Part 1: "Where do we live?" I'll first set up the geometry of democracy by looking at human geography and discussing how to model patterns of residential clustering and segregation.

Part 2: "How do we vote?" Next, we'll review some models for studying voting patterns and polarization. I'll discuss adding spatial dimensions to statistical models of voting behavior.

Part 3: "Systems and fairness" Then we'll bring the geometry and probability together to understand how systems of election can interact with political geography and complex preferences to give different kinds of representational outcomes. And we'll try to think in sophisticated ways about the big underlying question -- what is fair representation?

Growth functions of groups have been studied intensively in geometric group theory.  Regarded up to affine rescaling, they give a group invariant called the growth rate, and there are long-standing questions about possible growth rates for finitely generated groups.  Another stream of questions asks whether the growth values satisfy a recursion, a property which is called rational growth and depends on the choice of generators.  I will survey the area and discuss the classic proofs that hyperbolic groups and virtually abelian groups have rational growth in any generators

Growth functions of groups have been studied intensively in geometric group theory.  Regarded up to affine rescaling, they give a group invariant called the growth rate, and there are long-standing questions about possible growth rates for finitely generated groups.  Another stream of questions asks whether the growth values satisfy a recursion, a property which is called rational growth and depends on the choice of generators.  I will survey the area and discuss the classic proofs that hyperbolic groups and virtually abelian groups have rational growth in any generators