L6

Recent foundational work by Ben-Bassat, Kelly, and Kremnitzer describes a model for derived analytic geometry as homotopical geometry relative to the infinity category of simplicial commutative complete bornological rings. In this talk, Rhiannon Savage will discuss a representability theorem for derived stacks in these contexts and will set out some new foundations for derived smooth geometry. Rhiannon will also briefly discuss the representability of the derived moduli stack of non-linear elliptic partial differential equations by an object we call a derived C∞-bornological affine scheme.

In this talk, I will present a representation-theoretical approach to constructing a non-commutative analogue of the classical Laplace transform on Lie groups. I will begin by discussing the motivations for such a generalization, emphasizing its connections with harmonic analysis, probability theory, and the study of evolution equations on non-commutative spaces. I will also outline some of the key challenges that arise when extending the Laplace transform to the setting of Lie groups, including the non-commutativity of the group operation and the complexity of its dual space.

The main part of the talk will focus on an explicit construction of the Laplace transform in the framework of connected, simply connected nilpotent Lie groups. This construction relies on Kirillov’s orbit method, which provides a powerful bridge between the geometry of coadjoint orbits and the representation theory of nilpotent groups.

As an application, I will describe an operator-theoretic analogue of the classical Müntz–Szász theorem, establishing a density result for a family of generalized polynomials in associated with the group setting. This result highlights the strength of the representation-theoretical approach and its potential for solving classical approximation problems in a non-commutative context.

I'll discuss how to show 3D Poincaré duality and residual finiteness are together incompatible with property (T).

We give examples to show that theorems of S Fisher and D Kielak 
relating the vanishing of L^2 cohomology to fibering over the 
circle for RFRS groups cannot be extended to some larger classes
of groups, and we introduce an L^2-torsion invariant that may 
prove useful. 

(Joint with Sam Hughes and Wolfgang Lueck) 

The theory of JSJ decomposition plays a key role in the classification of hyperbolic groups, in analogy with the case of 3-manifolds. While this theory can be extended to larger families of groups, the JSJ decomposition displays significant flexibility in general, making a complete understanding of its behaviour more challenging. In this talk, Dario Ascari explores this flexibility, with an emphasis on the case of generalized Baumslag-Solitar groups.

For homogeneous, defect-free TQFTs, (1) n+\epsilon-dimensional versions of the theories are relatively easy to construct; (2) an n+\epsilon-dimensional theory can be extended to n+1-dimensional (i.e. the top-dimensional path integral can be defined) if certain more restrictive conditions related to handle cancellation are satisfied; and (3) applying this path integral construction to a handle decomposition of an n+1-manifold yields a state sum description of the path integral.  In this talk, I'll show that the same pattern holds for defect TQFTs.  The adaptation of homogeneous results to the defect setting is mostly straightforward, with the only slight difficulty being the purely topological problem of generalizing handle theory to manifolds with defects.  If time allows, I'll describe two applications: a Verlinde-like dimension formula for the dimension of the ground state of fracton systems, and a generalization, to arbitrary dimension, of Ostrik's theorem relating algebra objects to modules (gapped boundaries).

Abstract: This talk introduces the ZX-calculus, a powerful graphical language for reasoning about quantum computations. I will start with an overview of process theories, a general framework for describing how processes act upon different types of information. I then focus on the process theory of quantum circuits, where each function (or gate) is a unitary linear transformation acting upon qubits. The ZX-calculus simplifies the set of available gates in terms of two atomic operations: Z and X spiders, which generalize rotations around the Z and X axes of the Bloch sphere. I demonstrate how to translate quantum circuits into ZX-diagrams and how to simplify ZX diagrams using a set of seven equivalences. Through examples and illustrations, I hope to convey that the ZX-calculus provides an intuitive and powerful tool for reasoning about quantum computations, allowing for the derivation of equivalences between circuits. By the end of the talk listeners should be able to understand equations written in the ZX-calculus and potentially use them in their own work.