Related papers: Some conjectures regarding certain Schubert struct…

We consider tangent cones to Schubert subvarieties of the flag variety $G/B$, where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_6$, $E_7$ or $E_8$. We prove that if $w_1$ and $w_2$ form a good pair of…

This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable exponent $W^{s,p(.),\tilde{p}(.,.)}(\Omega)$, when $p$ and $\tilde{p}$ satisfies some conditions. As application, we study the existence of…

We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants $a_w$ are expressed as a sum of \emph{normalized} mixed Eulerian numbers indexed naturally by…

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi-Civita connections and explicit formulas for…

We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular, we give sufficient conditions for Chern…

A $\Bbbk$-configuration of type $(d_1,\dots,d_s)$ is a specific set of points in $\mathbb P^2$ that has a number of algebraic and geometric properties. For example, the graded Betti numbers and Hilbert functions of all…

The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the…

Let $X$ be a compact Riemann surface of genus at least $3$. We compute the Brauer groups of the moduli spaces of stable parabolic $\text{SL}(r,\mathbb{C})$-connections and stable strongly parabolic $\text{SL}(r,\mathbb{C})$-Higgs bundles…

We relate a certain category of sheaves of k-vector spaces on a complex affine Schubert variety to modules over the k-Lie algebra (for ch k>0) or to modules over the small quantum group (for ch k=0) associated to the Langlands dual root…

Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant…

The "back-stabilization number" for products of Schubert polynomials is the distance the corresponding permutations must be shifted before the structure constants stabilize. We give an explicit formula for this number and thereby prove a…

We investigate the toric geometry of two families of generalised determinantal varieties arising from permutations: Matrix Schubert varieties ($\overline{X_w}$) and Kazhdan-Lusztig varieties ($\mathcal{N}_{v,w}$). Matrix Schubert varieties…

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the…

Stimulated by the category theorems of Eisner and Ser\'eny in the setting of unitary and isometric $C_0$-semigroups on separable Hilbert spaces, we prove category theorems for Schr\"odinger semigroups. Specifically, we show that, to a given…

A permutation is called covexillary if it avoids the pattern $3412$. We construct an open embedding of a covexillary matrix Schubert variety into a Grassmannian Schubert variety. As applications of this embedding, we show that the…

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis…

Given a rational homogeneous manifold $S=G/P$ of Picard number one and a Schubert variety $S_0 $ of $S$, the pair $(S,S_0)$ is said to be homologically rigid if any subvariety of $S$ having the same homology class as $S_0$ must be a…

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…

We determine the structure of the equivariant cohomology and $K$-theory of Bott towers. By restriction, we obtain similar results for Bott-Samelson varieties. This results allow us to describe more precisely the equivariant cohomology and…

A group may be considered $C^*$-stable if almost representations of the group in a $C^*$-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are $C^*$-stable or only stable with…