Let $X$ be a topological space and $k$ a field of characteristic $p$. Let $A^\cdot$ be a bounded below complex of sheaves of differential graded commutative $\F_p$-algebras. We show that there exist Steenrod operations canonically defined on the sheaf hypercohomology groups, $\mathbf{H}^\cdot(X, A^\cdot)$. These Steenrod operations satisfy most of their usual properties, including the Cartan formula and the Adem relations. Suppose further that $A^\cdot$ is equipped with a filtration $F^\cdot$, which is finite in each degree, and compatible with the product on $A^\cdot$. The filtration on $F^\cdot A^\cdot$ induces a spectral sequence that converges to $\mathbf{H}^\cdot(X, A^\cdot)$, and we prove that the constructed Steenrod operations also have a compatible action on the $E_1$ and $E_\infty$ pages of this spectral sequence. When $X$ is a smooth projective variety over $k$, we obtain Steenrod operations on the algebraic De Rham cohomology groups, $H^\cdot_\text{DR}(X/k)$, as well as the Hodge cohomology groups. The Steenrod operations on $H^\cdot_\text{DR}(X/k)$ have a compatible action on the first and infinite pages of the Hodge to De Rham spectral sequence, as well as the spectral sequence from Katz and Oda related to the Gauss-Manin connection.

This dissertation studies the notion of simplicial distance on Bruhat-Tits buildings. That is a measure of proximity between vertices in the simplicial structure. The purpose of this research is three-fold: (i). to provide a concrete characterization of the simplicial distance; (ii). to better understand simplicial balls, and (iii). to derive a formula for the simplicial volume and explore its asymptotic growth.To accomplish these goals, a comprehensive examination of vertices becomes necessary. They are analyzed using three frameworks: root systems, norms, and lattices. By leveraging concave functions, we interpret simplicial balls as fixed-point sets of Moy-Prasad subgroups and deduce a formula for the simplicial volume. Additionally, the theory of ?-exponential polynomials is developed to facilitate the asymptotic study. Through this research, we focus on split classical types (namely, types of ??, ??, ??, ??, and any combination thereof) over a local field. The presented findings contribute to the advancement of our understanding of Bruhat-Tits buildings.