Cancelled

Minimal surfaces, which are critical points of the area functional, have long been a source of fruitful problems in geometry. In this talk, I will introduce a new approach, primarily coming from a recent paper of M. Struwe, to constructing free boundary minimal discs using a gradient flow of a suitable energy functional. I will discuss the uniqueness of solutions to the gradient flow, including recent work on the uniqueness of weak solutions, and also what is known about the qualitative behaviour of the flow, especially regarding the interpretation of singularities which arise. Time permitting, I will also mention ongoing joint work with M. Rupflin and M. Struwe on extending this theory to general surfaces with boundary.

In recent joint work with Pablo Cubides Kovacsics and Jinhe Ye on beautiful pairs in the unstable context, the amalgamation property (AP) for the class of global definable types plays a key role. In the talk, we will first indicate some important cases in which AP holds, and we will then present the construction of examples of theories, obtained in joint work with Rosario Mennuni, where AP fails.

I will talk about the behaviour of random maps on surfaces (for example, random triangulations) of given genus, when their size tends to infinity. Such questions can be asked from the viewpoint of the local behaviour (Benjamini-Schramm convergence) or global behaviour (diameter, Gromov Hausdorff convergence), and in both cases, much combinatorics is involved. I will survey the landmark results for the case of fixed genus, and state very recent results in which we manage to address the "high genus" regime, when the genus grows proportionally to the size – for this regime we establish isoperimetric inequalities and prove the long-suspected fact that the diameter is logarithmic with high probability.

Based on joint work with Thomas Budzinski and Baptiste Louf.

Coboundary expansion is a notion introduced by Linial and Meshulam, and by Gromov that combines combinatorics topology and linear algebra. Kaufman and Lubotzky observed its relation to "Property testing", and in recent years it has found several applications in theoretical computer science, including for error correcting codes (both classical and quantum), for PCP agreement tests, and even for studying polarization in social networks.

In the talk I will introduce this notion and some of its applications. No prior knowledge is assumed, of course.

Does the limit construction for inverse systems of first-order structures preserve elementary equivalence? I will give sufficient conditions for when this is the case. Using Karp's theorem, we explain the connection between a syntactic and formal-semantic approach to inverse limits of structures. We use this to give a simple proof of van den Dries' AKE theorem (in ZFC), a general AKE theorem for mixed characteristic henselian valued fields with no assumptions on ramification. We also recall a seemingly forgotten result of Feferman, that can be interpreted as a "saturated" AKE theorem in positive characteristic: given two elementarily equivalent $\aleph_1$-saturated fields $k$ and $k'$, the formal power series rings $k[[t]]$ and $k'[[t]]$ are elementarily equivalent as well. We thus hope to popularise some ideas from categorical logic.

Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+. As cosequences we obtain that ACFA+ is a simple theory, an explicit description of the connected component of the Kim-Pillay group and (weak) elimination of imaginaries. If time permits we present some results on higher amalgamation.