Math Calendar
Supervisors: prof. dr. Tristan van Leeuwen & dr. Palina Salanevich
Other supervisors: dr. Gyula Kotek (EMC), dr. Jukka Hirvasniemi(EMC), dr. Frans Vos (TU Delft)
Abstract:
Inthis thesis presentation for both my Mathematics and Physics degrees, I presenta novel pipeline for simulating anatomically realistic lesions in 18F-FDGPET/MR scans to overcome the scarcity of annotated data. Using dimensionalityreduction (PCA) and probabilistic sampling (GMM), synthetic lesions aregenerated and inserted at clinically relevant locations. These datasets arethen used to train nnU-NetV2 models, which outperform those trained on reallesions alone.
Experimentally,I identified which features influence lesion detectability – revealing thatbesides intensity, shape-based features like convexity affect detectability –and demonstrated how radiomics can be used to interpret the AI’sdecision-making.
Ialso introduce a method for generating diverse anatomical variations throughdeformation fields, enabling scalable data augmentation without manualannotation. This work strengthens the foundation for automated lesion detectionin PET/MR and supports future applications in diagnostic imaging and treatmentplanning.
Blowing up the diagonal of M^2 for a manifold M yields a configuration space that remembers the collision axis of collided configurations. Fulton and MacPherson famously generalized this construction to configurations of more than two points. As part of my thesis advised by Á. del Pino, we build configuration spaces for jets of maps in the same spirit, but adapted to the main structure on jet space: The Cartan distribution and the Lie filtration it generates. Higher jet orders necessitate the use of weighted blow-ups, which we tackle within the differential-geometric framework of weightings due to Loizides and Meinrenken. In this talk, I first want to intuitively illustrate this construction and then motivate why it is natural to consider when studying h-principles for differential constraints involving multiple points.
Title: Extremes in dynamical systems: max-stable and max-semistable laws
Abstract:
Extreme value theory for chaotic, deterministic dynamical systems is a rapidly expanding area of research. Given a dynamical system and a real-valued observable defined on its state space, extreme value theory studies the limit probabilistic laws for asymptotically large values attained by the observable along orbits of the system. Under suitable mixing conditions the extreme value laws are the same as those for stochastic processes of i.i.d. random variables.
Max-stable laws typically arise for probability distributions with regularly varying tails. However, in the context of dynamical systems, where the underlying invariant measure can be irregular, max-semistable distributions also have a natural place in studying extremal behaviour. In this talk I will first discuss a family of autoregressive processes with marginal distributions resembling the Cantor function. The resulting extreme value law can be proven to be a max-semistable distribution. Alternatively, we can describe the autoregressive process in terms of an iterated map with an invariant measure. Further examples of extreme value laws in dynamical systems are discussed as well.