I will discuss a formulation of an Abelian Chern-Simons theory on the lattice employing the modified Villain formalism. The theory suffers from a well-known problem of having extra zero modes in the Gaussian operator. I will argue that these zero modes are associated with a kind of subsystem symmetry which projects out almost all naive Wilson loops. The operators which survive are framed Wilson loops. These turn out to be topological charges of the associated one-form symmetry, and it has the correct topological spin and correlation functions.

A C*-algebra has the lifting property (LP) if any unital completely positive map into a quotient C*-algebra admits a completely positive lift. The local lifting property (LLP), introduced by Kirchberg in the early 1990s, is a weaker, local version of the LP.  I will present a method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C*-algebras of countable groups with (relative) property (T). This allows us to derive that the full C*-algebras of the groups $Z^2\rtimes SL_2(Z)$ and $SL_n(Z)$, for n>2, do not have the LLP. The same method allows us to prove that the full C*-algebras of a large class of groups with property (T), including those admitting a probability measure preserving action with non-vanishing second real-valued cohomology, do not have the LP.  In a different direction, we prove that the full C*-algebras of any non-finitely presented groups with property (T) do not have the LP. Time permitting, I will also discuss a connection with the notion of Hilbert-Schmidt stability for countable groups. This is based on a joint work with Pieter Spaas and Matthew Wiersma.

A self-similar group is a group $G$ acting on a regular, infinite rooted tree by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups, like Grigorchuk's 2-group of intermediate growth, are of this form. Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra. Nekrashevych showed that the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims, and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example). The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.

In this talk, we discuss a result of the speaker and Benjamin Steinberg characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like GGS groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.

Lie groupoids are closely connected to pseudo-differential calculus. On a vector bundle considered as a `commutative Lie groupoid' (i.e. as a family of commutative Lie groups), they can be treated using the Fourier transform. In this talk, we explore the extension of this idea to the noncommutative space by employing the tubular neighborhood construction and subsequently adopting a global approach through the introduction of deformation to the normal cone (groupoid). By utilizing this groupoid, we can construct the analytic index of pseudo-differential operators without relying on pseudo-differential calculus.

Furthermore, through the canonical construction of the space of functions with Schwartz decay, pseudo-differential operators on a manifold can be represented as an integral associated with smooth functions on the deformation to the normal cone. This perspective provides a geometric characterization that allows for the direct proof of fundamental properties of pseudo-differential operators.

I will discuss various definitions of quantum or noncommutative graphs that have appeared in the literature, along with motivating examples.  One definition is due to Weaver, where examples arise from quantum channels and the study of quantum zero-error communication.  This definition works for any von Neumann algebra, and is "spatial": an operator system satisfying a certain operator bimodule condition.  Another definition, first due to Musto, Reutter, and Verdon, involves a generalisation of the concept of an adjacency matrix, coming from the study of (simple, undirected) graphs.  Here we study finite-dimensional C*-algebras with a given faithful state; examples are perhaps less obvious.  I will discuss generalisations of the latter framework when the state is not tracial, and discuss various notions of a "morphism" of the resulting objects

In 1986, Katznelson and Tzafriri proved that, if $T$ is a power-bounded operator on a Banach space $X$, and the spectrum of $T$ meets the unit circle only at 1, then $\|T^n(I-T)\| \to 0$ as $n\to\infty$. Actually, they went further and proved that $\|T^nf(T)\| \to 0$ if $T$ and $f$ satisfy certain conditions. Soon afterward, analogous results were obtained for bounded $C_0$-semigroups $(T(t))_{t\ge0}$. Further extensions and variants were proved later. I will speak about several extensions to the Katznelson-Tzafriri theorem(s), including in particular a recent result(s) obtained by David Seifert and myself.