Related papers: Towards the Atiyah-Sutcliffe conjectures for copla…

Motivated by certain questions in physics, Atiyah defined a determinant function which to any set of $n$ distinct points $x_1,..., x_n$ in $\mathbb R^3$ assigns a complex number $D(x_1,..., x_n)$. In a joint work, he and Sutcliffe stated…

Given a configuration $\mathbf{x}$ of $n$ distinct points in hyperbolic $3$-space $H^3$, Michael Atiyah associated $n$ polynomials $p_1,\ldots,p_n$ of a variable $t \in \mathbb{C}P^1$, of degree $n-1$, and conjectured that they are linearly…

We present a general formula for the Atiyah-Sutcliffe determinant function, which holds for any integer $n \geq 2$, as a global factor times a sum of terms, with each term similar to a higher degree cross-ratio. The formula is to our…

In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any…

From any configuration of finitely many points in Euclidean three-space, Atiyah constructed a determinant and conjectured that it was always non-zero. Atiyah and Sutcliffe (hep-th/0105179) amass a great deal of evidence it its favour. In…

For the case of 4 points in Euclidean space, we present a computer aided proof of Conjectures II and III made by Atiyah and Sutcliffe regarding Atiyah's determinant along with an elegant factorization of the square of the imaginary part of…

In 2001 Sir M. F. Atiyah formulated a conjecture (C1) and later with P. Sutcliffe two stronger conjectures (C2) and (C3). These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for…

We present a direct proof of the second conjecture made by M. Atiyah and P. Sutcliffe for the case of convex quadrilaterals. Unlike previous work on this conjecture, our proof does not require any computer aided computations. The new proof…

We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.

We generalize the Atiyah problem on configurations and the related Atiyah--Sutcliffe conjectures 1 and 2 using finite graphs, configurations of points and tensors. Our conjectures are intriguing geometric inequalities, defined using the…

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of…

Atiyah's conjecture concerning configurations of N points in the Euclidean three-space is verified for the following nonplanar configurations: The first m points lie on a line L and the remaining n=N-m (>2) points are the vertices of a…

We settle the Hadwiger-Boltyanski Illumination Conjecture for all 1-unconditional convex bodies in ${\mathbb R}^3$ and in ${\mathbb R}^4$. Moreover, we settle the conjecture for those higher-dimensional 1-unconditional convex bodies which…

In Surveys in Differential Geometry, Volume 7, published in 2002 and Philosophical Transactions of the Royal Society A, Volume 359, published in 2001, Sir Michael Atiyah introduced what is known as the Atiyah problem on configurations of…

We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in "On the volume of a hyperbolic simplex", Stud. Sci. Math. Hung. 21, 243-249, 1986. These conjectures concern expressing the volume of an ideal…

General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the…

In the recent articles by Alper, Eastwood and Isaev, it was conjectured that all rational $GL_n({\mathbb C})$-invariant functions of forms of degree $d\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of…

In this paper, the Conley conjecture, which were recently proved by Franks and Handel \cite{FrHa} (for surfaces of positive genus), Hingston \cite{Hi} (for tori) and Ginzburg \cite{Gi} (for closed symplectically aspherical manifolds), is…

We prove a conjecture concerning the third Hankel determinant, proposed in ``Anal. Math. Phys., https://doi.org/10.1007/s13324-021-00483-7", which states that $|H_3(1)|\leq 1/9$ is sharp for the class $\mathcal{S}_{\wp}^{*}=\{zf'(z)/f(z)…

Let h_R denote an L ^{\infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{\infty} norm of the `hyperbolic' sums $$ \sum _{|R|=2 ^{-n}}…