We give a generators and relations presentation of the HOMFLYPT skein algebra $H$ of the torus $T^2$, and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that $H$ is isomorphic to the $\sigma ={\overline{\sigma }}^{-1}$ specialization of the elliptic Hall algebra of Burban and Schiffmann.

As an application, for an iterated cable $K$ of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the $\lambda$-colored HOMFLYPT polynomial of $K$. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko using the ${\mathfrak{gl}}_N$ double affine Hecke algebra. This proves one of the connection conjectures in their recent work.

Our interest in this article is a generalization of the additive Deligne–Simpson problem, which was originally defined for Fuchsian differential equations on the Riemann sphere. We extend this problem to differential equations having an arbitrary number of unramified irregular singular points, and we determine the existence of solutions of the generalized additive Deligne–Simpson problems. Moreover, we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere (namely, open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces). Furthermore, we detail the connectedness of the moduli spaces.

We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1<p<\infty$, a Hörmander-type holomorphic functional calculus on $L^p$ in the sector of angle ${\varphi }_p^{\ast }=arcsin|1-2/p|$. The obtained angle is optimal.

Let $\mathbf{k}$ be an algebraically closed field of characteristic $p>0$, and let $C$ be a nonsingular projective curve over $\mathbf{k}$. We prove that for any real number $x\ge 2$, there are minimal surfaces of general type $X$ over $\mathbf{k}$ such that (a) $c_1^2\left(X\right)>0$, $c_2\left(X\right)>0$, (b) ${\pi }_1^{\acute{e}t}\left(X\right)\simeq {\pi }_1^{\acute{e}t}\left(C\right)$, and (c) $c_1^2\left(X\right)/c_2\left(X\right)$ is arbitrarily close to $x$. In particular, we show the density of Chern slopes in the pathological Bogomolov–Miyaoka–Yau interval $(3,\infty )$ for any given $p$. Moreover, we prove that for $C={\mathbb{P}}^1$ there exist surfaces $X$ as above with $H^1(X,{\mathcal{O}}_X)=0$, that is, with Picard scheme equal to a reduced point. In this way, we show that even surfaces with reduced Picard scheme are densely persistent in $[2,\infty )$ for any given $p$.