Georgia Institute of Technology College of Sciences

Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called geometric bijections. Recently we have extended the geometric bijections to  subgraph-orientation correspondences. In this talk, I will introduce the bijections and the geometry behind them.

What are all the ways to draw the lines, when you're dividing up a state to get representation? If you can't find them all, can you choose a good sample? I'll discuss some surprisingly simple questions about graphs and geometry that can help us make advances in policy and civil rights.

Living and active systems exhibit various emergent dynamics necessary for system regulation, growth, and motility. However, how robust dynamics arises from stochastic components remains unclear. Towards understanding this, I develop topological theories that support robust edge states, effectively reducing the system dynamics to a lower-dimensional subspace. In particular, I will introduce stochastic networks in molecular configuration space that enable different phenomena from a global clock, stochastic growth and shrinkage, to synchronization. These out-of-equilibrium systems further possess uniquely non-Hermitian features such as exceptional points and vorticity. More broadly, my work  provides a blueprint for the design and control of novel and robust function in correlated and active systems.

The problem of classifying collections of objects (graphs, manifolds, operators, etc.) up to some notion of equivalence (isomorphism, diffeomorphism, conjugacy, etc.) is central in every domain of mathematical activity. Invariant descriptive set-theory provides a formal framework for measuring the intrinsic complexity of such classification problems and for deciding, in each case, which types of invariants are “too simple” to be used for a complete classification. It also provides a very interesting link between topological dynamics and the meta-mathematics of classification. In this talk I will discuss various forms of classification which naturally occur in mathematical practice (concrete classification, classification by countable structures, classification by cohomological invariants, etc.) and I will provide criteria for showing when some classification problem cannot be solved using these forms of classification.