2020  Exact solutions in LogConcave MLE 
in prepapration 
with Kaie Kubjas, Olga Kuznetsova, Alexander Heaton, Georgy Scholten, MirunaStefana Sorea  In nonparametric statistics one abandons the requirement that a probability density function belongs to a statistical model. Given a sample of points, we want to find the best logconcave distribution that fits this data. This problem was originally studied by statisticians, who found that the optimal solutions have probability density functions whose logarithm is piecewiselinear and used numerical methods to approximate the pdf. In this work we use algebraic and combinatorial methods to provide exact solutions to this problem. At the same time we use tools from algebraic geometry to test if the solutions provided by statistical software can be certified to be correct.
How many regions of linearity does the depicted tent function have? 

2018  Moment ideals of local Dirac mixtures  older arXiv version, the published journal version 
with Markus Wageringel  Algebraic methods can find wide use in parameter estimation from statistical moments. In this work we use tools from elimination theory, the extended Prony method, and numerical algebra to reconstruct parameters from mixtures of local Diracs. Computational algebra mixes successfully with applications in statistics and signal processing.
The picture shows the moment hypersurface of a first order local Dirac mixture, i.e. the variety parametrized by the moments. It is the tangent to the twisted cubic, in the same way that regular mixing gives rise to secants. 