A view on extending morphisms from ample divisors

The philosophy that ``a projective manifold is more special than any of its smooth hyperplane sections" was one of the classical principles of projective geometry. Lefschetz type results and related vanishing theorems were among the typically used techniques. We shall survey most of the problems, results and conjectures in this area, using the modern setting of ample divisors, and (some aspects of) Mori theory.Comment: To Appear in: Interactions of Classical and Numerical Algebraic Geometry, ed. by A. Bates, G. Besana and S. Di Rocco, Contemporary Mathematics, American Mathematical Societ

Nonlinear cross Gramians and gradient systems

We study the notion of cross Gramians for non-linear gradient systems, using the characterization in terms of prolongation and gradient extension associated to the system. The cross Gramian is given for the variational system associated to the original nonlinear gradient system. We obtain linearization results that precisely correspond to the notion of a cross Gramian for symmetric linear systems. Furthermore, first steps towards relations with the singular value functions of the nonlinear Hankel operator are studied and yield promising results.

ContextVP: Fully Context-Aware Video Prediction

Video prediction models based on convolutional networks, recurrent networks, and their combinations often result in blurry predictions. We identify an important contributing factor for imprecise predictions that has not been studied adequately in the literature: blind spots, i.e., lack of access to all relevant past information for accurately predicting the future. To address this issue, we introduce a fully context-aware architecture that captures the entire available past context for each pixel using Parallel Multi-Dimensional LSTM units and aggregates it using blending units. Our model outperforms a strong baseline network of 20 recurrent convolutional layers and yields state-of-the-art performance for next step prediction on three challenging real-world video datasets: Human 3.6M, Caltech Pedestrian, and UCF-101. Moreover, it does so with fewer parameters than several recently proposed models, and does not rely on deep convolutional networks, multi-scale architectures, separation of background and foreground modeling, motion flow learning, or adversarial training. These results highlight that full awareness of past context is of crucial importance for video prediction.Comment: 19 pages. ECCV 2018 oral presentation. Project webpage is at https://wonmin-byeon.github.io/publication/2018-ecc

Weighted Low-Regularity Solutions of the KP-I Initial Value Problem

In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper " Low regularity solutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper

Helioseismic analysis of the solar flare-induced sunquake of 2005 January 15

We report the discovery of one of the most powerful sunquakes detected to date, produced by an X1.2-class solar flare in active region 10720 on 2005 January 15. We used helioseismic holography to image the source of seismic waves emitted into the solar interior from the site of the flare. Acoustic egression power maps at 3 and 6 mHz with a 2 mHz bandpass reveal a compact acoustic source strongly correlated with impulsive hard X-ray and visible-continuum emission along the penumbral neutral line separating the two major opposing umbrae in the $\delta$-configuration sunspot that predominates AR10720. The acoustic emission signatures were directly aligned with both hard X-ray and visible continuum emission that emanated during the flare. The visible continuum emission is estimated at $2.0 \times 10^{23}$ J, approximately 500 times the seismic emission of $\sim 4 \times 10^{20}$ J. The flare of 2005 January 15 exhibits the same close spatial alignment between the sources of the seismic emission and impulsive visible continuum emission as previous flares, reinforcing the hypothesis that the acoustic emission may be driven by heating of the low photosphere. However, it is a major exception in that there was no signature to indicate the inclusion of protons in the particle beams thought to supply the energy radiated by the flare. The continued strong coincidence between the sources of seismic emission and impulsive visible continuum emission in the case of a proton-deficient white-light flare lends substantial support to the ``back -- warming'' hypothesis, that the low photosphere is significantly heated by intense Balmer and Paschen continuum-edge radiation from the overlying chromosphere in white-light flares.Comment: 12 pages, 7 figures, published in MNRA