## Student Number Theory Seminar 2019-20

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4/20/20: Jacob Hegna -- *Elliptic cohomology and topological modular forms*

**Abstract:** Chromatic homotopy theory studies the interplay between generalized cohomology theories and algebraic geometry via the geometry of formal groups. A distinguished class of one-dimensional formal groups are given by neighborhoods of infinity on elliptic curves, and to each elliptic curve we may thus associate a cohomology theory. These theories are known as "elliptic cohomology," and the universal example is TMF, or topological modular forms. This spectrum intertwines the primary object of interest in stable homotopy theory (the stable homotopy groups of spheres) and the classical ring of modular forms. In this talk, I will discuss the construction of TMF, applications of TMF, and further connections between number theory and homotopy theory.