The Turán number of an $r$-graph $G$, denoted by $ex(n,G)$, is the maximum number of edges in an $G$-free $r$-graph on $n$ vertices. The Turán density  of an $r$-graph $G$, denoted by $\pi(G)$, is the limit as $n$ tends to infinity of the maximum edge density of an $G$-free $r$-graph on $n$ vertices.

During this talk I will discuss a method, which we call  local stability method, that allows one to obtain exact Turán numbers from Turán density results. This method can be thought of as an extension of the classical stability method by  generically utilising the Lagrangian function. Using it, we obtained new hypergraph Turán numbers. In particular, we did so for a hypergraph called generalized triangle, for uniformities 5 and 6, which solved a conjecture of Frankl and Füredi from 1980's.

This is joint work with Sergey Norin.

Dstl are interested in removing liquid contaminants from capillary features (cracks in surfaces, screw threads etc.). We speculated that liquid decontaminants with low surface tension would have beneficial properties. The colloid literature, and in particular the oil recovery literature, discusss the properties of multiphase systems in terms of “Winsor types”, typically consisting of “brine” (water + electrolyte), “oil” (non-polar, water-insoluble solvent) and surfactant. Winsor I systems are oil-in-water microemulsions and Winsor II systems are water-in-oil microemulsions. Under certain circumstances, the mixture will separate into three phases. The middle (Winsor III) phase is surfactant-rich, and is reported to exhibit ultra-low surface tension. The glycol ethers (“Cellosolve” type solvents) consist of short (3-4) linked ether groups attached to short (3-4 carbon) alkyl chains. Although these materials would not normally be considered to be surfactants, their polar head, non-polar tail properties allow them to form a “surfactantless” Winsor III middle phase. We have found that small changes in temperature, electrolyte concentration or addition of contaminant can cause these novel colloids to phase separate. In our decontamination experiments, we have observed that contaminant-induced phase separation takes the form of droplets of the separating phase. These droplets are highly mobile, exhibiting behaviour that is visually similar to Brownian motion, which induces somewhat turbulent liquid currents in the vicinity of the contaminant. We tentatively attribute this behaviour to the Marangoni effect. We present our work as an interesting physics/ physical chemistry phenomenon that should be suitable for mathematical analysis.

Currently all medical x-ray imaging is performed using point-like sources which produce cone or fan beams. In planar radiology the source is fixed relative to the patient and detector array and therefore only 2D images can be produced. In CT imaging, the source and detector are rotated about the patient and through reconstruction (such as Radon methods), a 3D image can be formed. In Tomosynthesis, a limited range of angles are captured which greatly reduces the complexity and cost of the device and the dose exposure to the patient while largely preserving the clinical utility of the 3D images. Conventional tomosynthesis relies on mechanically moving a source about a fixed trajectory (e.g. an arc) and capturing multiple images along that path. Adaptix is developing a fixed source with an electronically addressable array that allows for a motion-free tomosynthesis system. The Adaptix approach has many advantages including reduced cost, portability, angular information acquired in 2D, and the ability to shape the radiation field (by selectively activating only certain emitters).

The proposed work would examine the effects of patient motion and apply suitable corrections to the image reconstruction (or raw data). Many approaches have been considered in the literature for motion correction, and only some of these may be of use in tomosynthesis. The study will consider which approaches are optimal, and apply them to the present geometry.

A related but perhaps distinct area of investigation is the use of “structured light” techniques to encode the x-rays and extract additional information from the imaging. Most conventional structured light approaches are not suitable for transmissive operation nor for the limited control available in x-rays. Selection of appropriate techniques and algorithms, however, could prove very powerful and yield new ways of performing medical imaging.

Adaptix is a start-up based at the Begbroke Centre for Innovation and Enterprise. Adaptix is transforming planar X-ray – the diagnostic imaging modality most widely used in healthcare worldwide. We are adding low-dose 3D capability – digital tomosynthesis - to planar X-ray while making it more affordable and truly portable so radiology can more easily travel to the patient. This transformation will enhance patient’s access to the world’s most important imaging technologies and likely increases the diagnostic accuracy for many high incidence conditions such as cardiovascular and pulmonary diseases, lung cancer and osteoporosis. 
 

Hotels.com is one of the world’s leading accommodation booking websites featuring an inventory of around 300.000 hotels and 100s of millions of users. A crucial part of our business is to act as an agent between these two sides of the market, thus reducing search costs and information asymmetries to enable our visitors to find the right hotel in the most efficient way.

From this point of view selling hotels is one large recommendation challenge: given a set of items and a set of observed choices/ratings, identify a user’s preference profile.  Over the last years this particular problem has been intensively studied by a strongly interdisciplinary field based on ideas from choice theory, linear algebra, statistics, computer science and machine learning. This pluralism is reflected in the broad array of techniques that are used in today’s industry applications, i.e. collaborative filtering, matrix factorization, graph-based algorithms, decision trees or generalized linear models.

The aim of this workshop is twofold.

Firstly we want to give some insight into the statistical modelling techniques and assumptions employed at hotels.com, the practical challenges one has to face when designing a flexible and scalable recommender system and potential gaps between current research and real-world applications.

Secondly we are going to consider some more advanced questions around learning to rank from partial/incomplete feedback (1), dealing with selection-bias correction (2) and how econometrics and behavioral theory (eg Luce, Kahneman /Tversky) can be used to complement existing techniques (3).

Tbd

A Kähler group is a group which can be realised as the fundamental group of a close Kähler manifold. We will prove that for a Kähler group $G$ we have that $G$ is residually free if and only if $G$ is a full subdirect product of a free abelian group and finitely many closed hyperbolic surface groups. We will then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler: We explain how to construct subgroups of direct products of surface groups which have even first Betti number but are not Kähler. All relevant notions will be explained in the talk.

We find exact solutions of Maxwell's equations that are the precise analog of plane waves, but in the case that the translation group is replaced by the Abelian helical group. These waves display constructive/destructive interference with helical atomic structures, in the same way that plane waves interact with crystals. We show how the resulting far-field pattern can be used for structure determination. We test the method by doing theoretical structure determination on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea is that the structure is the orbit of a group, and this group is a subgroup of the invariance group of the differential equations. Joint work with Dominik Juestel and Gero Friesecke. (Acta Crystallographica A72 and SIAM J. Appl Math).