In this seminar, we consider an isoperimetric problem for planar tilings with possibly unequal repeating cells. We present general existence and regularity results, and we study the classification of planar isoperimetric double tilings, namely tilings with two repeating cells of minimal perimeter. In this case, we explicitly determine the associated energy profile and provide a complete description of the phase transitions. We also comment on possible extensions and discuss some open problems. This is based on joint work with M. Novaga and E. Paolini.

The study of combinatorial designs includes some of the oldest questions at the heart of combinatorics. In a breakthrough result of 2014, Keevash proved the longstanding Existence Conjecture by showing the existence of (n,q,r)-Steiner systems (equivalently K_q^r-decompositions of K_n^r) for all large enough n satisfying the necessary divisibility conditions. Meanwhile, in recent decades, incremental progress has been made on the celebrated Nash-Williams' Conjecture of 1970, which posits that any large enough, triangle-divisible graph on n vertices with minimum degree at least 3n/4 admits a triangle decomposition. In 2021, Glock, Kühn, and Osthus proposed a generalization of these results by conjecturing a hypergraph version of the Nash-Williams' Conjecture, where their proposed minimum degree K_q^r-decomposition threshold is motivated by hypergraph Turán theory. By using the recently developed method of refined absorption and establishing a non-uniform Turán theory, we tie the K_q^r-decomposition threshold to its fractional relaxation. Combined with the best-known fractional decomposition threshold from Delcourt, Lesgourgues, and Postle, this dramatically closes the gap between what was known and the above conjecture. This talk is based on joint work with Luke Postle.

We reformulate the statement that the theory of the free group is stable in terms of simplicial diagram chasing and profinite sets, without any terminology from logic. This includes three characterisations of stability (via indiscernible sequences, counting types, and definable types), and the notions of a first order theory and a model.

We do so by generalising slightly and allowing the universe of a first order structure/model to be an arbitrary (symmetric) simplicial set: formulas and basic predicates now may denote sets of simplices of an arbitrary (symmetric) simplicial set rather than sets of tuples of elements of a set. In this generalised sense the type space functor of a theory is its universal model classifying its usual models: taking the type of a tuple gives a map from a usual model of a theory to its type space functor. We define a property of simplicial maps weaker then being a fibration, and find it appears in the conditions characterising which maps correspond to models, when the generalised semantics is well-behaved, and which symmetric simplicial profinite sets correspond to first order theories.