Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.

CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach — such as computer vision, playing Go, or protein folding — are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation.

While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This talk is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications.

Such a 'geometric unification' endeavour in the spirit of Felix Klein's Erlangen Program serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.

In this talk, we discuss the convergence of a sequence of random fields that generalise the Gaussian Free Field and bi-Laplacian field. Such fields are defined in terms of non-homogeneous elliptic operators which will be sampled at random. Under standard assumptions of stochastic homogenisation, we identify the limit fields as the usual GFF and bi-Laplacian fields up to a multiplicative constant. This is a joint work with W. Ruszel.

I will report on joint work in progress with Aleksandar Milivojevic (MPIM Bonn) on the elementary topology of the space of almost complex structures on a manifold. First I will describe a certain natural parametrization and associated stratification of the space of linear complex structures on a vector space and give a lower bound for the number of complex k-planes jointly preserved by two linear complex structures. Then I will focus on dimension 6 and prove a formula for the homological intersection of two orthogonal almost complex structures on a Riemannian 6-manifold when these are regarded as sections of the twistor space.

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation with universality, and it has been derived from several microscopic models through scaling limits. When the temperature of a system tends to infinity, we can often extract a heat diffusion part with some residual perturbation by a Taylor expansion argument, which decomposition is crucial for the derivation. We will show through some particular models that we can thereby obtain the KPZ equation as a limit in a robust way.

Diffeomorphism symmetry is an intrinsic difficulty in gravitational theory, which appears in almost all of the questions in gravity. As is well known, the diffeomorphism symmetries in gravity should be interpreted as gauge symmetries, so only diffeomorphism invariant operators are physically interesting. However, because of the non-linear effect of gravitational theory, the results for diffeomorphism invariant operators are very limited.

In this work, we focus on the Jackiw-Teitelboim gravity in classical limit, and use Peierls bracket (which is a linear response like computation of observables’ bracket) to compute the algebra of a large class of diffeomorphism invariant observables. With this algebra, we can reproduce some recent results in Jackiw-Teitelboim gravity including: traversable wormhole, scrambling effect, and $SL(2)$ charges. We can also use it to clarify the question of when the creation of an excitation deep in the bulk increases or decreases the boundary energy, which is of crucial importance for the “typical state” version of the firewall paradox.

In the talk, I will first give a brief introduction of Peierls bracket, and then use the Peierls bracket to study the brackets between diffeomorphism invariant observables in Jackiw-Teitelboim gravity. I will then give two applications of this algebra: reproducing the scrambling effect, and studying the energy change after creating an excitation in the bulk.