Related papers: Rationality of inner twisted flags of type $A_n$

Let $X$ be a ruled surface over a curve of genus $g$. We prove that $X$ has a scalar-flat Hermitian metric if and only if $g\geq 2$ and $m(X)>2-2g$ where $m(X)$ is an intrinsic number depends on the complex structure of $X$.

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure…

We investigate geometry of D-affine varieties. Our main result is that a D-affine rational projective surface over an algebraically closed field is a generalised flag variety of a reductive group.

Given a rational number $r$ such that $2r$ is not an integer, we prove that $\tan^2(r\pi)$ is irrational unless it is equal to $0$, $1$, $3$ or $\frac{1}{3}$, using only basic trigonometry and the Rational Root Theorem. Moreover, we deduce…

For any natural number $n \geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers $\mathbb{O}_n$.

A linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoicic conjectured that the equation $x_1 + 2x_2 + \cdots…

We provide an alternative proof that $\sqrt{2}$ is irrational that does not begin with the assumption that $\sqrt{2}$ is in fact rational.

We investigate the rationality problem for $\mathbf{Q}$-Fano threefolds of Fano index $\ge 2$.

We show that when $m>n$, the space of $m\times n$-matrix-valued rational inner functions in the disk is path connected.

Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with…

Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$…

We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of $x$ with a single pole of order 2 at $x=1$ and (b) the leading coefficient of the…

If the cosine of a rational multiple of $\pi$ is a rational number then it is an integral multiple of $\frac12$. For this fact, we give a proof accessible to an interested school student. We then discuss which quadratic and cubic…

We study symplectic geometry of rationally connected $3$-folds. The first result shows that rationally connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or $b_2(X)=2$, we…

We show that the coefficients of rational 2-functions are contained in an abelian number field. More precisely, we show that the poles of such functions are poles of order one and given by roots of unity and rational residue.

The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.

Let $G$ be a finite simple graph and let $I_G$ denote its associated toric ideal in the polynomial ring $R$. For each integer $n\geq 2$, we completely determine all the possible values for the tuple $({\rm reg}(R/I_G), {\rm…

Let $X$ be a finite, 2-dimensional cell complex. The curvature invariants $\rho_\pm(X)$ and $\sigma_\pm(X)$ were defined in [13], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities…

We prove that any three-point genus zero Gromov-Witten invariant on a type A Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these…

Let $A$ be an elliptic curve over $\Q$ of square free conductor $N$. We prove that if $A$ has a rational torsion point of prime order $r$ such that $r$ does not divide $6N$, then $r$ divides the order of the cuspidal subgroup of $J_0(N)$.