arXiv research

The non-vanishing conjecture implies the abundance conjecture in certain cases.

The paper proves conditions for the Abundance conjecture in minimal projective klt pairs.

The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.

We prove that Witten's Conjecture [arXiv:hep-th/9411102] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with $b_1=0$ and odd $b_2^+\geq 3$ follows from our $\mathrm{SO}(3)$-monopole cobordism formula [arXiv:math/0203047] when the four-manifold has $…

Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety…

The paper proves a version of the SYZ conjecture for hyperkahler manifolds.

We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds o…

The Wu-Yau theorem is verified for negative curvature, and new examples of Kähler-Einstein metrics are found.

In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of …

Complex projective manifolds without rational curves are quotients of Abelian varieties.

The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extensi…

Study of superintegrable systems linked to affine hypersurfaces.

Study symplectically aspherical Kähler manifolds with unique properties.

Let $(M,g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that $(M,g)$ has bounded geometry, or more generally is thick at infinity. Then th…

The study shows boundedness of certain fibered varieties in algebraic geometry.

This paper presents an unsupervised algorithm for nonlinear unmixing of hyperspectral images. The proposed model assumes that the pixel reflectances result from a nonlinear function of the abundance vectors associated with the pure spectral components. We assume that the spectral signatures of the pure components and t…

We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the …

Study shows torsion grows subexponentially in book of I-bundles but can grow exponentially in non-regular covers.

In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are gen…

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. …

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace …

Paper analyzes weak-to-strong generalization in CNNs, identifying data-scarce and data-abundant regimes.

Researchers prove existence of cscK metrics on smooth minimal models.

Superintegrable systems on curved manifolds found to have Hessian structures.

New product structures encode superintegrable Hamiltonian systems in Euclidean spaces.

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces d…

The study finds abundant normal generators for mapping class groups.

The Ekeland variational principle implies what can be regarded as a strong version, in the $C^1$ category, of the Yau minimum principle: under the appropriate hypotheses {\it every} minimizing sequence admits a {\it good shadow}, a second minimizing sequence that has good properties and is asymptotic to the original on…

Algorithm simplifies Khovanov homology computations for 4-strand torus links.

New method uses neural networks to infer dark matter subhalo abundance from stellar streams.

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.

Metagenomic studies have increasingly utilized sequencing technologies in order to analyze DNA fragments found in environmental samples.One important step in this analysis is the taxonomic classification of the DNA fragments. Conventional read classification methods require large databases and vast amounts of memory to…

Study semi-supervised learning with noisy proxy covariates, deriving bounds and showing gains.

The paper explores fully affine maximal curves and their properties.

PseudoH-type is a natural generalization of H-type to geometries with indefinite metric tensors. We give a complete determination of the conjugate locus including multiplicities. We also obtain a partial characterization in terms of the abundance of totally geodesic, 3-dimensional submanifolds.

Sequences of Nilsson type appear in abundance in Algebraic Geometry, Enumerative Combinatorics, Mathematical Physics and Quantum Topology. We give an elementary introduction on this subject, including the definition of sequences of Nilsson type and the uniqueness, existence, and effective computation of their asymptoti…

Method maps imperfect simulations to observed stellar spectra using unsupervised domain adaptation.

In this work we propose a method to compute continuous embeddings for kmers from raw RNA-seq data, without the need for alignment to a reference genome. The approach uses an RNN to transform kmers of the RNA-seq reads into a 2 dimensional representation that is used to predict abundance of each kmer. We report that our…

In a plethora of applications dealing with inverse problems, e.g. in image processing, social networks, compressive sensing, biological data processing etc., the signal of interest is known to be structured in several ways at the same time. This premise has recently guided the research to the innovative and meaningful …

CIBP models feature abundance in latent feature models.

Constructs deformations of Vaisman manifolds preserving foliations.

We prove that there are no pseudoholomorphic theories of anything other than curves, even if one allows more general spaces than almost complex manifolds. The proof is elementary, except for theories of pseudoholomorphic hypersurfaces, where topological techniques are needed. Surprisingly, hypersurface theories exist `…

Study reveals DNNs prefer easy-to-learn cues over essential ones in image recognition.

Solving real-world problems, particularly with deep learning, relies on the availability of abundant, quality data. In this paper we develop a novel framework that maximises the utility of time-series datasets that contain only small quantities of expertly-labelled data, larger quantities of weakly (or coarsely) labell…

New method models aptamer libraries as Boltzmann-weighted graph ensembles for better affinity predictions.

Two types of nonvanishing results are presented for compact Kähler varieties.

We exhibit large classes of local actions for the vacuum Einstein equations. In presence of fermions, or more generally of matter which couple to the connection, these actions lead to inequivalent equations revealing an arbitrary number of parameters. Even in the pure gravitational sector, any corresponding quantum the…

New knot classification based on SU(2) representations and instanton homology.