Title: On the geometry of moduli spaces of curves
Compact Riemann surfaces are two-dimensional real manifolds with a complex structure. In 1857, Riemann was studying how to vary a complex structure on a compact Riemann surface, and he first suggested that such variations should be called "moduli." A word that algebraic geometers use now habitually and like to study from many perspectives. Given the beautiful correspondence between compact Riemann surfaces and complex complete algebraic curves, or just by thinking of compact Riemann surfaces as $1$ dimensional complex manifolds, we talk in the modern language of "moduli of curves." In this seminar, I will introduce them as complex varieties, from which the terminology "moduli spaces" comes. I will convince you that studying holomorphic forms, namely sections of the cotangent bundle and its exterior powers, is a powerful tool for understanding some of their geometric features. To do this, I will survey an established and deeply studied perspective, suggest a new direction of research, and explain my first result in this direction.
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Meeting ID: 890 3249 0100