Dimension reduction is an overarching theme in data science: we enjoy finding informative patterns, features or substructures in large, complex data sets. Within the field of network science, an important problem of this nature is to identify core-periphery structure. Given a network, our task is to assign each node to either the core or periphery. Core nodes should be strongly connected across the whole network whereas peripheral nodes should be strongly connected only to core nodes. More generally, we may wish to assign a non-negative value to each node, with a larger value indicating greater "coreness." This type of problem is related to, but distinct from, commumnity detection (finding clusters) and centrality assignment (finding key players), and it arises naturally in the study of networks in social science and finance. We derive and analyse a new iterative algorithm for detecting network core-periphery structure.

Using techniques in nonlinear Perron-Frobenius theory we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core--periphery random graph model. This viewpoint also gives a new basis for quantitatively judging a core--periphery detection algorithm. We illustrate the algorithm on a range of synthetic and real networks, and show that it offers advantages over the current state-of-the-art.

This is joint work with Francesco Tudisco (Strathclyde)

The problem of decomposing a given dataset as a superposition of basic motifs arises in a wide range of application areas, including neural spike sorting and the analysis of astrophysical and microscopy data. Motivated by these problems, we study a "short-and-sparse" deconvolution problem, in which the goal is to recover a short motif a from its convolution with a random spike train $x$. We formulate this problem as optimization over the sphere. We analyze the geometry of this (nonconvex) optimization problem, and argue that when the target spike train is sufficiently sparse, on a region of the sphere, every local minimum is equivalent to the ground truth, up to symmetry (here a signed shift). This characterization obtains, e.g., for generic kernels of length $k$, when the sparsity rate of the spike train is proportional to $k^{-2/3}$ (i.e., roughly $k^{1/3}$ spikes in each length-$k$ window). This geometric characterization implies that efficient methods obtain the ground truth under the same conditions. 

Our analysis highlights the key roles of symmetry and negative curvature in the behavior of efficient methods -- in particular, the role of a "dispersive" structure in promoting efficient convergence to global optimizers without the need to explicitly leverage second-order information. We sketch connections to broader families of benign nonconvex problems in machine learning and signal processing, in which efficient methods obtain global optima independent of initialization. These problems include variants of sparse dictionary learning, tensor decomposition, and phase recovery.

Joint work with Yuqian Zhang, Yenson Lau, Han-Wen Kuo, Dar Gilboa, Sky Cheung, Abhay Pasupathy

The stabilization of thermo-mechanical systems is a classical problem in thermodynamics and well

understood in a context of gases. The objective of this talk is to indicate the role of null-Lagrangians and

certain transport/stretching identities in stabilizing thermomechanical systems associated with general

thermoelastic free energies. This allows to prove various convergence results among thermomechan-

ical theories, and suggests a variational scheme for the approximation of the equations of adiabatic

thermoelasticity.

The Kronecker-Weber theorem states that every finite abelian extension of the rationals is contained in some cyclotomic field. I will present a proof that emphasizes the standard local-global philosophy by first proving it for the p-adics and then deducing the global case.

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes according to the space-time scale in such a way that the system rescales to the SPDE in the large-scale limit. We justify this belief for a large class of continuous microscopic growth models (for KPZ) and phase co-existence models (for Phi^4_3), allowing microscopic nonlinear mechanisms far beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussians. This essentially allows one to reduce the problem of a general function to that of a polynomial. Based on a joint work with Martin Hairer, and another joint work in progress with Chenjie Fan and Jiawei Li. 

Neural Network algorithms have achieved unprecedented performance in image recognition over the past decade. However, their application in real world use-cases, such as self driving cars, raises the question of whether it is safe to rely on them.

We generally associate the robustness of these algorithms with how easy it is to generate an adversarial example: a tiny perturbation of an image which leads it to be misclassified by the Neural Net (which classifies the original image correctly). Neural Nets are strongly susceptible to such adversarial examples, but when the architecture of the target neural net is unknown to the attacker it becomes more difficult to generate these examples efficiently.

In this Black-Box setting, we frame the generation of an adversarial example as an optimisation problem solvable via derivative free optimisation methods. Thus, we introduce an algorithm based on the BOBYQA model-based method and compare this to the current state of the art algorithm.

Simplicial volume was first introduced by Gromov to study the minimal volume of manifolds. Since then it has emerged as an active research field with a wide range of applications. 

I will give an introduction to simplicial volume and describe a recent result with Clara Löh (University of Regensburg), showing that the set of simplicial volumes in higher dimensions is dense in $R^+$.

It is often fruitful to study an infinite discrete group via its finite quotients.  For this reason, conditions that guarantee many finite quotients can be useful.  One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.