C3

I will give an update on a proposed model theory for directed limits and colimits of first-order structures, originally motivated by applications to commutative algebra and the model theory of valued fields. To illustrate the usefulness of the formalism, I will prove a new general AKE theorem in mixed characteristic in a language with a cross-section of the value group and a lift of the residue field.

I will also discuss connections with other approaches to this topic, including pro- and ind-definable sets, infinitary logic, Feferman's local functors, accessible functors, and ultraproducts, some of which I have not discussed previously.

There has been substantial progress in the construction of eigenvarieties and $p$-adic families of automorphic forms, and their relationship with Selmer groups and ($p$-adic) $L$-functions. In this talk I will introduce some of these constructions, starting with modular forms, and the concept of complete $p$-adic rigidity: the non-existence of nontrivial $p$-adic deformations. I will explain some of the techniques used to study the geometry of eigenvarieties, and how these specialise to show that certain noncuspidal 'Saito—Kurokawa' points are completely $p$-adically rigid. If time permits, I will also briefly outline how similar strategies may be used to construct $p$-adic families through cuspidal, nonholomorphic Saito—Kurokawa points and to produce nontrivial Selmer classes predicted by the Bloch—Kato conjecture. 

I am going to talk on a work aimed at formalising approximation procedures in physics.  The main new model-theoretic tool in this work is the notion of the ultraproduct in the classes of emerging metric structures which generalises the ultraproduct  of general structures developed by J.Kiesler. In particular, the structure of Minkowski spacetime with the action of the Lorentz group is an emerging metric ultraproduct of certain finite structures invariant under the action of appropriate finite groups. Also, it is shown that any compact simple Lie group is representable as emerging metric ultraproduct of finite groups.
 

Nilsequences are sequences coming from Lie groups which play the role of additive characters in higher order Fourier analysis. In this talk, I will define these and give some basic examples without assuming any prior knowledge. I'll use this to state an equidistribution result due to Green and Tao, and compare what happens in this setting to the familiar case of sequences in the torus.
 

The Lindelöf hypothesis is known to be weaker than the Riemann hypothesis and one way to assess the difference in their strength is to consider what can be said about the zeroes of the zeta function under the assumption of the Lindelöf hypothesis. Viewing this question in the context of zero density estimates, we prove that $N(\sigma,T) \leq T^{\frac{4(5-6\sigma)}{3(3-2\sigma)} + o(1)}$. This improves the currently known estimate conditional on the Lindelöf hypothesis, $N(\sigma,T) \leq T^{2(1-\sigma)+o(1)}$ based on the mean value theorem, for $\sigma$ near $3/4$.

Abstract elementary classes (AECs) provide an extension of first order model theory in which we can still attempt a classification theory. The question of when limit models (a kind of surrogate for saturated models for AECs) are isomorphic has connections to important open problems in AECs, such as Shelah's categoricity conjecture. Most work in this area is towards 'positive' results - that is, showing limit models are isomorphic. The question of when limit models are not isomorphic is less explored.

In this talk we give a full characterisation of the spectrum of limit models under reasonable assumptions in a stable AEC - that is, describe completely which limit models are isomorphic and which are not. In particular this applies to the first order stable setting. Given time we will discuss applications, a more general result in the 'positive' direction, and touch on a recent result which says that all high cofinality limit models are disjoint amalgamation bases. Based largely on joint work with Marcos Mazari-Armida.

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.

The ε-energy is a regularisation of the Dirichlet energy introduced by Tobias Lamm. Like the famous Sacks-Uhlenbeck regularisation this greatly improves the existence and regularity theory. When we take the limit of a sequence of ε-harmonic maps with the parameter ε decreasing to 0 these converge, in the standard bubbling sense, to harmonic maps, which we hope to extract information about. I will talk about some recent results for these sequences, being when we might hope to have no loss of energy and no neck forming and what sort of harmonic maps we can obtain in the limit.