The goal of this paper is to prove that a random polynomial with independent and identically distributed random coefficients taking values uniformly in $\{1,\dots ,210\}$ is irreducible with probability tending to $1$ as the degree tends to infinity. Moreover, we prove that the Galois group of the random polynomial contains the alternating group, again with probability tending to $1$.

We construct a noncommutative desingularization of the discriminant of a finite reflection group $G$ as a quotient of the skew group ring $A=S\ast G$. If $G$ is generated by order $2$ reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement $\mathcal{A}\left(G\right)$ viewed as a module over the coordinate ring $S^G/(\Delta )$ of the discriminant of $G$. This yields, in particular, a correspondence between the nontrivial irreducible representations of $G$ to certain maximal Cohen–Macaulay modules over the coordinate ring $S^G/(\Delta )$. These maximal Cohen–Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement $\mathcal{A}\left(G\right)$ viewed as a module over $S^G/(\Delta )$. We identify some of the corresponding matrix factorizations, namely, the so-called logarithmic (co-)residues of the discriminant.

We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently proposed Riemann–Hilbert representation of the rogue wave solution of arbitrary order $k$, we establish the existence of a limiting profile of the rogue wave in the large-$k$ limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables—the rogue wave of infinite order—which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving Riemann–Hilbert problems. In a certain transitional region for the asymptotics, the near-field limit function is described by a specific globally defined tritronquée solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.

We use the recently established existence and regularity of area and energy minimizing disks in metric spaces to obtain canonical parameterizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parameterizations of linearly locally connected, Ahlfors $2$-regular metric $2$-spheres. Generalizations and applications to the geometry of such surfaces are described.

We correct a definition in “Discriminants in the Grothendieck ring,” Duke Math. J. 164 (2015), no. 6, 1139–1185.