The GEARS seminar

Alexandra Ciotau (Edinburgh)

Title: Growth and Noetherianity of U(W_+).

Abstract: Motivated by the question of whether any infinite-dimensional Lie algebra admits a Noetherian universal enveloping algebra, we study one of the best-understood cases, U(W_+).

We introduce Gelfand–Kirillov dimension as a measure of algebraic growth and work through examples of algebras of different GK-dimension, before showing that the partition-number growth rate (Hardy–Ramanujan’s formula) forces "GKdim"(U(W_+))=∞. Despite this, U(W_+) has a controlled two-sided ideal structure: too large to be Noetherian but more structured than a free algebra. We present the theorem of Iyudu and Sierra that U(W_+) has “just infinite” GK-dimension: every nonzero proper quotient has polynomial growth, making U(W_+) infinite-dimensional in the most minimal way.

Simeon Hellsten (Glasgow)

Title: Cobordism and Concordance of Surfaces in 4-Manifolds

Abstract: The goal of a mathematician is often to classify phenomena up to the “natural” notion of equivalence. Unfortunately, said phenomena are often impossible to classify, and so we force ourselves to be happy with a classification up to weaker equivalence relations.

One such phenomenon is embedded surfaces in 4-manifolds, where we are forced to weaken the natural notion of isotopy in various ways to obtain a complete classification. In this talk, I will discuss my completion of this classification up to cobordism, and up to concordance in simply-connected 4-manifolds. I will also discuss the status of the concordance classification in general 4-manifolds, and how much actually seems within reach.

Ervin Hadžiosmanović (Scuola Normale Superiore)

Title: Bounded cohomology of negatively curved manifolds and differential forms

Abstract: Bounded cohomology is a functional-analytical variant of singular cohomology developed by Gromov in the 80s in his work on Riemannian geometry. Since then, it has developed into an independent and rich research field. As we will see, the definition may seem innocuous, but it turns out that it behaves very differently from ordinary cohomology, leading to some strange phenomena (for example, it can be infinite-dimensional even for closed manifolds).

A general principle is that it tends to vanish for "flat" manifolds and be very big for "negatively curved" ones, even if its definition is purely topological. In this talk, I will focus on the second part of this principle by discussing a result of Barge and Ghys, which allows one to build quite concretely an infinite-dimensional subspace of the second-degree bounded cohomology of closed hyperbolic surfaces via differential forms.

If time permits, I will also discuss some possible related questions and answers.