Nowadays, there are different medical imaging techniques to collect data about the progression of Alzheimer’s disease in a patient’s brain. These data describe different phenomena which are still not understood from a biological point of view. How can these data sets be combined in a mathematical model to simulate the evolution of such a neurodegenerative disease in a computer? In this snapshot, we present one possible approach to address this task with the help of graph theory and partial differential equations.

The famous 4-Color Theorem from graph theory states that the vertices of any planar graph can be colored with four colors, so that no neighboring vertices have the same color. The 4-Sample Theorem from algebraic statistics says that the maximum likelihood estimator for a Gaussian graphical model of a planar graph exists with probability 1 if one has at least four samples.

A bijection transforms one type of mathematical object into another. Such transformations provide new perspectives on these objects, revealing surprising properties and uncovering new mysteries. We discuss bijections from alternating sign matrices to other objects in mathematics and physics and recent progress in the search for a missing bijection.

Nonlocal models consider interactions over a range of distances, not just at a single point. In this snapshot, we give a short introduction to nonlocal modeling, explain how it differs from its local counterpart, and present an application: fracture mechanics.

We introduce Tadao Oda’s famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the two-dimensional case – including a proof of Pick’s Theorem, which elegantly relates the area of a lattice polygon to the number of lattice points it contains in its interior and on its boundary.