Let \(R\) be a commutative Noetherian domain with identity and finite Krull dimension \(d\). If \(R\) is non-singular, then for every ideal \(I \subseteq R\) and \(n \in \mathbb{N}\), the symbolic power \(I^{(dn)}\) is contained in the ordinary power \(I^n\). This property is known as the Uniform Symbolic Topology Property, reflecting a uniform comparison between symbolic and ordinary powers of ideals in \(R\).

Historically, this property was first established for rings over complex numbers by Ein, Lazarsfeld, and Smith, then extended to non-singular rings containing a field by Hochster and Huneke. Later, Ma and Schwede proved it for reduced ideals in excellent non-singular rings of mixed characteristic, and Murayama extended it to all regular rings, even non-excellent ones. Their proofs have profound connections with the subjects of multiplier/test ideal theory, closure operations, and constructions of big Cohen-Macaulay algebras.

The study of symbolic powers becomes significantly more challenging in the presence of singularities. In this talk, we focus on prime characteristic strongly \(F\)-regular singularities that arose from Hochster and Huneke's tight closure theory. We show that an \(F\)-finite strongly \(F\)-regular domain satisfies the Uniform Symbolic Topology Property, meaning there exists a constant \(C\) such that \(I^{(Cn)} \subseteq I^n\) for all ideals \(I \subseteq R\) and \(n \in \mathbb{N}\). We will explore the role of splitting ideals and Cartier linear maps in establishing this result.