arXiv research

We briefly describe each of the four topics: Schubert Calculus, Schubert Cell, Schubert Cycle, and Schubert Polynomials.

We relate Kostant's theorem on the cohomology of a flag manifold $G/B$ with the geometry of the Bruhat-Poisson structure. We express Kostant's harmonic forms in terms of the moment maps (for the torus action) and the Liouville volume forms for the symplectic structures on the Schubert cells induced by the Bruhat-Poisso…

The paper computes the cellular homology of real flag manifolds.

Study lifts Schubert stratification to $Spin_{n+1}$, revealing new Bruhat cells.

Algorithm to compute cohomology groups of real flag manifolds, proving torsion and Schubert varieties.

K. Ding studied a class of Schubert varieties X_λin type A partial flag manifolds, corresponding to integer partitions λand in bijection with dominant permutations. He observed that the Schubert cell structure of X_λis indexed by maximal rook placements on the Ferrers board B_λ, and that the integral cohomology groups …

The paper combinatorizes spaces of nondegenerate spherical curves.

The paper concerns a compactification of the isospectral varieties of nilpotent Toda lattices for real split simple Lie algebras. The compactification is obtained by taking the closure of unipotent group orbits in the flag manifolds. The unipotent group orbits are called the Peterson varieties and can be used in the co…

Suppose that $S^n$ is given a generic Riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures. This pinching condition is satisfied by manifolds whose real sectional curvatures $K_r(σ)$ satisfy $1/2 < K_r(σ) \leq 1.$ Then the number o…

We (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual the Schubert VHS form a basis of the invariant characteristic cohomol…

Schubert varieties are irreducible subvarieties of homogeneous manifold, which are important to understand the geometry of homogeneous manifold G/P and the action of the semisimple Lie group G. Consider the space of effective cycles in G/P with homology class equal to an integral multiple of the homology class of a Sch…

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class [S] is Schur rigid if the only irreducible subvarieties Y of X with homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles and The identifie…

Given a singular Schubert variety Z in a compact Hermitian symmetric space it is a longstanding question to determine when Z is homologous to a smooth variety Y. We identify those Schubert varieties for which there exist first-order obstructions to the existence of Y. This extends (independent) work of M. Walters, R. B…

Smooth Schubert varieties are rigid in rational homogeneous manifolds.

Let a=(p_1^{q_1}, ..., p_r^{q_r}) be a partition and a'=({p_1'}^{q_1'}, >..., {p_r'}^{q_r'}) be its conjugate. We will prove that if q_i, q_i > 1 for all i, then any irreducible subvariety X of Gr(m,n) whose homology class is an integral multiple of the Schubert class [σ_a] of type a is a Schubert variety of type a.

The paper examines when real matrix Schubert varieties are minimal submanifolds.

We introduce the Schubert form a $3$-bridge link diagram, as a generalization of the Schubert normal form of a $3$-bridge link. It consists of a set of six positive integers, written as $\left( p/n,q/m,s/l\right)$, with some conditions and it is based on the concept of $3$-butterfly. Using the Schubert normal form of …

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of Fox colors is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove …

The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus \mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space which has a Morse theoretic origin. We prove that these strata define closed subana…

Study characteristic classes of a specific type of determinantal varieties.

The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Ω^i_r(X) of the Grassmannian. We present here a method, essent…

A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parameterization of this family of knots by a four-tuple of integers, which they call Schubert's normal form. This article presents an algorithm for construct…

In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand pa…

A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal f…

We characterize the harmonic forms on a flag manifold $K/T$ defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson harmonic" with respect to the so-called Bruhat Poisson structure on $K/T$. This enables us to give Poisson geometrical proofs of many of the special properties of these harm…

We investigate the poset of strata of a Schubert-like stratification of certain natural compactification of the space of hermitian $n\times n$ matrices. We prove that this poset is a modular ortholattice, we compute its Möbius function and we describe the topology of its order intervals.

Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a projective space, which is obtained by identifying $G/P$ with a coadjoint orbit of the co…

Let X=G/P be a homogeneous space of a complex semisimple Lie group G equipped with a hermitian metric. We study the action of the Hodge star operator on the space of harmonic differential forms on X. We obtain explicit combinatorial formulas for this action when X is an irreducible hermitian symmetric space of compact …

The affine Grassmannian generalizes Euclidean and linear subspaces with rich geometric properties.

We consider the T-equivariant cohomology of Bott-Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi-simple complex algebraic group of adjoint type with maximal torus T. We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of …

In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond to points in the Euclidean Weyl chamber. We can hence assign vector-valued side-lengths to segments. Our main result is a system of homogeneous linear inequalities describing the restrictions on the side -lengths of closed polygons. T…

Kernel testing compares cell states in single-cell data.

The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we give an algorithm for computing the multiplicative structure constants of this alge…

Framework detects and classifies multi-label RBC images from microscopic images.

This study reviews and evaluates clustering methods for single-cell RNA-seq data.

Forest Fire Clustering discovers cell types from single-cell data.

Proposes CCCVAE for better single-cell clustering with cell-cell communication.

Matching cells over time has long been the most difficult step in cell tracking. In this paper, we approach this problem by recasting it as a classification problem. We construct a feature set for each cell, and compute a feature difference vector between a cell in the current frame and a cell in a previous frame. Then…

Thin position for knots in the 3-sphere was introduced by Gabai and has been used in a variety of contexts. We conjecture an analogue to a theorem of Schubert and Schultens concerning the bridge number of satellite knots. For a satellite knot K, we use the companion torus T to provide a lower bound for w(K), proving th…

Improved GPLVM model for single-cell RNA-seq data.

The study identifies all possible vector field structures on specific 2D shapes.

New model identifies cell-specific genes for cancer prognosis.

MarkerMap selects key genes for cell type analysis in single-cell RNA-seq.

Proposes CXNs for neural network computations on cell complexes.

New metric scores perturbations across populations, not cells, improving model comparison.

Cataloging the neuronal cell types that comprise circuitry of individual brain regions is a major goal of modern neuroscience and the BRAIN initiative. Single-cell RNA sequencing can now be used to measure the gene expression profiles of individual neurons and to categorize neurons based on their gene expression profil…

The process of morphogenesis, which can be defined as an evolution of the form of an organism, is one of the most intriguing mysteries in the life sciences. It is clear, that gene expression patterns cannot explain the development of the precise geometry of an organism and its parts in space. Here, we suggest a set of …