You are cordially invited to our next Monday Lecture & Colloquium on February 4th at 14:15 h & 16:00 h at FU Berlin. Location: Room 005 - Ground Floor Freie Universität Berlin Takustr. 9 14195 Berlin Time: Monday, February 4th - 14:15 h Lecture: Karim Adiprasito (Hebrew University of Jerusalem) Title: Triangulated manifolds, Lefschetz conjectures and the revenge of marriages Abstract: Stanley gave us necessary conditions that f-vectors of simplicial polytopes must satisfy by relating the problem to the hard Lefschetz theorem in algebraic geometry, an insurmountably deep and intimidating theorem in algebraic geometry. McMullen conjectured that these conditions are necessary in general, posing before us the problem of proving the hard Lefschetz theorem beyond what algebraic geometers would dream of. I will talk about the revenge of combinatorics, and in particular discuss how Hall's marriage theorem (or rather, one of its proofs) provides a way to this deep algebraic conjecture. Based on arxiv:1812.10454 Coffee & Tea Break: Room 134 Time: Monday, February 4th - 16:00 h s.t. Colloquium: Patrick Morris (Freie Universität Berlin) Title: Clique tilings in randomly perturbed graphs Abstract: A major theme in both extremal and probabilistic combinatorics is to find the appearance thresholds for certain spanning structures. Classical examples of such spanning structures include perfect matchings, Hamilton cycles and H-tilings, where we look for vertex disjoint copies of H covering all the vertices of some host graph G. In this talk we will focus on H-tilings in the case that H is a clique, a natural generalisation of a perfect matching. On the one hand there is the extremal question, how large does the minimum degree of an n-vertex graph G have to be to guarantee the existence of a clique factor in G? On the other hand, there is the probabilistic question. How large does p need to be to almost surely ensure the appearance of a clique factor in the Erdős-Rényi random graph G(n,p)? Optimal answers to these questions were given in two famous papers. The extremal question was answered by Hajnal and Szemerédi in 1970 and the probabilistic question by Johansson, Kahn and Vu in 2008. In this talk we bridge the gap between these two results by approaching the following question which contains the previous questions as special cases. Given an arbitrary graph of some fixed minimum degree, how many random edges need to be added on the same set of vertices to ensure the existence of a clique tiling? We give optimal answers to this question in all cases. Such results are part of a recent research trend studying properties of what is known as the randomly perturbed graph model, introduced by Bohman, Frieze and Martin in 2003. This is joint work with Jie Han and Andrew Treglown.