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What is the maximum proportion of d-dimensional space that can be covered by disjoint, identical spheres? In this talk I will discuss a new lower bound for this problem, which is the first asymptotically growing improvement to Rogers’ bound from 1947. Our proof is almost entirely combinatorial and reduces to a novel theorem about independent sets in graphs with bounded degrees and codegrees.

This is based on joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.

**Special Location**

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For more info, see https://math.mit.edu/combin/

The bordered Floer homology of Lipshitz, Ozsvath, and Thurston was interpreted by Auroux as defining an element in the partially wrapped Fukaya category of a symmetric product of the boundary. We naturally expect that this assignment should be functorial; e.g. a cobordism between two manifolds with torus boundary should induce a morphism between the corresponding Lagrangians. I’ll describe a framework for thinking about functoriality in terms of sutured manifolds and describe what it looks like for Heegaard Floer homology.