Uniform dimension results of level-sets of stable sheets

In this thesis, we investigate the geometric and dimensional properties of the zero sets of Gaussian processes. We begin by analyzing a specific Gaussian process, the Brownian sheet, and examining the Marstrand-type projection properties of its zero set in the multidimensional setting. Resolving an open problem posed in [\cite{khoshnevisan2007slices}], we prove that orthogonal projections of sufficiently high rank preserve the Hausdorff dimension of the zero set, whereas lower-rank projections result in sets with positive Lebesgue measure in the ambient subspace. Subsequently, we extend these results to general stable processes, relaxing the assumption of the stochastic Hölder condition. We develop a decomposition method that partitions the parameter space into small hypercubes, allowing us to estimate the zero set via hitting probabilities. Analogous to the Brownian sheet, we establish a Marstrand-type theorem for stable processes: high-rank projections preserve dimension, while lower-rank projections fill the target space. Finally, we investigate slices of the zero set by studying the self-intersections of sliced Brownian sheets, addressing two additional open problems. We establish a correspondence between the $p$-fold self-intersections of sliced Brownian sheets and the intersection of $p$ independent Brownian loops. Furthermore, we derive a construction for points of infinite multiplicity and determine the critical conditions for the existence of infinitely many zeros.