https://media.medfarm.uu.se/play/kanal/920svCopyright (C) 2026 Uppsala universitet, Enheten för verksamhetsstödFirst I will focus on the main ideas in "Experiments with the Site Frequency Spectrum" that appeared in: The Inaugural Issue in Algebraic Biology, Bulletin of Mathematical Biology, 73.4, 829-872 (2011), https://doi.org/10.1007/s11538-010-9605-5 (jointly with eight other co-authors) with the following Abstract: "Evaluating the likelihood function of parameters in highly-structured population genetic models from extant deoxyribonucleic acid (DNA) sequences is computationally prohibitive. In such cases, one may approximately infer the parameters from summary statistics of the data such as the site-frequency-spectrum (SFS) or its linear combinations. Such methods are known as approximate likelihood or Bayesian computations. Using a controlled lumped Markov chain and computational commutative algebraic methods, we compute the exact likelihood of the SFS and many classical linear combinations of it at a non-recombining locus that is neutrally evolving under the infinitely-many-sites mutation model. Using a partially ordered graph of coalescent experiments around the SFS, we provide a decision-theoretic framework for approximate sufficiency. We also extend a family of classical hypothesis tests of standard neutrality at a non-recombining locus based on the SFS to a more powerful version that conditions on the topological information provided by the SFS." Then, I will present some generalisations of these ideas (jointly with Tanja Stadler and Amandine Veber) in "Finding the best resolution for the Kingman–Tajima coalescent: theory and applications", Journal of Mathematical Biology, 70, 1207–1247 (2015), https://doi.org/10.1007/s00285-014-0796-5 and in "Full likelihood inference from the site frequency spectrum based on the optimal tree resolution", Theoretical Population Biology, 124, 1–15 (2018), https://doi.org/10.1016/j.tpb.2018.07.002 that seem to be having some impact today in the field of theoretical and applied population genomics (including machine-learning and "AI" approaches in predictive tasks). In the last few minutes, I will outline some mathematical challenges in combinatorial stochastic control processes for statistical inference over the same probability space for empirical genome-wide phenomena based on work done jointly with Bhalchandra Thatte and Amandine Veber (and current work with Berzunza-Ojeda) on population pedigree processes with coalescence, recombination and breeding behaviour in "Ancestries of a Recombining Diploid Population", Journal of Mathematical Biology, 72.1, 363-408 (2016) http://dx.doi.org/10.1007/s00285-015-0886-zhttps://media.medfarm.uu.se/play/video/15027Thu, 12 May 2022 10:06:20 +0000Counting objects up to symmetry is a classical subject in combinatorics. In this talk, we take a probabilistic viewpoint and study the automorphism group of two types of random trees: Galton-Watson trees (a family of random rooted trees that include, for example, plane, binary and labelled trees) as well as Pólya trees (rooted, unordered and unlabelled trees). Specifically, we prove that, in both cases, the size of the automorphism group follows a log-normal distribution, asymptotically as the size of the tree goes to infinity. Our proofs use both probabilistic tools and methods from analytic combinatorics.https://media.medfarm.uu.se/play/video/15000Tue, 26 Apr 2022 16:31:08 +0000In analytic combinatorics, the problem of singularity analysis aims at deducing the asymptotic expansion of an array of numbers from the properties of their generating functions near its singularities. In the univariate case (where the array is a sequence), this problem is completely solved for all algebraic functions and the solution is captured by the classical transfer theorem of Flajolet and Odlyzko. In the multivariate case however, the solution to the problem remains elusive. In the past decade, Robin Pemantle and collaborators carried out a substantial amount of work which tackles the case of multivariate rational functions in the diagonal limit. In the bivariate setting, this means computing the asymptotic expansion of the coefficients $a_{m,n}$ of a rational function in the limit where $m,nto infty$ and $m/nto lambda$ for fixed $lambda >0$. In this talk, I will present a new result which treats a different class of multivariate functions in a more general limit regime. In the bivariate case, it provides the asymptotic expansion of $a_{m,n}$ when $m,nto infty$ and $m/n^thetato lambda$ for fixed $theta, lambda >0$. This result generalizes formally Flajolet and Odlyzko's transfer theorem. In particular, it applies to all algebraic functions which are in some sense emph{$Delta$-analytic}. (It can be shown that in the multivariate setting, the class of $Delta$-analytic algebraic functions and the class of rational functions are essentially disjoint.) Applications of the result include Ising-decorated triangulations and parking processes on random trees.https://media.medfarm.uu.se/play/video/14996Tue, 26 Apr 2022 16:25:00 +0000A random recursive tree is a rooted tree constructed by successively choosing a vertex uniformly at random and adding a child to the selected vertex. A random preferential attachment tree is grown in a similar manner, but the vertex selection is made proportional to a linear function of the number of children of a vertex. Preferential attachment trees are the tree version of the Barabasi-Albert preferential attachment model. We consider a red-blue colouring of the vertices of preferential attachment trees, which we call a broadcasting induced colouring: the root is either red or blue with equal probability, while for a fixed value p between 0 and 1, every other vertex is assigned the same colour as its parent with probability p and the other colour otherwise. In this talk I will discuss properties of preferential attachment trees with broadcasting induced colourings, including limit laws for the number of vertices, clusters (maximal monochromatic subtrees) and leaves of each colour. The main focus of the talk will be on the size of the root cluster, that is, the maximal monochromatic subtree containing the root.https://media.medfarm.uu.se/play/video/14732Thu, 25 Nov 2021 11:33:16 +0000https://media.medfarm.uu.se/play/video/14641Wed, 20 Oct 2021 23:42:17 +0000https://media.medfarm.uu.se/play/video/14642Thu, 21 Oct 2021 00:44:50 +0000