Related papers: A congruence modulo four in real Schubert calculus

The foundations of matrix geometry are discussed, which provides the basis for recent progress on the effective geometry and gravity in Yang-Mills matrix models. Basic examples lead to a notion of embedded noncommutative spaces (branes)…

For all integers $g \geq 6$ we prove the existence of a metric graph $G$ with $w^1_4=1$ such that $G$ has Clifford index 2 and there is no tropical modification $G'$ of $G$ such that there exists a finite harmonic morphism of degree 2 from…

The Grassmannian model represents harmonic maps from Riemann surfaces by families of shift-invariant subspaces of a Hilbert space. We impose a natural symmetry condition on the shift-invariant subspaces that corresponds to considering an…

A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the Dirac operator. Examples include fuzzy…

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds $G_k(\mathbb{R}^n)$ to compute zero-divisor cup-length and estimate topological complexity of motion planning for $k$-linear subspaces in $\mathbb{R}^n$. In…

The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology of G/P. The Schubert's problem of…

We investigate the moduli space of sheaves supported on space curves of degree 4 and having Euler characteristic 1. We give an elementary proof of the fact that this moduli space consists of three irreducible components.

We present a geometric construction of a new class of hyper-Kahler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kahler geometry with torsion associated with the NS-5-brane along quaternionic planes…

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this…

In this paper, we study the relation of the sign of the Gaussian and mean curvature of modular surfaces in Lorentz-Minkowski $3$-space to the zeroes of the associated complex analytic functions and its derivatives. Further, we completely…

Given two nonsingular real algebraic varieties V and W, we consider the problem of deciding whether a smooth map f: V -> W can be approximated by regular maps in the space of smooth maps from V to W. Our main result is a complete solution…

Using the relation between four manifolds and topological modular form (TMF) from the six dimensional approach, we exhibit fiber sum formulas for infinite families of smooth spin four manifolds associated to compactifications of free and…

We define a weighted multiplicity function for closed geodesics of given length on a finite area Riemann surface. These weighted multiplicities appear naturally in the Selberg trace formula, and in particular their mean square plays an…

Metrics on Grassmannians have a wide array of applications: machine learning, wireless communication, computer vision, etc. But the available distances between subspaces of distinct dimensions present problems, and the dimensional asymmetry…

One hundred years ago, Hilbert gave a list of important open problems in mathematics. His 15th problem asked for the development of a rigorous calculus explaining Schubert's enumerative results for intersecting varieties defined by rank…

In this, the last of three papers about $C_2$-equivariant complex quadrics, we complete the calculation of the equivariant ordinary cohomology of smooth symmetric quadrics in the cases where the fixed sets have more than two components.…

Let $M$ be a topological spherical space form, i.e. a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature…

We consider the problem of determining Gr\"obner bases of binomial ideals associated with linear error correcting codes. Computation of Gr\"obner bases of linear codes have become a topic of interest to many researchers in coding theory…

The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the…

The coincidence problem for planar patterns with $N$-fold symmetry is considered. For the N-fold symmetric module with $N<46$, all isometries of the plane are classified that result in coincidences of finite index. This is done by…