Joint work Marta Betcke and Bolin Pan

In photoacoustic tomography (PAT) with flat sensor, we routinely encounter two types of limited data. The first is due to using a finite sensor and is especially perceptible if the region of interest is large relatively to the sensor or located farther away from the sensor. In this talk we focus on the second type caused by a varying sensitivity of the sensor to the incoming wavefront direction which can be modelled as binary i.e. by a cone of sensitivity. Such visibility conditions result, in Fourier domain, in a restriction of the data to a bowtie, akin to the one corresponding to the range of the forward operator but further narrowed according to the angle of sensitivity. 

We show how we can separate the visible and invisible wavefront directions in PAT image and data using a directional frame like Curvelets, and how such decomposition allows for decoupling of the reconstruction involving application of expensive forward/adjoint solvers from the training problem. We present fast and stable approximate Fourier domain forward and adjoint operators for reconstruction of the visible coefficients for such limited angle problem and a tailored UNet matching both the multi-scale Curvelet decomposition and the partition into the visible/invisible directions for learning the invisible coefficients from a training set of similar data.

The massive planar Gaussian free field (GFF) is a random distribution defined on a subset of the complex plane. As a random distribution, this field a priori does not have well-defined level lines. In this talk, we give a meaning to this concept by constructing a coupling between a massive GFF and a random collection of loops, called massive CLE_4, in which the loops can naturally be interpreted as the level lines of the field. This coupling is constructed by appropriately reweighting the law of the standard GFF-CLE_4 coupling and this construction can be seen as a conditional version of the path-integral formulation of the massive GFF. We then relate massive CLE_4 to a massive version of the Brownian loop soup. This provides a more direct construction of massive CLE_4 and proves a conjecture of Camia.

Brown-Erdős-Sós initiated the study of the maximum number of edges in an $n$-vertex $r$-graph such that no $k$ edges span at most $s$ vertices. If $s=rk-2k+2$ then this function is quadratic in $n$ and its asymptotic was previously known for $k=2,3,4$. I will present joint work with Stefan Glock, Jaehoon Kim, Lyuben Lichev and Shumin Sun where we resolve the cases $k=5,6,7$.