Genetic differences are a critical driver of disease risk and healthy variation, across the tree of life. Mutations arise and spread in our distant, genealogical ancestors, and so genetic variation data can provide a window into our evolutionary past, allowing us to understand processes such as population size changes, admixture, natural selection, and even evolution of the mutation and recombination processes that generate the variation itself. It has long been recognised that knowledge of genealogical relationships among individuals would allow us to capture almost all the information available from such data. However, only in recent years has it become computationally feasible to infer such genealogies, genome-wide, from variation patterns. One such method, Relate, developed in our lab, allows approximate inference of genealogical trees under coalescent-like models, for up to tens of thousands of samples. Here, we will show that a powerful approach for inference is to identify and characterise departures from the relatively simple models used to build these trees. By defining a 'population' as a set of coalescence rates between labelled individuals backwards in time, we can uncover variability in these rates, and use a single collection of trees to identify ancient mixing events among populations - including 'ghost' groups we have never sampled - natural selection favouring the descendents of particular branches of the genealogy, and departures from mathematical expectations under clock-like behaviour, indicating disruption of recombination or mutation.[-]

We consider a population model in which the season alternates between winter and summer, and individuals can acquire mutations either that are advantageous in the summer and disadvantageous in the winter, or vice versa. Also, we assume that individuals in the population can either be active or dormant, and that individuals can move between these two states. Dormant individuals do not reproduce but do not experience selective pressures. We show that, under certain conditions, over time we see two waves of adaptation. Some individuals repeatedly acquire mutations that are beneficial in the summer, while others repeatedly acquire mutations that are beneficial in the winter. Individuals can survive the season during which they are less fit by entering a dormant state. This result suggests that, for populations in fluctuating environments, dormancy may have the potential to induce speciation. This is joint work with Fernando Cordero and Adrian Gonzalez Casanova.[-]

The main question that we will investigate in this talk is: what does the spectrum of a quantum channel typically looks like? We will see that a wide class of random quantum channels generically exhibit a large spectral gap between their first and second largest eigenvalues. This is in close analogy with what is observed classically, i.e. for the spectral gap of transition matrices associated to random graphs. In both the classical and quantum settings, results of this kind are interesting because they provide examples of so-called expanders, i.e. dynamics that are converging fast to equilibrium despite their low connectivity. We will also present implications in terms of typical decay of correlations in 1D many-body quantum systems. If time allows, we will say a few words about ongoing investigations of the full spectral distribution of random quantum channels. This talk will be based on: arXiv:1906.11682 (with D. Perez-Garcia), arXiv:2302.07772 (with P. Youssef) and arXiv:2311.12368 (with P. Oliveira Santos and P. Youssef).[-]

In this talk, we discuss functional inequalities and gradient bounds for the heat kernel on the Vicsek set. The Vicsek set has both fractal and tree structure, whereas neither analogue of curvature nor obvious differential structure exists. We introduce Sobolev spaces in that setting and prove several characterizations based on a metric, a discretization or a weak gradient approach. We also obtain $L^{p}$ Poincaré inequalities and pointwise gradient bounds for the heat kernel. These properties have important applications in harmonic analysis like Sobolev inequalities and the Riesz transform. Moreover, several of our techniques and results apply to more general fractals and trees.[-]

We investigate the Dirichlet problem for a non-divergence form elliptic operator $L=a^{i j}(x) D_{i j}+b^{i}(x) D_{i}-c(x)$ in a bounded domain of $\mathbb{R}^{d}$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's function in a ball and derive two-sided pointwise estimates for it. Utilizing these results, we demonstrate the equivalence of regular points for $L$ and those for the Laplace operator, characterized via the Wiener test. This equivalence facilitates the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Furthermore, we construct the Green's function for $L$ in regular domains and establish pointwise bounds for it.[-]

