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In this talk, I will discuss recent results which characterize the long-time behavior of a conditional McKean-vlasov equation related to interacting diffusions on regular trees. This is joint work with Kavita Ramanan.

We study a problem arising in the management of staff positions in institutions with a cameralistic budgeting system, for example in public universities in Germany. When a new employee is hired, she needs to be assigned to one or (partially) to several of the available positions. These position may already be (partially) assigned to other staff members during certain time periods. Some positions are better suited for the new hire due to, e.g., their associated pay grades or other administrative reasons. One seeks a solution with assignments to suitable open positions and wants few changes of those assignments over time. This yields a multi-objective optimization problem with a lexicographic objective function which can be seen as a scheduling problem with non-availability periods for the machines. We derive structural insights into this problem and present several MIP-formulations for it. Their solutions are optimal w.r.t. the three most important objectives and optimal or near-optimal w.r.t. the least important objective, respectively. In particular, we are able to solve our problem faster than with a straight-forward approach if one is willing to potentially sacrifice a bit of accuracy in the least important objective. In addition, we present very fast combinatorial algorithms for important special cases of the problem. Overall, we can solve most practically relevant instances in less than a few seconds. Our optimization tool was developed in collaboration with the administration of the School of Computation, Information, and Technology (CIT) at the Technical University of Munich where it is now used on a regular basis.

Providing guarantees on the reproducibility of discoveries is essential when drawing inferences from high-dimensional data. Such data is common in numerous scientific domains, for example, in biomedicine, it is imperative to reliably detect the genes that are truly associated with the survival time of patients diagnosed with a certain type of cancer, or in finance, one aims at determining a sparse portfolio to reliably perform index tracking. This talk introduces the Terminating-Random Experiments (T-Rex) selector, a fast multivariate variable selection framework for high-dimensional data. The T-Rex selector provably controls a user-defined target false discovery rate (FDR) while maximizing the number of selected variables. It scales to settings with millions of variables. Its computational complexity is linear in the number of variables, making it more than two orders of magnitude faster than, e.g., the existing model-X knockoff methods. An easy-to-use open-source R package that implements the TRexSelector is available on CRAN. The focus of this talk lies on high-dimensional linear regression models, but we also describe extensions to principal component analysis (PCA) and Gaussian graphical models (GGMs).

We introduce a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic PDE system describing the evolution of the population density and the chemical signal. The key novelty lies in the transport term for the population, which depends on the second-order derivatives of the chemical field. This term is derived as a limiting anticipation-reaction mechanism for an infinitesimally small ant. We establish global existence and uniqueness of solutions, as well as the propagation of regularity of the initial data. We then analyze the long-time behavior of the system: we prove the existence of a compact global attractor and show that the homogeneous steady state becomes nonlinearly unstable under an inviscid instability criterion. Additionally, we provide a lower bound on the dimension of the attractor. Conversely, we prove that for sufficiently small interaction strength, the homogeneous steady state is globally asymptotically stable. Finally, we present several numerical simulations illustrating the model's dynamics. References: Curvature in chemotaxis: A model for ant trail pattern formation, Charles Bertucci, Matthias Rakotomalala, Milica Tomasevic Existence and dimensional lower bound for the global attractor of a PDE model for ant trail formation, Matthias Rakotomalala, Oscar de Wit