Some thoughts on mathematics

  In this post, we describe how ASD connections on four-manifolds relate to Chern-Simons theory on three-manifolds. Suppose that we have a smooth four-dimensional cobordism $W : Y_1 \to Y_2$ between homology three-spheres and consider the trivial $SU(2)$ bundle on $W$.  Just like for closed manifolds, we can count ASD connections on $W$.  The very rough idea is that an ASD connection restricts to a flat connection on the $Y_i$, and we can use it to relate the Chern-Simons invariants of $Y_1$ and $Y_2$.  (I lied a little, but I explain the issue in a technical point below.)  At the end of this post, we'll show that there is no simply-connected homology cobordism from the Poincare sphere to itself.  Get pumped! Let's be a little more precise with our gauge theory now.  For a connection $A$ on $W$, we can measure its  energy:    $$ \mathcal{E}(A) = \int_W F_A \wedge F_A. $$  If you don't like the formula, there are two relevant facts: if $A$ is ASD, then $\mathcal{E}(A) \

This popped in my head, so let me make this post short. I am anti-colloquium.   I never remember the content of the non-geometry/topology colloquia.  Not even general guiding principles.  The only exception is I remember that a lot of people care about modular forms, whatever those are.  Plenty of colloquium talks are poorly executed by the speaker, but many I actually enjoy and understand some amount of until I fall asleep.  But then it's immediately gone when I leave the room.  I personally don't think colloquium enhances my research or teaching; I don't think it helps my mentees either, unless it's in geometry/topology or maybe demonstrating how to give a good (or bad) talk.     I recognize that there's other value to the colloquium.  As one example, there aren't many departmental events that bring together all students and faculty, especially if tea is tied entirely to the colloquium.  It is important to keep the department from being too siloed.  I do think