Much of the important combinatorial information about an arrangement of hyperplanes can be understood by studying the cohomology ring of its complement. Given an arrangement with a group action G, a fundamental but difficult question is: how does one describe the G-representations on the (graded pieces of the) cohomology ring of its complement? In this talk, I will discuss some methods of answering this question for some of my favorite hyperplane arrangements, as well as challenges that arise. This is partially based on joint work with Megan Chang-Lee and Trevor Karn.
When does a system of coupled oscillators synchronize? This central question in dynamical systems arises in applications ranging from power grids to neuroscience to biology: why do fireflies sometimes begin flashing in harmony? Perhaps the most studied model is due to Kuramoto (1975); we analyze the Kuramoto model from the perspectives of algebra and topology. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks); our work also tackles more general situations.
We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. The talk will include a surprising (at least to us!) connection to Segre varieties, and close with examples of computations using the Macaulay2 software package "Oscillator"
Joint work with Heather Harrington (Oxford/Dresden) and Mike Stillman (Cornell).
For a hyperplane arrangement in a real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. We will introduce the Varchenko-Gel’fand ring and use it to study hyperplane arrangements and cones defined by intersections of halfspaces defined by some of the hyperplanes. Then, we will highlight a recent application Coxeter groups.
In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called “Wasserstein polytopes”or“Kantorovich-Rubinstein polytopes” in the literature. Recently such polytopes have been shown to play an important role in a host of different contexts – however, little is known to date about their structure. In particular, Vershik asked about the stratification of the metric cone according to the combinatorial type of such polytopes.
After stating the definitions and some examples, in this talk I will define an arrangement of hyperplanes that describes the stratification sought by Vershik. I will show some computational results on enumerative invariants in the case of metrics on up to six points and outline some combinatorial open problems arising in this context.
Time permitting, we will compare Wasserstein polytopes with the more classical "Tight spans” of metric spaces, showing that the stratifications of the metric cone induced by these two combinatorial invariants are not related by refinement.
The talk is based on joint work with Lukas Kühne and Leonie Mühlherr.
Scattering amplitudes are functions used to predict the outcome of particle scattering experiments in physics. We will explain a notion of scattering amplitudes for matroids based on the twisted intersection forms of hyperplane arrangement complements. For the case of the complete graphic matroid, these functions agree with the "biadjoint scalar amplitudes at tree level". An important ingredient in the construction is a notion of "canonical form" for a tope of an oriented matroid, developed in joint work with Chris Eur.
We will explore volume polynomials in three distinct settings: convex bodies, Hermitian matrices, and divisor classes on compact Kahler manifolds. After establishing their interrelations, we will leverage these connections to derive new inequalities for mixed volumes, mixed discriminants, and intersection numbers. Our results generalize classical inequalities, including the Alexandrov-Fenchel inequality and the reverse Khovanskii-Teissier inequality. Joint work with June Huh and Mateusz Michalek.