In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev–Slobodeckij spaces, in part motivated by a simple counter-example that illustrates how certain multiplication theorems fail in Sobolev–Slobodeckij spaces when a bounded domain is replaced by R^n. We identify the source of the failure, and examine why the same failure is not encountered in Bessel potential spaces. To analyze the situation, we begin with a survey of the classical multiplication results stated and proved in the 1977 article of Zolesio, and carefully distinguish between the case of spaces defined on the all of R^n and spaces defined on a bounded domain (with e.g. a Lipschitz boundary). However, the survey we give has a few new wrinkles; the proofs we include are based almost exclusively on interpolation theory rather than Littlewood–Paley theory and Besov spaces, and some of the results we give and their proofs, including the results for negative exponents, do not appear in the literature in this form. We also include a particularly important variation of one of the multiplication theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The conditions for multiplication to be continuous in the case of Sobolev–Slobodeckij spaces are somewhat subtle and intertwined, and as a result, the multiplication theorems of Zolesio in 1977 have been cited (more than once) in the standard literature in slightly more generality than what is actually proved by Zolesio, and in cases that allow for construction of counter-examples such as the one included here.

We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family contains skew shapes, consisting of disjoint rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier results by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in the form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occurs with the same frequency.

We study the existence of Artin–Schreier curves with large a‑number. We show that Artin–Schreier curves with large a‑number can be written in certain forms and discuss their supersingularity. We also give a basis of the de Rham cohomology of Artin–Schreier curves. By computing the rank of the Hasse–Witt matrix of the curve, we also give bounds on the a‑number of trigonal curves of genus 5 in small characteristic.

Let f be a holomorphic automorphism of a compact Kähler manifold. Assume that f admits a unique maximal dynamic degree d_p with only one eigenvalue of maximal modulus. Let μ be its equilibrium measure. In this paper, we prove that μ is exponentially mixing for all d.s.h. test functions.

Let X be a smooth complex projective variety. Using a construction devised by Gathmann, we present a recursive formula for some of the Gromov–Witten invariants of X. We prove that, when X is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of X. This generalizes the classical well known pairs of inflection (asymptotic) lines for surfaces in P^3 of Salmon, as well as Darboux’s 27 osculating conics.