C1

Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.

In this informal talk, I describe the kinds of functions relevant to scattering amplitudes in perturbative, four-dimensional quantum field theories. In particular, I will argue that generic amplitudes are non-polylogarithmic (beyond one loop), but that there is an upper bound to their geometric complexity. Moreover, I show a veritable `bestiary' of examples which saturate this bound in complexity---including three, all-loop families of integrals defined in massless $\phi^4$ theory which can, at best, be represented as dilogarithms integrated over (2L-2)-dimensional Calabi-Yau manifolds. 

Many protein interaction databases provide confidence scores based on the experimental evidence underpinning each in- teraction. The databases recommend that protein interac- tion networks (PINs) are built by thresholding on these scores. We demonstrate that varying the score threshold can re- sult in PINs with significantly different topologies. We ar- gue that if a node metric is to be useful for extracting bio- logical signal, it should induce similar node rankings across PINs obtained at different thresholds. We propose three measures—rank continuity, identifiability, and instability— to test for threshold robustness. We apply these to a set of twenty-five metrics of which we identify four: number of edges in the step-1 ego network, the leave-one-out dif- ference in average redundancy, average number of edges in the step-1 ego network, and natural connectivity, as robust across medium-high confidence thresholds. Our measures show good agreement across PINs from different species and data sources. However, analysis of synthetically gen- erated scored networks shows that robustness results are context-specific, and depend both on network topology and on how scores are placed across network edges. 

The restricted isometry property is arguably the most prominent tool in the theory of compressive sensing. In its classical version, it features l_2 norms as inner and outer norms. The modified version considered in this talk features the l_1 norm as the inner norm, while the outer norm depends a priori on the distribution of the random entries populating the measurement matrix.  The modified version holds for a wider class of random matrices and still accounts for the success of sparse recovery via basis pursuit and via iterative hard thresholding. In the special case of Gaussian matrices, the outer norm actually reduces to an l_2 norm. This fact allows one to retrieve results from the theory of one-bit compressive sensing in a very simple way. Extensions to one-bit matrix recovery are then straightforward.
 

This will be a discussion of the paper https://arxiv.org/abs/1607.06978.

In 1897 J.J. Thomson 'discovered' the electron. The previous year, he and his research student Ernest Rutherford (later to 'discover' theatomic nucleus), collaborated in experiments to work out why gases exposed to x-rays became conducting. 

This talk will discuss the very different mathematical educations of the two men, and the impact these differences had on their experimental investigation and the theory they arrived at. This theory formed the backdrop to Thomson's electron work the following year. 

This will be a discussion of the paper https://arxiv.org/abs/1604.02618

This will be a quick introduction to tropical algebra and the main results from the paper https://arxiv.org/pdf/1604.00113.pdf

Finding implementable descriptions of the possible configurations of a knotted DNA molecule has remarkable importance from a biological point of view, and it is a hard and well studied problem in mathematics.
Here we present two newly developed mathematical tools that describe the configuration space of knots and model the action of Type I and II Topoisomerases on a covalently closed circular DNA molecule: the Reidemeister graphs.
We determine some local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring.
Finally, we indicate how the Reidemeister graphs can be used to infer information about the proteins' action.