BF theory does not quite fit into (strict) Atiyah’s axioms. The space of states it assigns to a boundary is typically infinite-dimensional (which implies that the partition function of $S^1 \times X$ is infinite). This can be seen (a) as a consequence of noncompactness of the phase space of the theory or (b) as a manifestation of the problem of zero-modes. The BV-BFV formalism is an approach to gauge theories (in particular, topological ones) combining the Atiyah-Segal functorial picture with the idea of Wilson’s effective action. In this talk I will sketch the construction of BF theory in the BV-BFV language and will explain how it assigns meaningful partition functions (satisfying an appropriate gluing property) to all cobordisms.

Let H denote a class of manifolds (such as SO (oriented), O (unoriented), Spin, Pin+, Pin-, manifolds with spin defects, etc.). We define a 2+1-dimensional H-modular TQFT to be one which lives on the boundary of a bordism-invariant 3+1-dimensional H-TQFT. Correspondingly, we define a H-modular tensor category to be a H-premodular category which leads to a bordism-invariant 3+1-dimensional TQFT. When H = SO, this reproduces the familiar Witten-Reshetikhin-Turaev TQFTs and corresponding modular tensor categories. For other examples of H, non-zero H-bordism groups in dimensions 4 or lower lead to interesting complications (anomalies, mapping class group extensions, obstructions to defining the H-modular theory on all H-manifolds).

The main idea behind noncommutative geometry is to “algebralize” geometric notions and then generalize them to noncommutative algebras. This way noncommutative geometry offers a generalised notion of the geometry. Quantum groups or Hopf algebras play the role of ‘group objects’ in noncommutative geometry and they provide an approach to the development of the theory much as Lie groups do in differential geometry.

I will give an introduction to the topic and briefly mention results on classification of all bialgebras and Hopf algebras of dimension ≤ 4 over the field $F_2 = \{0, 1\}$. These results can be summarized as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras and, among them, 25 Hopf algebras, with at most one of these from one vertex to another. We found a unique smallest noncommutative and noncocommutative quantum group, which is moreover self-dual and resembles a digital version of $u_q(\operatorname{sl}_2)$.

The pure spinor superfield formalism gives a systematic and geometric technique to construct supersymmetric field theories from algebro-geometric input data. Crucially, this procedure provides superfield descriptions where the actions of the supersymmetries are strict and ompatible with twisting. In this talk, I will demonstrate the merits of the formalism using the example of eleven-dimensional supergravity. In particular, I present a uniform construction of the interacting theory and all its twists realizing them as generalizations of Poisson–Chern–Simons theory. In addition to simplifying the computation of twists immensely, this also sheds some new light on the supergeometric origin of the supergravity theory. The talk is based on joint work with Ingmar Saberi.

Let $G$ be a group and $k$ an algebraically closed field of characteristic $p$. If $V$ is a finite-dimensional representation of $G$ over $k$, then by the classical Krull–Schmidt theorem, the $n$th tensor power of $V$ can be uniquely decomposed into a direct sum of indecomposable representations. But we know very little about this decomposition, even for very small groups, such as $G = (\Bbb Z/2)^3$ for $p = 2$ or $G = (\Bbb Z/3)^2$ for $p = 3$.

For example, what can we say about the number $d_n(V)$ of summands with dimension coprime to $p$? It is easy to show that there is a finite limit $d(V) := \lim_{n \to \infty} d_n(V)^{1/n}$, but what kind of number is this? Is it algebraic or transcendental? Until recently, there were no techniques to solve such questions (and in particular the same question about the sum of dimensions of these summands is still wide open). Remarkably, a new subject which may be called "Lie theory in tensor categories" gives methods to show that $d(V)$ is indeed an algebraic number, which moreover has the form \[ d(V) = \sum_{1 \leq j \leq p/2} n_j(V)[j]_q, \] where $n_j(V)$ is a natural number, $q := \exp(\pi i/p)$ is a particular root of unity, and $[j]_q := \frac{q^j-q^{-j}}{q-q^{-1}}$ is a $q$-number. Moreover, $d(V \oplus W) = d(V) + d(W)$ and $d(V \otimes W) = d(V) d(W)$, so $d$ is a character of the Green ring of $G$ over $k$. Finally, $d_n(V) \geq C_V d(V)^n$, for some $0 < C_V \leq 1$, and we can give lower bounds for $C_V$. In the talk, I will explain what Lie theory in tensor categories is and how it can be applied to such problems. This is joint work with K. Coulembier and V. Ostrik.