Moon Duchin (Tufts University)
Title: Mathematical models in social context
Abstract: It's really hard to get mathematical modeling right! And when you are modeling social and political phenomena, there are inevitably going to be hidden assumptions in the model design and unexpected impacts of model choices. This is hard—but fundamentally important—to study. I'm a pure mathematician who has ventured into a very political area: electoral redistricting. I'll use examples from gerrymandering to make the broader point about the human interfaces of model-making.
Sam Payne (University of Texas at Austin)
Title: Graph complexes
Abstract: Consider a vector space generated by isomorphism classes of finite graphs, graded by the number of edges. Equip this with a differentia $d$, defined as a signed sum of edge contractions, with signs chosen so that $d^2 = 0$. The resulting complex of vector spaces is a graph complex, and its homology $\mathrm{ker}(d)/\mathrm{im}(d)$ is a graded group, called graph homology. Many variations are possible, depending on choices of signs, or by decorating the graphs with additional data, such as a ribbon structure. Such complexes are elementary to define, and suitable for direct computation by hand or by computer, yet they encode deep mathematical structures, including the homology of moduli spaces of curves and Feynman amplitudes in Chern-Simons topological quantum field theory.
In this talk, I will focus on one relatively simple graph complex, known as the “commutative graph complex” and discuss some of its known features, conjectured properties, and open problems for future research.
Tatiana Toro (University of Washington)
Title: The world through different mathematical lenses: rough vs smooth.
Abstract: This lecture will introduce some of the tools used in mathematics to distinguish between rough and smooth objects. Then it will explore some of their wide-ranging applications. One of the goals is to illustrate how the development of deep mathematical concepts is often motivated by real life observations.