We tie together two natural but, a priori, different themes. As a starting point consider Erdős and Simonovits's classical edge stability for an $(r + 1)$-chromatic graph $H$. This says that any $n$-vertex $H$-free graph with $(1 − 1/r + o(1)){n \choose 2}$ edges is close to (within $o(n^2)$ edges of) $r$-partite. This is false if $1 − 1/r$ is replaced by any smaller constant. However, instead of insisting on many edges, what if we ask that the $n$-vertex graph has large minimum degree? This is the basic question of minimum degree stability: what constant $c$ guarantees that any $n$-vertex $H$-free graph with minimum degree greater than $cn$ is close to $r$-partite? $c$ depends not just on chromatic number of $H$ but also on its finer structure.

Somewhat surprisingly, answering the minimum degree stability question requires understanding locally colourable graphs -- graphs in which every neighbourhood has small chromatic number -- with large minimum degree. This is a natural local-to-global colouring question: if every neighbourhood is big and has small chromatic number must the whole graph have small chromatic number? The triangle-free case has a rich history. The more general case has some similarities but also striking differences.

Latent Dirichlet Allocation (LDA) is capable of analyzing thousands of documents in a short time and highlighting important elements, recurrences and anomalies. It is generally used in linguistics to study natural language: its word analysis reveals the theme(s) of a document, each theme being identified by a specific vocabulary or, more precisely, by a particular statistical distribution of word frequency.
In the climatologists' use of LDA, the document is a daily weather map and the word is a pixel of the map. The theme with its corpus of words can become a cyclone or an anticyclone and, more generally, a 'pattern'  that the scientists term motif. Artificial intelligence – a sort of incredibly fast robot meteorologist – looks for correlations both between different places on the same map, and between successive maps over time. In a sense, it 'notices' that a particular location is often correlated with another location, recurrently throughout the database, and this set of correlated locations constitutes a specific pattern.
The algorithm performs statistical analyses at two distinct levels: at the word or pixel level of the map, LDA defines a motif, by assigning a certain weight to each pixel, and thus defines the shape and position of the motif;  LDA breaks down a daily weather map into all these motifs, each of which is assigned a certain weight.
In concrete terms, the basic data are the daily weather maps between 1948 and nowadays over the North Atlantic basin and Europe. LDA identifies a dozen or so spatially defined motifs, many of which are familiar meteorological patterns such as the Azores High, the Genoa Low or even the Scandinavian Blocking. A small combination of those motifs can then be used to describe all the maps. These motifs and the statistical analyses associated with them allow researchers to study weather phenomena such as extreme events, as well as longer-term climate trends, and possibly to understand their mechanisms in order to better predict them in the future.

The preprint of the study is available as:
 Lucas Fery, Berengere Dubrulle, Berengere Podvin, Flavio Pons, Davide Faranda. Learning a weather dictionary of atmospheric patterns using Latent Dirichlet Allocation. 2021. ⟨hal-03258523⟩ https://hal-enpc.archives-ouvertes.fr/X-DEP-MECA/hal-03258523v1
 

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.