In dimensions greater than four, the classification of smooth manifolds is an unsolvable problem, but manifolds can still be classified up to cobordism.

From this perspective, Liouville cobordisms provide a powerful tool for studying contact manifolds in high dimensions. In this talk, I will explain how Liouville cobordisms can be used to construct exact locally conformally symplectic (LCS) manifolds, in particular the LCS mapping tori associated with a contactomorphism. I will then use this construction to study the isomorphism classes of LCS mapping tori and explore their connections with the contact mapping class group.

We consider a renewal process which models a cumulative shock model that fails when the accumulation of shocks up-crosses a certain threshold. The ratio limit properties of the probabilities of non-failure after n cumulative shocks are studied. We establish that the ratio of survival probabilities converges to the probability that the renewal epoch equals zero. This limit holds for any renewal process, subject only to mild regularity conditions on the individual shock random variable. Precision on the rates of convergence are provided depending on the support structure and the regularity of the distribution. Arguments are provided to highlight the coherence between this new results and the well known Theory of Large Deviation.

Conway’s approach to enumerating knots from the 1970s highlighted an interesting ambiguity wherein pairs of tangles assembled in different ways can give rise to different knots. The process relating the resulting knots has come to be known as knot mutation, and because this often leads to a subtle and difficult-to-detect change to a knot, has received considerable attention ever since. This talk will focus on the history of knot mutation in the context of the Jones polynomial and its categorification known as Khovanov homology. The former highlights how one might view mutation as pointing to hidden symmetries in the definition of a knot invariant; the proof that the Jones polynomial is unchanged under mutation is surprisingly simple from this perspective. By contrast, the latter is a much more subtle story, which ultimately makes a surprising appeal to the homological mirror symmetry of the 3-punctured sphere. This last step is part of a project with Artem Kotelskiy and Claudius Zibrowius.

My presentation is set within the framework of inverse problems. The main objective is to determine initial conditions, the state, or parameters of a system from available observations, with a particular focus on biological applications. We concentrate on sequential methods in data assimilation, where observations are incorporated as they become available. In this context, I present two examples: the reconstruction of a source term in a wave equation, and the determination of both state and parameters in a PDE system modeling tumor growth. For the first problem, we define a Kalman estimator in infinite dimensions that sequentially estimates the source term. We show that this sequential estimator is equivalent to minimizing a functional, which allows us to perform convergence analysis under observability conditions. The second project studies the evolution of non-spherical tumor growth by combining mathematical modeling with data assimilation from biological measurements. The general strategy is to extract relevant information from images of spheroids, formulate a PDE model for tumor evolution, and then reduce it to an ODE model. A reduced ROUKF coupled with a Luenberger observer is then used to estimate both the state and the parameters.