The aim of this project is a deeper investigation of off-diagonal estimates. In the ISem lectures, in Theorem 11.16, it has already been shown that off-diagonal estimates in combination with Sobolev embeddings lead to $L^{p}$-extrapolation for the resolvents of an elliptic operator $L$ in divergence form on $\mathbb{R}^{n}$. More precisely, if $\left|\frac{1}{p}-\frac{1}{2}\right|<\frac{1}{n}$, then there exists $C>0$ such that $\left\|(1+t L)^{-1} u\right\|_{p} \leqslant C\|u\|_{p}$ for all $t>0, u \in L^{p} \cap L^{2}\left(\mathbb{R}^{n}\right)$.
A related (more difficult!) question is for what range of $p \in(1, \infty)$ the norm equivalence $\|\sqrt{L} u\|_{2} \simeq\|\nabla u\|_{2}$ from Theorem 12.1 (the Kato square root property for $L$ !) extrapolates to $L^{p}\left(\mathbb{R}^{n}\right)$. It turns out that there are different ranges of $p$ for the two estimates $\|\sqrt{L} u\|_{p} \lesssim\|\nabla u\|_{p}$ and $\|\nabla u\|_{p} \lesssim\|\sqrt{L} u\|_{p}$. The latter estimate is generally known as $L^{p}$-boundedness of the Riesz transform, and this is what shall be the core of the project.
Starting point of the project is the AMS memoir [1], which starts with an excellent introduction into the topic; you can find a preprint version of the memoir on the arXiv (with different numbering of theorems than in the published version, unfortunately). An important abstract $L^{p}$-extrapolation result is Theorem 1.1 in [1], the application Riesz transforms on $L^{p}$ can be found in Section 4.1. This approach is due to Blunck and Kunstmann [2, 3]. If time permits, we can also study the approach of Shen [4] to Riesz transforms. The precise selection of topics will be decided among the participants of the project.[-]

The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modelled by nonlinear diffusion, and long-range attraction modelled by nonlocal interaction. In this course, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint with Carrillo, Hittmeir and Volzone). Once the symmetry is known, we further investigate whether steady states are unique within the radial class, and show that for a given mass, the uniqueness/non-uniqueness of steady states is determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint with Delgadino and Yan). I'll also discuss some properties on the long-time behavior of aggregation-diffusion equation with linear diffusion (joint with Carrillo, Gomez-Castro and Zeng), and global-wellposedness if Keller-Segel equation when coupled with an active advection term (joint with Hu and Kiselev).[-]

Sampling is a fundamental task in Machine Learning. For instance in Bayesian Machine Learning, one has to sample from the posterior distribution over the parameters of a learning model, whose density is known up to a normalizing constant. In other settings such as generative modelling, one has to sample from a distribution from which some samples are available (e.g. images). The task of sampling can be seen as an optimization problem over the space of probability measures. The mathematical theory providing the tools and concepts for optimization over the space of probability measures is the theory of optimal transport. The topic of this course will be the connection between optimization and sampling, more precisely, how to solve sampling problems using optimization ideas. The goal of the first part of the course will be to present two important concepts from optimal transport: Wasserstein gradient flows and geodesic convexity. We will introduce them by analogy with their euclidean counterpart that is well known in optimization. The goal of the second part will be to show how these concepts, along with standard optimization techniques, enable to design, improve and analyze various sampling algorithms. In particular. we will focus on several interacting particles schemes that achieve state-of-the-art performance in machine learning.[-]

The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modelled by nonlinear diffusion, and long-range attraction modelled by nonlocal interaction. In this course, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint with Carrillo, Hittmeir and Volzone). Once the symmetry is known, we further investigate whether steady states are unique within the radial class, and show that for a given mass, the uniqueness/non-uniqueness of steady states is determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint with Delgadino and Yan). I'll also discuss some properties on the long-time behavior of aggregation-diffusion equation with linear diffusion (joint with Carrillo, Gomez-Castro and Zeng), and global-wellposedness if Keller-Segel equation when coupled with an active advection term (joint with Hu and Kiselev).[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]