L3

3d N=2 Chern-Simons-matter theories have a large variety of boundary conditions that preserve 2d N=(0,2) supersymmetry, and support chiral algebras. I'll discuss some examples of how the chiral algebras transform across dualities. I'll then explain how to construct duality interfaces in 3d N=2 theories, and relate dualities *of* duality interfaces to "Pachner moves" in triangulations of 4-manifolds. Based on recent and upcoming work with K. Costello, D. Gaiotto, and N. Paquette.

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about variational models for image analysis and their connection to partial differential equations, and go all the way to the challenges of their mathematical analysis as well as the hurdles for solving these - typically non-smooth - models computationally. The talk is furnished with applications of the introduced models to image de-noising, motion estimation and segmentation, as well as their use in biomedical image reconstruction such as it appears in magnetic resonance imaging.

 I will summarise old and recent developments on the classification and solution of Rational Conformal Field Theories in 2 dimensions using the method of Modular Differential Equations. Novel and exotic theories are found with small numbers of characters and simple fusion rules, one of these being the Baby Monster CFT. Correlation functions for many of these theories can be computed using crossing-symmetric differential equations.

The central charges “c” and “a” in two and four dimensional conformal field theories (CFTs) have a central organizing role in our understanding of quantum field theory (QFT) more generally.  Appearing as coefficients of curvature invariants in the anomalous trace of the stress tensor, they constrain the possible relationships between QFTs under renormalization group flow.  They provide important checks for dualities between different CFTs.  They even have an important connection to a measure of quantum entanglement, the entanglement entropy.  Less well known is that additional central charges appear when there is a boundary, four new coefficients in total in three and four dimensional boundary CFTs.   While largely unstudied, these boundary charges hold out the tantalizing possibility of being as important in the classification of quantum field theory as the bulk central charges “a” and “c”.   I will show how these charges can be computed from displacement operator correlation functions.  I will also demonstrate a boundary conformal field theory in four dimensions with an exactly marginal coupling where these boundary charges depend on the marginal coupling.  The talk is based on arXiv:1707.06224, arXiv:1709.07431, as well as work to appear shortly.