A Discussion on the Description of Geometric Models in the Langlands Program Using Laurent Series

This paper explores the use of Laurent series in describing geometric models within the Langlands Program, with emphasis on the geometric (function field) perspective. We discuss how Laurent expansions aid in understanding local behavior near singularities and how these local methods can be “glued” to yield global information. Through various examples and recent results, we highlight the advantages and limitations of employing Laurent series for local analysis in geometric settings and touch on future research directions, including connections with the Fargues-Fontaine curve and \( p \)-adic methods.