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Abstract: I'll discuss Stiefel's combinatorial formula and Whitney's characteristic singularity cycles, which are common threads of my work with Tom Banchoff over thirty years. I will explain some of my favorite topics: singularities and the Euler characteristic, combinatorial Steenrod homology operations, binocular visual topology of surfaces, and cusps of Gauss maps. I'll conclude with a new version of Stiefel's formula for toric varieties.
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