Codenominator is a function defined recursively on the Farey tree that yields an extension of the Fibonacci sequence to rationals as an integral-valued map. After discussing its arithmetic properties, we will show how it relates to Dyer's outer automorphism of the group PGL(2,Z) and the modular involution Jimm of the real line. Automorphisms of the cubic tree studied by Djokovic-Miller and Conder provide analogues of the codenominator. We will also explain its connection to quantizations of real numbers recently introduced by Ovsienko and Mourier-Genoud. The talk will also present some questions and conjectures related to the codenominator and the Jimm

The asymptotic theory of partitions is a classical topic in number theory, where an asymptotic formula for the number of unrestricted partitions was famously obtained by Hardy and Ramanujan over 100 years ago. In this talk, we will discuss a recent result, obtaining a rather precise asymptotic formula for the number of partitions of \(n \) into parts which are Fibonacci numbers (each Fibonacci number is allowed to be used more than once). As in the case of partitions into powers of a single number, the main term oscillates. We will also discuss a generalisation of this result to other recurrence sequences. This is joint work with Michael Coons and Mathias Løkkegaard Laursen.

Brownian motion tree models are used to describe the evolution of a continuous trait along a phylogenetic tree under genetic drift. Such a model is obtained by placing linear constraints on a mean-zero multivariate Gaussian distribution according to the topology of the underlying tree. We investigate the enumerative geometry of the standard and dual maximum likelihood estimation problems in these models. In particular, we study the number of complex critical points of the log-likelihood and dual log-likelihood functions, known as the ML-degree and dML-degree, respectively. We use the toric geometry of Brownian motion tree models to give a formula for the dML-degree for all trees. We also prove a formula for the ML-degree of a star tree and show that for general trees, the ML-degree does not depend on the location of the root.

Abstract. There is long history in the relation between the critical points of distance function and concur- rent normals to a submanifold in Euclidean space. The study of caustics and counting the number of normals play a important role. In this talk we will give a general approach to the study of crit- ical points of the distance function to a PL submanifold X. Examples are: polygons in the plane and in space and polygonal surfaces in 3-space (not necessarily convex), etc. What is the relation between normals and critical points ? Are generic singularities Morse and if so what is the index ? We will discuss the bifurcation set and show that for a knotted closed PL-curve there are at least 10 concurrent normals. Also for a convex simple polytope there is a point at least 10 concurrent normals. What can be said about the ED-degree?

I will give a brief introduction to well-rounded lattices and to their utility in (post-quantum) security. We will see how the lattice theta series naturally arises in these contexts and discuss its connections to well-rounded lattices.

For integer \(k\geq4 \) Somos--\(k \) sequence is a sequence generated by quadratic recurrence relation of the form $$s_{n+k}s_n=\sum_{j=1}^{[k/2]}\alpha_js_{n+k-j}s_{n+j},$$ where \(\alpha_j \) are constants and \(s_0 , \dots, s_{k-1} \) are initial data. Among them exist an important class of sequences with many properties. This class consists of finite rank sequences. The sequence \(\{s_n\}_{n=-\infty}^\infty \) has a (finite) rank \(r \) if maximal rank of two infinite matices $$\left.\vphantom{\sum}(s_{m+n}s_{m-n})\right|_{m,n=-\infty}^\infty,\qquad \left.\vphantom{\sum}(s_{m+n+1}s_{m-n})\right|_{m,n=-\infty}^\infty$$ is \(r \). If \(r=2 \) then general term of Somos sequence can be expressed in terms of elliptic function. One can consider a general finite rank sequence as a sequence admitting more complicated addition theorem. Presumably the following properties are more or less equivalent: finitness of the rank, Laurent phenomenon, periodicity \(\pmod N \), solvability in theta-functions. The talk will be devoted to periodicity \(\pmod N \) of general integer finite rank sequences.

The medial axis of a closed subset X of the Euclidean space is the set of points admitting more than one closest point in X. It has been extensively studied since the late 60's as a central concept in pattern recognition. On the other hand, its peculiar relation to the singularities of X was observed only a decade ago. Focusing on the question which singular points of X are `reached' by the medial axis (i.e. belong to the closure of the latter) we will give an overview of what is known and discuss the latest developments in this matter.

In this talk, I will discuss generalized Markov numbers arising from semigroups of reduced integer matrices. This construction allows us to find these numbers by counting perfect matchings of a new family of bipartite graphs, which we call wug-snake graphs. We also show how this relates to the geometry of numbers and the classical theory of Markov minima.

Let A and B be two Heisenberg translations in Sp(2, 1) or SU(2, 1) with distinct fixed points. Here, Sp(2, 1) and SU(2, 1) act isometrically on the quaternionic and complex hyperbolic spaces, respectively. We provide sufficient conditions that guarantee that the subgroup ⟨A, B⟩ is discrete and free, using Klein’s combination theorem.