Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [Cohen-Cont-Rossier-Xu, "Scaling Properties of Deep Residual Networks”, 2021]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. Based on joint work with C. Bayer and P. K. Friz.
 

The Uniswap v3 ecosystem is built upon liquidity pools, where pairs of tokens are exchanged subject to a fee. We propose a systematic workflow to extract a meaningful but tractable sub-universe out of the current > 6,000 pools. We filter by imposing minimum levels on individual pool features, e.g. liquidity locked and agents’ activity, but also maximising the interconnection between the chosen pools to support broader dynamics. Then, we investigate liquidity consumption behaviour on the most relevant pools for Jan-June 2022. We propose to describe each liquidity taker by a transaction graph, which is a complete graph where nodes are transactions on pools and edges have weights from the time elapsed between pairs of transactions. Each graph is embedded into a vector by our own variant of the NLP rooted graph2vec algorithm. Thus, we are able to investigate the structural equivalence of liquidity takers behaviour and extract seven clusters with interpretable features. Finally, we introduce an ideal crypto law inspired from the ideal gas law of thermodynamics. Our model tests a relationship between variables that govern the mechanisms of each pool, i.e. liquidity provision, consumption, and price variation. If the law is satisfied, we say the pool has high cryptoness and demonstrate that it constitutes a better venue for the activity of market participants. Our metric could be employed by regulators and practitioners for developing pool health monitoring tools and establishing minimum levels of requirements.

(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.