NOTE CHANGE OF TIME AND PLACE

It is known by results of Macintyre and Chatzidakis-Hrushovski that the theory ACFA of existentially closed difference fields is decidable. By developing techniques of difference algebraic geometry, we view quantifier elimination as an instance of a direct image theorem for Galois formulae on difference schemes. In a context where we restrict ourselves to directly presented difference schemes whose definition only involves algebraic correspondences, we develop a coarser yet effective procedure, resulting in a primitive recursive quantifier elimination. We shall discuss various algebraic applications of Galois stratification and connections to fields with Frobenius.

I will speak about weak Fano 3-folds, K3 surfaces and their Picard lattices, and explain how to solve the matching problem in various situations

Motivated by the study of supersymmetric backgrounds with non-trivial fluxes, we provide a formulation of supergravity in the language of generalised geometry, as first introduced by Hitchin, and its extensions. This description both dramatically simplifies the equations of the theory by making the hidden symmetries manifest, and writes the bosonic sector geometrically as a direct analogue of Einstein gravity. Further, a natural analogue of special holonomy manifolds emerges and coincides with the conditions for supersymmetric backgrounds with flux, thus formulating these systems as integrable geometric structures.
 

Dellamonica, Kohayakawa, Rödl and Ruciński showed that for $p=C(\log n/n)^{1/d}$ the random graph $G(n,p)$ contains asymptotically almost surely all spanning graphs $H$ with maximum degree $d$ as subgraphs. In this talk I will discuss a resilience version of this result, which shows that for the same edge density, even if a $(1/k-\epsilon)$-fraction of the edges at every vertex is deleted adversarially from $G(n,p)$, the resulting graph continues to contain asymptotically almost surely all spanning $H$ with maximum degree $d$, with sublinear bandwidth and with at least $C \max\{p^{-2},p^{-1}\log n\}$ vertices not in triangles. Neither the restriction on the bandwidth, nor the condition that not all vertices are allowed to be in triangles can be removed. The proof uses a sparse version of the Blow-Up Lemma. Joint work with Peter Allen, Julia Ehrenmüller, Anusch Taraz.

Erdős asked the following question: given a positive integer $n$, what is the largest integer $k$ such that any set of $n$ points in a plane, with no $4$ on a line, contains $k$ points no $3$ of which are collinear? Füredi proved that $k = o(n)$. Cardinal, Toth and Wood extended this result to $\mathbb{R}^3$, finding sets of $n$ points with no $5$ on a plane whose subsets with no $4$ points on a plane have size $o(n)$, and asked the question for the higher dimensions. For given $n$, let $k$ be largest integer such that any set of $n$ points in $\mathbb{R}^d$ with no more than $d + 1$ cohyperplanar points, has $k$ points with no $d + 1$ on a hyperplane. Is $k = o(n)$? We prove that $k = o(n)$ for any fixed $d \geq 3$.