Related papers: On the Conjectures Regarding the 4-Point Atiyah De…
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…
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 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…
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…
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…
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 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…
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…
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…
The purpose of this note is to announce our proof of the Atiyah-Jones conjecture concerning the topology of the moduli spaces of based SU(2)-instantons over S^4. Full details and proofs appear in our paper [BHMM1].
The Atiyah-Sutcliffe normalized determinant function $D$ is a smooth complex-valued function on $C_n(H^3)$, where $C_n(H^3)$ denotes the configuration space of $n$ distinct points in hyperbolic $3$-space $H^3$. The hyperbolic version of the…
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…
We show that if the Atiyah Jones conjecture holds for a surface $X,$ then it also holds for the blow-up of $X$ at a point. Since the conjecture is known to hold for ${\mathbb P}^2$ and for ruled surfaces, it follows that the conjecture is…
We prove a degree-six inequality on convex quadrilaterals. This inequality originated from work on the Atiyah-Sutcliffe conjectures on configurations of points in \Bbb{R} ^{3}.
We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does.
We show that a certain conjecture by Atiyah and Sutcliffe implies the existence of an $ E_3 $-algebra (respectively $ E_2 $-algebra) structure on the disjoint union of all complex (respectively real) full flag manifolds modulo symmetric…
We prove a formula which generalizes both Onn's colorful determinantal formula, related to Rota's basis conjecture, and Svrtan's $n!$ formula, related to the Atiyah-Sutcliffe problem. In some cases, our formula allows us to prove some…
A version of the Atiyah-Floer conjecture, adapted to admissible SO(3)-bundles, is established.
In this short note, we show that the Atiyah-Sutcliffe conjectures for $n = 2m$, related to the unitary groups $U(2m)$, imply the author's analogous conjectures, which are associated with the symplectic groups $Sp(m)$. The proof is based on…
It is shown that the strong Atiyah conjecture and the L\"uck approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also…