We study the motion of miscible liquids in porous media with the speed determined by Darcy's law. The two basic examples are the displacement of viscous liquids and the motion induced by gravity. Such motion often is unstable and creates patterns called viscous fingers (picture attached).

We concentrate on important for applications property of viscous fingers - speed of their propagation. The work is inspired by the results of F. Otto and G. Menon for a simplified model, called transverse flow equilibrium (TFE). In this work a rigorous upper bound was proved using the comparison principle. At the same time numerical experiments suggest that the actual speeds are better than Otto-Menon estimates.

We consider a two-tubes model -- the simplest model we were able to construct which includes transverse liquid flow. For this model for the gravitational fingers we were able to find families of travelling waves and found the relation between original model and TFE simplification. The main tool in the proof in normal hyperbolicity. For viscous liquid it seems that the phenomenon is the same but up to now it is work in progress.

This is a joint work with Yulia Petrova and Yalchin Efendiev.

It is a well-known conjecture that integer programs on an integer constraint matrix whose subdeterminants are bounded in absolute value by a constant could be solvable in polynomial time. Despite some recent progress in special cases, the question is still wide open. We consider the special case consisting of a TU constraint matrix with one additional row, which in general encompasses different hard and interesting problems. We present partial progress to the resolution of the question of polynomial solvability of such problems in particular for transposed network matrices. The applied techniques range from Seymour's decomposition of TU matrices over certain types of proximity results for integer programs to graph and minor theory.

This is joint work with Manuel Aprile, Samuel Fiorini, Stefan Weltge and Yelena Yuditsky.

An important objective function in the scheduling literature is to minimize the sum of weighted flow times. We are given a set of jobs, where each job is characterized by a release time, a processing time, and a weight. Our goal is to find a preemptive schedule on a single machine that minimizes the sum of the weighted flow times of the jobs, where the flow time of a job is the time between its completion time and its release time. We answer this question in the affirmative and present a polynomial time (1 + 𝜀)-approximation algorithm for weighted flow time on a single machine. We use a reduction of the problem to a geometric covering problem, which was introduced in previous approaches and which loses only a factor of 1+𝜀 in the approximation ratio. However, unlike the previous algorithm, we solve the resulting instances of the covering problem exactly, rather than losing a factor 2 + 𝜀. Key for this is to identify and exploit structural properties of instances of that problem covering problem which arise in the reduction from weighted flow time.

This talk is based on joint work with Lars Rohwedder and Andreas Wiese.

We consider the random connection model with bounded edges which is generated by a Poisson point process with density $\lambda$ in $\mathbb{R}^d$. We prove that this model undergoes a sharp phase transition, i.e. we prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in $n$, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to ($\lambda-\lambda_c)$, $\lambda_c$ being the critical density. This proof uses newly developed methods by Last, Peccati and Yogeshwaran in their recent work, in particular a continuous version of the OSSS inequality for Poisson functionals, relying on stopping sets and continuous-time decision trees. This approach simplifies an earlier result of Faggionato and Mimun, who proved sharp phase transition in the random connection model via the discrete OSSS inequality.

This talk is concerned with the Cauchy-Dirichlet problem for fast diffusion equations on bounded domains. It is well known that every weak solution vanishes in finite time at the unique power rate, and therefore, asymptotic profiles for such vanishing solutions are defined as a limit of rescaled solutions, which solve the Cauchy-Dirichlet problem for a fast diffusion equation with a blow-up reaction. Asymptotic profiles are characterized as nontrivial equilibria of the rescaled problem (see pioneer works of Berryman and Holland in 1980s and subsequent results for qualitative results). Recently, Bonforte and Figalli (CPAM, 2021) established a quantitative result on the convergence of rescaled solutions to nondegenerate positive asymptotic profiles. More precisely, they proved an exponential convergence of nonnegative rescaled solutions to nondegenerate positive asymptotic profiles in a weighted L2 space with a sharp rate (in view of some linearized analysis) by developing a nonlinear entropy method. In this talk, we present a different approach to prove exponential convergence with rates for nondegenerate asymptotic profiles. In particular, we can directly verify an H1 0 convergence with the sharp rate. Our method of proof is based on an energy method rather than entropic one, and a key ingredient is a quantitative gradient inequality established based on an eigenvalue problem with weights

Instead of a classical research talk, Andreas Wiese will give a short presentation on the topic "How to be productive".

The talk will be followed by an open discussion, where everyone is invited to participate and give their own input, opinions and ideas on the topic.

Fast-slow dynamical systems are useful tools to describe various real life scenarios. The presence of slow variables that can have strong influence on fast ones makes the study of early warning signs an important field. This talk on my master's thesis seeks to deepen the study of early warning signs of stochastic partial differential equations (SPDEs), focusing on multiplication operators defined by analytic functions $f$. The studied early warning sign takes the form of changes in the variance as the bifurcation point is approached, especially divergence and its rate. The focus is on the study of analytic functions with domains in different dimensions. For a one-dimensional domain we obtain precise rates of divergence. In the case of the two- and three-dimensional domain an upper bound is obtained. These results are cross-validated by numerical simulations. Such theory can be very useful for several applications, such as epileptic seizures or voltage collapses in power systems