Multimodal reasoning in advanced mathematics: Argumentation and proof through simplicial complexes

In this paper we examine how simplicial complexes serve as a paradigmatic example of multimodal mathematical reasoning in advanced university mathematics, particularly regarding argumentation and proof. We analyse how an undergraduate mathematics student engaged with this topological concept through visual-geometric, computational, and formal-axiomatic reasoning modes while exploring rich connections to STEM disciplines. By examining strategic choices in presenting key theorems, we aim to show how the constructive nature of simplicial complexes enables progression through levels of mathematical complexity. Our analysis reveals how different proof strategies and representational modes support distinct forms of mathematical argumentation across disciplinary boundaries, contributing to understanding how advanced mathematical topics can foster integrated development of multiple reasoning competencies essential for contemporary mathematical practice.