International Journal of Mathematics

In this paper, we study a so-called Condition C1 on square matrices with complex coefficients and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. We show that these conditions satisfy many good properties and in particular are satisfied by a dense subset of the set of square matrices of a given rank $r$. Based on this, we propose a heuristic argument for the truth of the Jacobian conjecture. We propose some new equivalent formulations and some generalizations of the Jacobian conjecture, and some approaches (including computer algebra and numerical methods) toward resolving it. We show that some of these equivalent formulations are automatically satisfied by generic Druzkowski matrices. Applications and experimental results are included.

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras ${𝔣}_4$ and ${𝔣}_4^{\ast }$, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fiber bundles. They are known to have positive signature divisible by $4$. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyze by studying the monodromy action for bundles of punctured curves. As a by-product, we obtain a classification of all fix-point-free automorphisms on curves of genus at most $9$.

For Banach spaces $X$ and $Y$, we establish a natural bijection between preduals of $Y$ and preduals of ${ℒ}(X,Y)$ that respect the right ${ℒ}(X)$-module structure. If $X$ is reflexive, it follows that there is a unique predual making ${ℒ}(X)$ into a dual Banach algebra. This removes the condition that $X$ have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement $Y$ in its bidual and ${ℒ}(X)$-linear projections that complement ${ℒ}(X,Y)$ in its bidual. It follows that $Y$ is complemented in its bidual if and only if ${ℒ}(X,Y)$ is (either as a module or as a Banach space). Our results are new even in the well-studied case of isometric preduals.

Given a vector bundle $E$ on a complex reduced curve $C$ and a subspace $V$ of $H^0(E)$ which generates $E$, one can consider the kernel of the evaluation map ${\mathrm{\text{ev}}}_V:V\otimes {{𝒪}}_C\to E$, i.e. the kernel bundle $M_{E,V}$ associated to the pair $(E,V)$. Motivated by a well-known conjecture of Butler about the semistability of $M_{E,V}$ and by the results obtained by several authors when the ambient space is a smooth curve, we investigate the case of a reducible curve with one node. Unexpectedly, we are able to prove results which goes in the opposite direction with respect to what is known in the smooth case. For example, $M_{E,H^0(E)}$ is actually quite never $w$-semistable. Conditions which gives the $w$-semistability of $M_{E,V}$ when $V\subset H^0(E)$ or when $E$ is a line bundle are then given.

We will give a new proof of Ramanujam’s topological characterization of the affine $2$-space $C^2$ using the Abhyankar–Moh–Suzuki theorem on embeddings of the affine line in $C^2$ and some ideas from the theory of open algebraic surfaces.

The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions.

Ito–Takimura recently defined a splice-unknotting number $u^{-}(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.