Given a smooth geometrically connected curve C over a perfect field k and a smooth commutative group scheme G defined over the function field K of C, one can consider isomorphism classes of G-torsors locally trivial at completions of K coming from closed points of C. They form a generalized Tate-Shafarevich group which specializes to the classical one in the case when k is finite. Recently, these groups have been studied over other base fields k as well, for instance p-adic or number fields. Surprisingly, finiteness can be proven in some cases but there are also quite a few open questions which I plan to discuss  in my talk.

Keywords: SDE on the plane, Brownian sheet, path by path uniqueness, space time local time integral, Malliavin calculus

In this talk, we discuss the existence, uniqueness, and regularisation by noise for stochastic differential equations (SDEs) on the plane. These equations can also be interpreted as quasi-linear hyperbolic stochastic partial differential equations (HSPDEs). More specifically, we address path-by-path uniqueness for multidimensional SDEs on the plane, under the assumption that the drift coefficient satisfies a spatial linear growth condition and is componentwise non-decreasing. In the case where the drift is only measurable and uniformly bounded, we show that the corresponding additive HSPDE on the plane admits a unique strong solution that is Malliavin differentiable. Our approach combines tools from Malliavin calculus with variational techniques originally introduced by Davie (2007), which we non-trivially extend to the setting of SDEs on the plane.

This talk is based on a joint works with A. M. Bogso, M. Dieye and F. Proske.

The accurate semantic segmentation of individual tree crowns within remotely sensed data is crucial for scientific endeavours such as forest management, biodiversity studies, and carbon sequestration quantification. However, precise segmentation remains challenging due to complexities in the forest canopy, including shadows, intricate backgrounds, scale variations, and subtle spectral differences among tree species. While deep learning models improve accuracy by learning hierarchical features, they often fail to effectively capture fine-grained details and long-range dependencies within complex forest canopies.

This seminar introduces PerceptiveNet, a novel model that incorporates a Logarithmic Gabor-implemented convolutional layer alongside a backbone designed to extract salient features while capturing extensive context and spatial information through a wider receptive field. The presentation will explore the impact of Log-Gabor, Gabor, and standard convolutional layers on semantic segmentation performance, providing a comprehensive analysis of experimental findings. An ablation study will assess the contributions of individual layers and their interactions to overall model effectiveness. Furthermore, PerceptiveNet will be evaluated as a backbone within a hybrid CNN-Transformer model, demonstrating how improved feature representation and long-range dependency modelling enhance segmentation accuracy.