Schedule, Notes/Handouts & Videos

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?

This mini-course will focus on the central role of mathematics in Data Science and machine learning. An overview of unsupervised learning and clustering will be presented along with the importance of linear algebra in our data-informed world. We will also discuss the mathematics behind recommendation systems - the algorithms behind the next video you see, the next song playing, and the products you are suggested. Different filtering and similarity metrics will be defined, applied, and implemented on different data sets. By the end of the mini-course, you will leave with a theoretical and practical (in Python) understanding of the role of mathematics in data science and its applications in biology, social change, and more.

How does the dimension of the first cohomology grow in a tower of covering spaces? After a tour of examples of behaviors for low-dimensional spaces, I will focus on arithmetic manifolds. Specifically, for towers of complex hyperbolic manifolds, I will describe how to bound the rates of growth using results from Langlands functoriality.

Based on the tile method for DNA self-assembly, which involves branched junction molecules whose flexible k-arms are double strands of DNA, a collection of tiles can be designed to construct a nanostructure shaped like a target graph. A critical design step is finding minimal sets of branched junction molecules that will self-assemble into target structures subject without unwanted substructures forming. We apply tools from graph theory to address this problem and we show that finding optimal design strategies for this method is generally NP-complete. Additionally, we provide pragmatic solutions in the form of programs for special settings and provably optimal solutions for natural assembly targets such as platonic solids and regular lattices.

In the real world, behaviors of the people in the society cannot be directly controlled, instead they react to policies according to their own best interests while they interact with other individuals. In this talk, we discuss how we can utilize the power of mathematics and game theory to design policies that incentivize people to react in preferable ways. We first start with discussing the importance of game theory and related notions such as the Nash equilibrium. Then we discuss how to quantify social problems with the use of mathematical modeling. After discussing game theoretical ideas and mathematical modeling, we look at some examples where we can apply these approaches, such as controlling the carbon emission by implementing a carbon tax to slow down the climate change and deciding on the best non pharmaceutical policies (such as social distancing) to decrease the spread of an epidemic.

Participants with different backgrounds and trajectories will discuss their experiences navigating the joys and struggles of graduate school and finding ways to thrive, personally and professionally.