Topic: Presenting an open problem

Abstract: Given a permutation w, we consider three sets:
PD(w), BPD(w), and KD(w). There are some recent bijections
F: BPD(w) to PD(w) and G: PD(w) to KD(w).
We request an explicit bijection H: BPD(w) to KD(w) such that H=G◦F.
This talk will expose you to these very interesting sets of objects & maps between them, and along the way we’ll see a few fundamental combinatorial objects to know about, like ‘reduced words’.

Topic: Introduction to asymmetric hypergraphs

Abstract: A graph G is called asymmetric if it admits no non-identity automorphisms. Erdős and Rényi famously showed that almost all graphs are asymmetric. The graph G is said to be minimal asymmetric if it is asymmetric and every non-trivial induced subgraph of G is symmetric. Confirming a conjecture of Nešetřil, Schweitzer and Schweitzer proved that there are only finitely many, namely 18, minimal asymmetric graphs up to isomorphism. Jiang and Nešetřil later constructed infinitely many minimal asymmetric k-uniform hypergraphs for k > 2. Afterward, Bohnert and Winter modified their construction so that the hypergraphs further satisfy a few desirable properties. In this talk, we will discuss their results and present relevant open problems. No familiarity with hypergraphs will be assumed.

Topic: Boltzmann Samplers

Abstract: Given a combinatorial family of objects, it is helpful to visualize what a typical large element would look like. However, uniformly selecting an element of a fixed size can be difficult, even if the generating function for the family is known. Boltzmann samplers provides an alternative approach to sampling. It allows for a quick way to sample a family with a known generating function (or even just its functional recurrence) at the cost of controlling the size. This talk will provide an overview of Boltzmann samplers, the advantages of them, and ways to alleviate their limitations.