Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will summarize what is known about the noetherianity of enveloping algebras, with a focus on Lie algebras of derivations of associative algebras.

Using amalgamated free products of lamplighter groups, we construct masas that have exactly k maximal amenable extensions which are factors. Moreover, in this setting we are able to identify the k-simplex with the space of all maximal amenable extensions. The proof uses the concepts of simultaneous relative asymptotic orthogonality, an extension of relative asymptotic orthogonality, and the uniform flattening strategy, which was introduced by the authors in a previous paper to compute lower bounds on sequential commutation path lengths. This work is joint with Srivatsav Kunnawalkam Elayavalli.

Let $\mathrm{Mat}_{n \times n}(\mathbb{C})$ be the affine space of $n \times n$ complex matrices with coordinate ring $\mathbb{C}[{\mathbf x}_{n \times n}]$. We define graded quotients of $\mathbb{C}[{\mathbf x}_{n \times n}]$ where each quotient ring carries a group action. These quotient rings are obtained by applying the orbit harmonics method to matrix loci corresponding to the permutation matrix group $S_n$, the colored permutation matrix group $S_{n,r}$, the collection of all involutions in $S_n$, and the conjugacy classes of involutions in $S_n$ with a given number of fixed points. In each case, we explore how the algebraic properties of these quotient rings are governed by the combinatorial properties of the matrix loci. Based on joint work with Yichen Ma, Brendon Rhoades, and Hai Zhu.

The canonical van der Waerden theorem states that, for large enough $n$, any colouring of $[n]$ gives rise to monochromatic or rainbow $k$-APs. In joint work with Alvarado, Kohayakawa, Morris and Mota, we study sparse random versions of this result. More concretely, we determine the threshold at which the binomial random set $[n]_p$ inherits the canonical van der Waerden properties of $[n]$, using the container method.

We will discuss an inductive approach to determining the asymptotic number of G-extensions of a number field with bounded discriminant, and outline the proof of Malle's conjecture in numerous new cases. This talk will include discussions of several examples demonstrating the method.

[pre-talk at 3:00PM]

We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. This is a joint work with Subhajit Jana.

In Bernoulli percolation, events such as "two vertices are connected by an open path" naturally emerge. In this talk, I will explore the dependencies between these events for various vertex pairs and derive key FKG-type inequalities governing their probabilities and explain the relevance of these inequalities to the recent disproof of the Bunkbed Conjecture.

Self-expanders are a special class of solutions to the mean curvature flow, in which a later time slice is a scale-up copy of an earlier one. They are also critical points for a suitable weighted area functional. Self-expanders model the asymptotic behavior of a mean curvature flow when it emerges from a cone singularity. The nonuniqueness of self-expanders presents challenges in the study of cone-like singularities in the flow. In this talk, I will discuss some recent development on a variational theory for self-expanders and an application to the question on lower density bounds for minimal cones.

In the 1960s Thoma developed a theory of characters which generalizes the classical Fourier/Pontryagin theory of abelian groups, and at the same time Frobenius' theory on finite (and compact) groups.

After presenting the general theory, I will focus on arithmetic groups, or similarly, lattices in (semi)simple Lie groups, and tell about my work with Levit and Slutsky regarding the geometry/topology of the space of characters of such groups. Our main result is that for lattices in higher rank simple Lie groups (e.g for the group SL3(Z)), any sequence of distinct characters must converge pointwise to the dirac character at the identity.  This implies character bounds of finite groups of Lie type (e.g SL3(Fp)).

Does every infinite group admit a generating set such that the corresponding Cayley graph has infinite diameter? While there are examples of uncountable groups that fail to satisfy this property (e.g., the group of all permutations of the integers), the question for countable groups remains open. After reviewing the necessary background and some known results, I will discuss an attempt to solve this problem by choosing a random generating set. For a wide class of countable groups, this approach answers the question affirmatively and reveals a surprising phenomenon: random generating sets yield the same Cayley graph, independent of the group. Depending on the randomness model, this is either the familiar Rado graph (which has diameter 2) or a certain mysterious graph of infinite diameter.

Not the factorizations you're thinking of! A 1-factorization of a graph is a partition of its edges into perfect matchings. In this talk I hope to share some of the many questions about 1-factorizations that I find interesting, explain how I came to care about a rather obscure fact, and prove it. Along the way we will draw some pretty pictures, of course. This talk should be totally accessible to anyone who has ever heard of graphs, and is based on collaboration with Michael Orrison and Rohan Chauhan.