The classical Turán problem asks: given a graph $H$, how many edges can an $4n$-vertex graph have while containing no isomorphic copy of $H$? By viewing $(k+1)$-uniform hypergraphs as $k$-dimensional simplicial complexes, we can ask a topological version (first posed by Nati Linial): given a $k$-dimensional simplicial complex $S$, how many facets can an $n$-vertex $k$-dimensional simplicial complex have while containing no homeomorphic copy of $S$? Until recently, little was known for $k > 2$. In this talk, we give an answer for general $k$, by way of dependent random choice and the combinatorial notion of a trace-bounded hypergraph. Joint work with Jason Long and Bhargav Narayanan.

Introduced by Fomin and Zelevinsky in 2002, cluster algebras have become ubiquitous in algebra, combinatorics and geometry. In this talk, I'll introduce the notion of a cluster algebra and present the approach of Kang-Kashiwara-Kim-Oh to categorify a large class of them arising from quantum groups. Time allowing, I will explain some recent developments related to the coherent Satake category.

Point counting on definable sets in non-archimedean settings has many faces. For sets living in Q_p^n, one can count actual rational points of bounded height, but for sets in C((t))^n, one rather "counts" the polynomials in t of bounded degree. What if the latter is of infinite cardinality? We treat three settings, each with completely different behaviour for point counting : 1) the setting of subanalytic sets, where we show finiteness of point counting but growth can be aribitrarily fast with the degree in t ; 2) the setting of Pfaffian sets, which is new in the non-archimedean world and for which we show an analogue of Wilkie's conjecture in all dimensions; 3) the Hensel minimal setting, which is most general and where finiteness starts to fail, even for definable transcendental curves! In this infinite case, one bounds the dimension rather than the (infinite) cardinality. This represents joint work with Binyamini, Novikov, with Halupczok, Rideau, Vermeulen, and separate work by Cantoral-Farfan, Nguyen, Vermeulen.