## 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

Semi-algebraic sets of integer points.

We look at sets of integer points in the plane,

and discuss possible definitions of when such a set

is “complicated” — this might be the case if it

is not the set of integer solutions

to some system of polynomial equations and inequalities.

Let’s together work out lots of examples,

and on the way let’s try to develop criteria

and proof techniques …

The examples that motivated our study come

from polytope theory: Many question of the type

“What is the possible pairs of

(number of vertices, number of edges)

for 5-dimensional polytopes?”

have been asked, many of them with simple and complete

answers, but in other cases the answer looks complicated.

Our main result says: In some cases it IS complicated!

(Joint work with Hannah Sjöberg.)

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, a new approach to the theory of discriminants for complete intersection curves in the 3-dimensional projective space will be discussed.