## Section: Research Program

### Discrete geometric structures

Our work on discrete geometric structures develops in several directions, each one probing a different type of structure. Although these objects appear unrelated at first sight, they can be tackled by the same set of probabilistic and topological tools.

A first research topic is the study of *Order types.* Order types
are combinatorial encodings of finite (planar) point sets, recording
for each triple of points the orientation (clockwise or
counterclockwise) of the triangle they form. This already determines
properties such as convex hulls or half-space depths, and the
behaviour of algorithms based on orientation predicates. These
properties for all (infinitely many) $n$-point sets can be studied
through the finitely many order types of size $n$. Yet, this finite
space is poorly understood: its estimated size leaves an exponential
margin of error, no method is known to sample it without concentrating
on a vanishingly small corner, the effect of pattern exclusion or VC
dimension-type restrictions are unknown. These are all directions we
actively investigate.

A second research topic is the study of *Embedded graphs and
simplicial complexes.* Many topological structures can be
effectively discretized, for instance combinatorial maps record
homotopy classes of embedded graphs and simplicial complexes represent
a large class of topological spaces. This raises many structural and
algorithmic questions on these discrete structures; for example, given
a closed walk in an embedded graph, can we find a cycle of the graph
homotopic to that walk? (The complexity status of that problem is
unknown.) Going in the other direction, some purely discrete
structures can be given an associated topological space that reveals
some of their properties (*e.g.* the Nerve theorem for
intersection patterns). An open problem is for instance to obtain
fractional Helly theorems for set system of bounded topological
complexity.

Another research topic is that of *Sparse inclusion-exclusion formulas.*
For any family of sets ${A}_{1},{A}_{2},...,{A}_{n}$, by the principle of
inclusion-exclusion we have

${\mathbb{1}}_{{\bigcup}_{i=1}^{n}{A}_{i}}=\sum _{I\subseteq \{1,2,...,n\}}{(-1)}^{\left|I\right|+1}{\mathbb{1}}_{{\bigcap}_{i\in I}{A}_{i}}$ | (1) |

where ${\mathbb{1}}_{X}$ is the indicator function of $X$. This formula is universal (it applies to any family of sets) but its number of summands grows exponentially with the number $n$ of sets. When the sets are balls, the formula remains true if the summation is restricted to the regular triangulation; we proved that similar simplifications are possible whenever the Venn diagram of the ${A}_{i}$ is sparse. There is much room for improvements, both for general set systems and for specific geometric settings. Another interesting problem (the subject of the PhD thesis of Galatée Hemery) is to combine these simplifications with the inclusion-exclusion algorithms developed, for instance, for graph coloring.