We discuss linear convolutional neural networks (LCNs) and their critical points. We observe that the function space (that is, the set of functions represented by LCNs) can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the function space.

For instance, for LCNs with one-dimensional convolutions having stride one and arbitrary filter sizes, we provide a full description of the boundary of the function space. We further study the optimization of an objective function over such LCNs: We characterize the relations between critical points in function space and in parameter space and show that there do exist spurious critical points. We compute an upper bound on the number of critical points in function space using Euclidean distance degrees and describe dynamical invariants for gradient descent.

This talk is based on joint work with Thomas Merkh, Guido Montúfar, and Matthew Trager.

In the aftermath of the Thirty Years War (1618–1648), the mining regions of Central Europe underwent numerous technical and political evolutions. In this context, the role of underground geometry expanded considerably: drawing mining maps and working on them became widespread in the second half of the seventeenth century. The new mathematics of subterranean surveyors finally realized the old dream of “seeing through stones,” gradually replacing alternative tools such as written reports of visitations, wood models, or commented sketches.

I argue that the development of new cartographic tools to visualize the underground was deeply linked to broad changes in the political structure of mining regions. In Saxony, arguably the leading mining region, captain-general Abraham von Schönberg (1640–1711) put his weight and reputation behind the new geometrical technology, hoping that its acceptance would in turn help him advance his reform agenda. At-scale representations were instrumental in justifying new investments, while offering technical road maps to implement them.

In 2013, Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees found a remarkable correspondence between SCFTs in 4d with N ≥ 2 and vertex algebras. The chiral algebras of class S, i.e. the vertex algebras associated to theories of Class S, are of particular interest as they exhibit rich algebraic structures arising from the requirement of generalised S-duality. I will explain a mathematical construction of these vertex algebras, due to Arakawa, that is remarkably uniform and requires no knowledge of the underlying SCFT. Time permitting, I will detail a recent generalisation of this construction to the case of the chiral algebras of class S with outer automorphism twist lines.

Understanding the impact of societal and economic change on the labour market is important for many causes, such as automation or the post-carbon transition. Occupational mobility plays a role in how these changes impact the labour market because of indirect effects, brought on by the different levels of direct impact felt by individual occupations. We develop an agent-based model which uses a network representation of the labour market to understand these impacts. This network connects occupations that workers have transitioned between in the past, and captures the complex structure of relationships between occupations within the labour market. We develop these networks in both space and time using rich survey data to compare occupational mobility across the United States and through economic upturns and downturns to start understanding the factors that influence differences in occupational mobility.

For almost calibrated Lagrangian mean curvature flow it is known that all singularities are of Type II. To understand the finer structure of the singularities forming, it is thus necessary to understand the structure of general ancient solutions arising as potential limit flows at such singularities. We will discuss recent progress showing that ancient solutions with a blow-down a pair of static planes meeting along a 1-dimensional line are translators. This is joint work with J. Lotay and G. Szekelyhidi.

Let $N$ be a compact Riemannian manifold of dimension 3 or higher, and $g$ a Lipschitz non-negative (or non-positive) function on $N$. In joint works with Neshan Wickramasekera we prove that there exists a closed hypersurface $M$ whose mean curvature attains the values prescribed by $g$. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface $M$ is $C^2$ immersed and two-sided (it admits a global unit normal); the scalar mean curvature at $x$ is $g(x)$ with respect to a global choice of unit normal. More precisely, the immersion is a quasi-embedding, namely the only non-embedded points are caused by tangential self-intersections: around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially (as in the case of two spherical caps touching at a point). A special case of PMC (prescribed-mean-curvature) hypersurfaces is obtained when $g$ is a constant, in which the above result gives a CMC (constant-mean-curvature) hypersurface for any prescribed value of the mean curvature.

Recent advances in Deep Learning have enabled us to explore new application domains in molecular biology and drug discovery - including those driven by complex processes that defy analytical modelling.  However, despite the combined forces of increased data, improving compute resources and continuous algorithmic innovation all bringing previously intractable problems into the realm of possibility, many advances are yet to make a tangible impact for life science discovery.  In this talk, Dr Marcin J. Skwark will discuss the challenge of bringing machine learning innovation to tangible real-world impact.  Following a general introduction of the topic, as well as newly available methods and data, he will focus on the modelling of COVID-19 variants and, in particular, the DeepChain Early Warning System (EWS) developed by InstaDeep in collaboration with BioNTech.  With thousands of new, possibly dangerous, SARS-CoV-2 variants emerging each month worldwide, it is beyond humanities combined capacity to experimentally determine the immune evasion and transmissibility characteristics of every one.  EWS builds on an experimentally tested AI-first computational biology platform to evaluate new variants in minutes, and is capable of risk monitoring variant lineages in near real-time.  This is done by combining an AI-driven protein structure prediction framework with large, spike protein sequence-oriented Transformer models to allow for rapid simulation-free assessment of the immune escape risk and expected fitness of new variants, conditioned on the current state of the world.  The system has been extensively validated in cooperation with BioNTech, both in terms of host cell infection propensity (including experimental assays of receptor binding affinity), and immune escape (pVNT assays with monoclonal antibodies and real-life donor sera). In these assessments, purely unsupervised, data-first methods of EWS have shown remarkable accuracy. EWS flags and ranks all but one of the SARS-CoV-2 Variants of Concern (Alpha, Beta, Gamma, Delta… Omicron), discriminates between subvariants (e.g. BA.1/BA.2/BA.4 etc. distinction) and for most of the adverse events allows for proactive response on the day of the observation. This allows for appropriate response on average six weeks before it is possible for domain experts using domain knowledge and epidemiological data. The performance of the system, according to internal benchmarks, improves with time, allowing for example for supporting the decisions on the emerging Omicron subvariants on the first days of their occurrence. EWS impact has been notable in general media [2, 3] for the system's applicability to a novel problem, ability to derive generalizable conclusions from unevenly distributed, sparse and noisy data, to deliver insights which otherwise necessitate long and costly experimental assays.