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?

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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.