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The commutative graph complex, defined by contracting edges in graphs, was introduced by Kontsevich in 1993. Recently, Chan-Galatius-Payne have shown that its cohomology can be identified with a piece of the cohomology of  moduli spaces of curves, and Willwacher showed that graph cohomology in degree zero it is related to the Grothendieck-Teichmuller Lie algebra.  Nevertheless, the cohomology of the graph complex remains mysterious, and little is known explicitly.

In this talk I will explain how to use invariant trace forms to construct differential forms on a moduli space of metric graphs. By integrating these forms, one can canonically assign integrals to graphs, which are very closely related to Feynman integrals in perturbative quantum field theory, but have the property that they always converge.  A Stokes formula enables us to deduce some information about the cohomology of the graph complex, and leads to new predictions about its structure, and its relation to physics.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Liquid crystals are an intermediate state of matter which flow like liquids but retain molecular ordering similar to that of crystals. Their physical properties make them ideal for a wide range of technological applications. The molecules in smectic liquid crystals form well-defined layers which slide across one another. In this talk we will discuss a model for smectics from the physics literature based on the minimization of a suitable energy and present recent results obtained jointly with Xiaodong Yan. No prior knowledge of liquid crystals or the relevant mathematics will be assumed.

Zoom: https://harvard.zoom.us/j/98248914765?pwd=Q01tRTVWTVBGT0lXek40VzdxdVVPQT09

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