## November – Dezember 2017

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2017/Berlin-LE-WS/index.html

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2017/Berlin-LE-WS/index.html

In some approaches to the reconstruction of phylogenetic trees, Markov chain Monte Carlo methods are used. These in turn use simple base-chains on the set of (binary) trees of a given size $N$. It is at least of mathematical interest (but might also help to understand properties of such Markov chains when the trees are large) to consider limit processes as $N$ tends to infinity and the time is suitably sped up. Here, we have to decide in which state space we are working, i.e., what kind of objects we want to consider as “continuum trees” in the limit, and what we mean by “limit”.

One by now almost-classical approach is to work in a space of metric measure spaces, but while it has proven successful in some situations, it seems difficult to prove convergence in others. Motivated by a particular Markov chain, the Aldous chain on cladograms, where existence of a limit process has been conjectured almost two decades ago, we introduce an alternative state space. We define the objects by a “tree structure” (formalized by a branch-point map) instead of a metric structure and call them algebraic measure trees. In this new state space, we are able to

prove convergence of the Aldous chain to a limit process with continuous paths.

(joint work with Anita Winter and Leonid Mytnik)

Faces of the December:

Ulrich Menne

Max Pfeffer

Andre Uschmajew

In this talk we will present a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic.This formula also provides a new method for evaluating and computing this discriminant more efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we derive new properties and we show that this new definition of the discriminant satifises to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. More precisely, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY