Maximally even sets, which were discovered by Clough and Douthett (1991) in their work on music theory, are sets of vertices in cycle graphs that are "spaced out as evenly as possible." Maximally even sets have become popularized due to Toussaint’s discovery (2005) that such sets, which Toussint called Euclidean rhythms, manifest as rhythms in musical traditions from all over the world; it is worth noting that Toussaint’s work has led to the widespread use of Euclidean rhythm generators by composers and electronic music producers in many musical genres. In this talk we will generalize the notion of maximally even sets to address the question: which sets of vertices in a finite connected graph are "spaced out as evenly as possible?" We will describe how the mathematics of Issai Schur (1923) as well as Hardy, Littlewood and Polya (1929) is related to maximal evenness and present several new theorems.

While the classical cluster problem, where the goal is to find a configuration of N regions with fixed finite areas and an exterior region of infinite area so that the total perimeter of the interfaces between the regions is minimal, has been studied extensively in the past,  the study of locally isoperimetric partitions with more than one infinite region has only recently been initiated. In this talk I will present on the state-of-the-art regarding locally perimeter minimizing partitions and on recent work regarding the stability of these partitions in the spirit of the quantitative isoperimetric inequality. This is a joint work with Marco Bonacini and Riccardo Cristoferi.

In this talk we discuss a classical linear algebra problem arising in topology and Lie theory. We show how this problem is essentially equivalent to one known to be complete for a quantum complexity class of problems believed to be hard for a quantum computer to solve. 

The spectrum of the Laplace operator under Robin boundary conditions on bounded domains is a well-explored topic, with implications for various physical phenomena. For example, in analyzing the long-term behavior of Brownian motion with particle creation at the boundary, the principal eigenvalue provides information about the expected number of particles within the domain.

However, when the domain is unbounded - particularly the complement of a compact set - many open questions and intriguing challenges arise, which may also have applications in quantum theory. In my talk, I would like to discuss the existence of discrete eigenvalues, as well as the geometric optimization of the lowest point of the spectrum. We will find that while the complement of a ball represents a local optimum among domains of prescribed measure, it is not necessarily a global optimum.