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.

A class C of abelian groups is called a torsion class if it is closed under homomorphic images, group extensions, and (possibly infinite) direct sums. A torsion class is singly generated if it is the closure of a single group under such operations. For example, the entire class of all abelian groups is a singly-generated torsion class (generated by the group of integers). It is difficult to come up with examples of torsion classes that are NOT singly-generated, and there is good reason: whether every torsion class is singly generated is independent of the axioms of mathematics. In fact, the scheme "every torsion class of abelian groups is singly generated" is equivalent to the large cardinal principle known as Vopenka's Principle.

Recent work on generalizing algebraic K-theory to apply its techniques beyond the abelian setting (due to Campbell—Zakharevich and Sarazola—S.) has focused on how simple objects like sets can be treated analogously to R-modules by regarding complementary inclusions as short exact sequences. Based on this analogy, we can define chain complexes of sets and their associated sequences of homology sets. These chain complexes satisfy many theorems from homological algebra, whose proofs can be demonstrated visually using Venn diagram-like pictures. These pictures, in which the snake lemma and related long exact sequence of a pair actually look like snakes, can be useful tools for researchers and may have pedagogical potential for homological algebra more broadly.

Woodin's striking HOD dichotomy states that, under the presence of an extendible cardinal, either HOD is close to the mathematical universe or it is far apart from it. Specifically, if δ is an extendible cardinal, either HOD is a weak extender model for the supercompactness of δ or every regular cardinal κ≥δ is Ω-strongly measurable in HOD. Woodin's HOD hypothesis is the postulate asserting that the second of the above alternatives cannot occur.

In this presentation I will report on various recent results on the optimality of the HOD dichotomy. For instance, we will show that even under the HOD hypothesis the first extendible cardinal may not be extendible in HOD. Also, we show that the HOD hypothesis is consistent with the first supercompact cardinal carrying a club of HOD-regular cardinals whose successors are incorrectly computed by HOD. As a bi-product of our analysis, we solve questions by Cheng--Friedman--Hamkins and Cummings--Friedman--Golshani, respectively. (This is joint work with Gabe Goldberg).

Let K be a field. A basis B of a K-algebra R is multiplicative provided that for all b, b' ∈ B either b.b' ∈ B or b.b' = 0. Multiplicative bases are abundant: full matrix algebras, group algebras, polynomial algebras, and path algebras of quivers all possess multiplicative bases. One of the highlights of classic representation theory of finite dimensional algebras is the proof in [1] that for algebraically closed fields K, each K-algebra R of finite representation type possesses a normed multiplicative basis B. Here, normed means that B contains a complete set of primitive orthogonal idempotents of R as well as a K-basis of each power of the Jacobson radical of R.

However, there are many settings, notably of infinite dimensional algebras, where no multiplicative bases exist. This is the case when R is a field and K ⊊ R, when R is the endomorphism ring of an infinite dimensional K-linear space, or R is an infinite product of fields.

In [2], the class of infinite dimensional semiartinian algebras R of countable type was introduced and proved to possess conormed multiplicative bases. A K-basis B of R is conormed, if B contains a complete set of orthogonal idempotents of R as well as a K-basis of each layer of the socle sequence of R.

Though the latter result is in a sense dual to [1], its proof is quite different: We assign to each semiartinian von Neumann regular algebra R an invariant called the dimension sequence, and show that for algebras of countable type, this dimension sequence determines R up to a K-algebra isomorphism. Finally, we construct the unique representatives of these isomorphism classes together with their conormed multiplica- tive bases by induction on the socle length.

[1] R. Bautista, P. Gabriel, A.V. Roiter, L. Salmerón, Representation-finite algebras and multiplicative bases, Invent. Math. 81(1985), 217–285.

[2] K. Fukov ́a, J.Trlifaj, Multiplicative bases and commutative semiartinian von Neumann regular algebras.

This talk is jointly hosted by the Analysis, Logic and Physics Seminar and the Geometry and Topology Seminar.

We show that assuming the background periodic wave is diffusively stable, a stronger form of spectral stability, then the wave is nonlinearly stable even in the presence of conservation laws. The key difference with the case without conservation law analyzed by Melinand-Rodrigues is that even for extremely nice perturbations the linearized semigroup decays at a slow rate and so phase modulations play a deeper role. This work is joint with L. Miguel Rodrigues

In this talk I will describe a recently discovered dynamics called rippled almost periodic behavior. I will go through the process of the discovery of this behavior and give recent results of a model that exhibits this behavior. I will also discuss open problems involving this model as well as this behavior that has a structure that appears to be pseudo-periodic however is also "unpredictable."