In this thesis, we consider some aspects of$noncommutative classical invariant theory$, i.e., noncommutative invariants of$the classical group SL(2, k)$. We develop a$symbolic method$for invariants and covariants, and we use the method to compute some invariant algebras. The subspace$Ĩ$_{d}^{m}of the noncommutative invariant algebra$Ĩ$_{$d$}consisting of homogeneous elements of degree$m$has the structure of a module over the$symmetric group S$_{$m$}. We find the explicit decomposition into irreducible modules. As a consequence, we obtain the$Hilbert series$of the commutative classical invariant algebras. The$Cayley—Sylvester theorem$and the$Hermite reciprocity law$are studied in some detail. We consider a new power series H($Ĩ$_{d},$t$) whose coefficients are the number of irreducible$S$_{$m$}-modules in the decomposition of$Ĩ$_{d}^{m}, and show that it is rational. Finally, we develop some analogues of all this for covariants.

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We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when$h$is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of$h$joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −$a$< 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4;$a$/λ - 1,$b$/λ - 1), a variant of SLE(4).

Let$E$ç$S$^{1}be a set with Minkowski dimension$d(E)1$. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions of$E$. The main result of this paper shows that these operators are bounded on$L$_{$rad$}^{$p$}(R^{2}) for$p>1+d(E)$and unbounded when$p<1+d(E)$. We also give some end-point results.

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