L3

Motile cells inside living tissues often encounter junctions, where their path branches into several alternative directions of migration. We present a theoretical model of cellular polarization for cells migrating along one-dimensional lines, exhibiting diverse migration modes. When arriving at a symmetric Y-junction and extending protrusions along the different paths that emanate from the junction. The model predicts the spontaneous emergence of deterministic oscillations between competing protrusions, whereby the cellular polarization and growth alternates between the competing protrusions. These predicted oscillations are found experimentally for two different cell types, noncancerous endothelial and cancerous glioma cells, migrating on patterned network of thin adhesive lanes with junctions. Finally we present an analysis of the migration modes of multicellular "trains" along one-dimensional tracks.

I will discuss logarithmic corrections to various CFT partition functions in the context of the AdS4/CFT3 correspondence for theories arising on the worldvolume of M2-branes. I will use four-dimensional gauged supergravity and heat kernel methods and present general expressions for the logarithmic corrections to the gravitational on-shell action or black hole entropy for a number of different supergravity backgrounds. I will outline several subtleties and puzzles in these calculations and contrast them with a similar analysis of logarithmic corrections performed directly in the eleven-dimensional uplift of a given four-dimensional supergravity background. This analysis suggests that four-dimensional supergravity consistent truncations are not the proper setting for studying logarithmic corrections in AdS/CFT. These results have important implications for the existence of scale-separated AdS vacua in string theory and for effective field theory in AdS more generally.

I will show that the massless irreducible representations of the Poincare group are precisely Corrolian field living on I^+. I will also show that the analogous massless irreducible representation of E11 are just the degrees of freedom of maximal supergravity. Finally I will speculate how spacetime could emerge from an underlying fundamental theory.

In this talk I will review how the BV formalism is used in quantizing theories with local gauge symmetries within the framework of perturbative algebraic quantum field theory. The latter is a mathematically rigorous approach to QFT that combines the locality idea going back to Haag and Kastler with Epstein-Glaser renormalization. In my talk I will also show how these methods can also lead to the construction of a factorization algebra.

I will recall the Zilber-Pink conjecture for Shimura varieties and give my perspective on current progress towards a proof.

Consider the following "controlled" random graph process: edges of the complete graph are revealed one by one in random order to an online algorithm, which immediately decides whether to retain each observed edge. The algorithm's objective is to construct a graph property within specified constraints on the total number of observed edges ("time") and the total number of retained edges ("budget").

During this talk, I will present results in this model for natural graph properties, such as connectivity, Hamiltonicity, and containment of fixed-size subgraphs. Specifically, I will describe a strategy to construct a Hamilton cycle at the hitting time for minimum degree 2 by retaining a linear number of edges. This extends the classical hitting time result for Hamiltonicity originally established by Ajtai–Komlós–Szemerédi and Bollobás.

The talk is based on joint work with Alan Frieze and Michael Krivelevich.

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves contains a Zariski-dense set of special points if and only if it is special. I'll explain this result, and discuss an effective version that Gal Binyamini, Harry Schmidt, Margaret Thomas and I proved.

I will present a generalisation of the Elekes-Szabó result, that any ternary algebraic relation in characteristic 0 having large intersections with (certain) finite grids must essentially be the graph of a group law, to a version where one obtains an algebraic group action. In the end the conclusion will be similar, but with weaker assumptions. This is recent work with Tingxiang Zou.

A key ingredient in the proof of the model completeness of the real exponential field was a valuation inequality for polynomially bounded o-minimal structures. I shall briefly describe the argument, and then move on to the complex exponential field and Zilber's quasiminimality conjecture for this structure. Here, one can reduce the problem to that of establishing an analytic continuation property for (complex) germs definable in a certain o-minimal expansion of the real field and in order to study this question I propose notions of "complex Hardy fields" and "complex valuations".   Here, the value group is not necessarily ordered but, nevertheless, one can still prove a valuation inequality.