Notes

Universal Coefficient Theorem for Homology Definition of homology with coefficients (in an abelian group) and a proof of the universal coefficient theorem, modulo some facts from homological algebra. Made in preparation of a lecture I did for my algebraic topology reading group.

Schrödinger-Lichnerowicz-Weitzenböck Formula These notes are an (unsuccessful) attempt to write down basic facts about spin connections, Dirac operators, and curvature in notation that conforms to my aesthetic sensibilities (no proofs, sorry!).

Group C*-Algebras I made these slides for a talk I gave as part of a course on C*-algebras at SISSA. Includes a proof of the fact that every unitary representation of a group induces a nondegenerate unitary representation of the group C*-algebra and vice versa.

Analysis on the Euclidean Motion Group A classification of representations of \(M(2)\), the Euclidean motion group in two dimensions. Also: Fourier transforms, Parseval identity, Plancherel theorem for square-integrable functions on \(M(2)\).

Hilbert Spaces, Tensor Products and Representations of \(\mathfrak{sl}(2, \mathbb C)\) Includes a proof that every Hilbert space has an orthonormal basis, a short discussion of tensor product of Hilbert spaces, classification of unitary representations of the Lie algebra \(\mathfrak{sl}(2, \mathbb C)\), and addition of angular momentum.