7/12/2021 – Randy Van Why (Northwestern): Disk bundle plumbings, lens spaces, and continued fractions
I will attempt to state and prove a theorem showing a strange connection between continued fractions, lens spaces, and disk bundle plumbings. I will start by introducing the plumbing construction and surgery and then sketch a proof of the theorem.
6/14/2021 – Hari Rau-Murthy: The matrix exponential, the Bismut Chern character, and the character of a representation
The matrix exponential can be used to solve the differential equation dγ/dt = A(t)γ(t)$. We will discuss a cool trick involving this matrix exponential. This trick will be used to define the Bismut Chern character, which is the trace of a certain matrix exponential associated to a loop, γ(t), in a manifold. The Bismut Chern character has striking connections to the group theoretic character of a representation, which is trace of a matrix that represents an element, g, of a group. The loop γ(t) will end up corresponding to the conjugacy class of g. Thus we relate a differential geometric construction to an algebraic construction.