, we reduce the computation of Berkovich–Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya and to Balakrishnan and Besser for regular and meromorphic |$1$| -forms, respectively. We then employ tropical geometric techniques due to the 1st-named author with Rabinoff and Zureick-Brown to convert the Berkovich–Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish. [ABSTRACT FROM AUTHOR]

In this paper we introduce several geometric flows that evolve primarily non-degenerate 2-forms, with the motivation to develop a geometric flow to approach the existence of the symplectic forms on a compact manifold that supports a non-degenerate 2-form. In particular, we introduce |$\mathrm{d}^{\ast }\mathrm{d}$| -flow and |$\mathrm{d}^{\ast }\mathrm{d}$| -Ricci flow for a compatible pair |$(\omega , J)$| of an almost Hermitian structure. We prove the short time existence and uniqueness of these flows with smooth initial data, and give some examples of long time existence and convergence. [ABSTRACT FROM AUTHOR]

We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system may be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. Various sequences correspond to various discrete parametrizations. The coordinate functions of discrete confocal quadrics are computed explicitly. The theory is illustrated with a variety of examples in two and three dimensions. These include confocal coordinate systems parametrized in terms of Jacobi elliptic functions. Connections with incircular nets and a generalized Euler–Poisson–Darboux system are established. [ABSTRACT FROM AUTHOR]

This article presents a geometric approach to some similarity problems involving metric arguments in the non-positively curved space of positive invertible operators of an operator algebra and the canonical isometric action by invertible elements on the cone given by |$g\cdot a=gag^*$|⁠. Through this approach, we extend and put into a geometric framework results by Pisier and partially answer a question by Andruchow et al. about minimality properties of canonical unitarizers. [ABSTRACT FROM AUTHOR]

We develop an approach through geometric functional analysis to reconstruction of signals from few linear measurements and to error-correcting codes. An error-correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We show that most linear orthogonal transformations Q : ℝn → ℝm form efficient and robust error-correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the ℓ1-norm. This yields robust error-correcting codes over reals (and over alphabets of polynomial size), with a Gilbert-Varshamov type bound, and with quadratic time encoders and polynomial time decoders. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L1-norm minimization. This is an improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of “neighborly” symmetric polytopes, that is, polytopes with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of a cube form such polytopes. Our work thus belongs to a common ground of coding theory, signal processing, combinatorial geometry, and geometric functional analysis. Our argument, which is based on concentration of measure and improving Lipschitzness by random projections, may be of independent interest in geometric functional analysis. [ABSTRACT FROM AUTHOR]

We study elliptic solutions to difference analogs of soliton equations and their connections with finite-gap Schrödinger difference operators. We generalize the algebro geometric approach of elliptic (KDV) solitons, developed jointly with J.-L. Verdier, to the difference case. This approach allows us to characterize the corresponding spectral curves and find new classes of difference elliptic finite-gap operators which are difference analogs of the currently known elliptic finite-gap Schrödinger operators. [ABSTRACT FROM AUTHOR]

We prove a quantitative version of Frobenius reciprocity in the theory of automorphic functions. Namely, given an automorphic representation we show how to estimate the norm of the corresponding Γ-invariant functional with respect to some norms on the representation space. The answer is given by relative traces. [ABSTRACT FROM AUTHOR]

For quasilinear wave equations with analytic coefficients, we construct singular solutions u such that u;u0 remain continuous, while u00 is singular like (T − t)−1 and has rank 2.This work extends our previous work on geometric blowup to mere multidimensional solutions. [ABSTRACT FROM AUTHOR]

The article presents a report on the cobordism theory and localization formulas for hamiltonian group actions. It describes some symplecto-geometric applications that M, compact oriented 2d-dimensional manifold on which the group S1 acts, which is a is cobordant to a disjoint union of N copies of CPd where N is the number of fixed points. It exploits the fact that certain structures on M that are preserved by the S1-action can be incorporated into the cobordism.