L5

The main object of study of the talk is the balanced triple product p-adic L-function; this is a p-adic L-function associated with a triple of families of (quaternionic) modular forms. The first instances of these functions appear in the works of Darmon-Lauder-Rotger, Hsieh, and Greenberg-Seveso. They have proved to be effective tools in studying cases of the p-adic equivariant Birch & Swinnerton-Dyer conjecture. With this aim in mind, we discuss the construction of a new p-adic L-function, extending Hsieh's construction, and allowing classical weight one modular forms in the chosen families. Such improvement does not come for free, as it coincides with the increased dimension of certain Hecke-eigenspaces of quaternionic modular forms with non-Eichler level structure; we discuss how to deal with the problems arising in this more general setting. One of the key ingredients of the construction is a p-adic extension of the Jacquet-Langlands correspondence addressing these more general quaternionic modular forms. This is joint work in progress with Aleksander Horawa.

Many problems in number theory are equivalent to determining all of the rational points on some curve or family of curves. In general, finding all the rational points on any given curve is a challenging (even unsolved!) problem. 

The focus of this talk is rational points on so-called Erdős-Selfridge curves. A deep conjecture of Sander, still unproven in many cases, predicts all of the rational points on these curves. 

I will describe work-in-progress proving new cases of Sander's conjecture, and sketch some ideas in the proof. The core of the proof is a `mass increment argument,' which is loosely inspired by various increment arguments in additive combinatorics. The main ingredients are a mixture of combinatorial ideas and quantitative estimates in Diophantine geometry.

The closest thing we have to a general method for finding primes in sets is to use sieve methods to turn the problem into some other (hopefully easier) arithmetic questions about the set.

Unfortunately this process is still poorly understood - we don’t know ‘how much’ arithmetic information is sufficient to guarantee the existence of primes, and how much is not sufficient. Often arguments are rather ad-hoc.

I’ll talk about work-in-progress with Kevin Ford which shows that many of our common techniques are not optimal and can be refined, and in many cases these new refinements are provably optimal.

Hoheisel used zero-density results to prove that for all x large enough there is a prime number in the interval $[x−x^{\theta}, x]$ with $θ < 1$. The connection between zero-density estimates and primes in short intervals was explicitly described in the work of Ingham in 1937. The approach of Ingham combined with the zero-density estimates of Huxley (1972) provides us with the distribution of primes in $[x−x^{\theta}, x]$ with $\theta > 7/12$. Further improvement upon the value of \theta was achieved by combining sieves with the weighted zero-density estimates in the works of Iwaniec and Jutila, Heath-Brown and Iwaniec, and Baker and Harman. The last work provides the best result achieved using zero-density estimates. We will discuss the main ideas of the paper by Baker and Harman and simplify some parts of it to show a more explicit connection between zero-density results and the sieved sums, which are used in the paper. This connection will provide a better understanding on which parts should be optimised for further improvements and on what the limits of the methods are. This project is still in progress.

Coarse embeddings (maps between metric spaces whose distortion can be controlled by some function) occur naturally in various areas of pure mathematics, most notably in topology and algebra. It may therefore come as a surprise to discover that it is not known whether there is a coarse embedding of three-dimensional real hyperbolic space into the direct product of a real hyperbolic plane and a 3-regular tree. One reason for this is that there are very few invariants which behave monotonically with respect to coarse embeddings, and thus could be used to obstruct coarse embeddings.

In 1986, Katznelson and Tzafriri proved that, if $T$ is a power-bounded operator on a Banach space $X$, and the spectrum of $T$ meets the unit circle only at 1, then $\|T^n(I-T)\| \to 0$ as $n\to\infty$. Actually, they went further and proved that $\|T^nf(T)\| \to 0$ if $T$ and $f$ satisfy certain conditions. Soon afterward, analogous results were obtained for bounded $C_0$-semigroups $(T(t))_{t\ge0}$. Further extensions and variants were proved later. I will speak about several extensions to the Katznelson-Tzafriri theorem(s), including in particular a recent result(s) obtained by David Seifert and myself.

1:30-2:15 Deborah Miori, CDT student, University of Oxford

DeFi: Data-Driven Characterisation of Uniswap v3 Ecosystem & an Ideal Crypto Law for Liquidity Pools

Uniswap is a Constant Product Market Maker built around liquidity pools, where pairs of tokens are exchanged subject to a fee that is proportional to the size of transactions. At the time of writing, there exist more than 6,000 pools associated with Uniswap v3, implying that empirical investigations on the full ecosystem can easily become computationally expensive. Thus, we propose a systematic workflow to extract and analyse a meaningful but computationally tractable sub-universe of liquidity pools.

Leveraging on the 34 pools found relevant for the six-months time window January-June 2022, we then investigate the related liquidity consumption behaviour of market participants. We propose to represent each liquidity taker by a suitably constructed transaction graph, which is a fully connected network where nodes are the liquidity taker’s executed transactions, and edges contain weights encoding the time elapsed between any two transactions. We extend the NLP-inspired graph2vec algorithm to the weighted undirected setting, and employ it to obtain an embedding of the set of graphs. This embedding allows us to extract seven clusters of liquidity takers, with equivalent behavioural patters and interpretable trading preferences.

We conclude our work by testing for relationships between the characteristic mechanisms of each pool, i.e. liquidity provision, consumption, and price variation. We introduce a related ideal crypto law, inspired from the ideal gas law of thermodynamics, and demonstrate that pools adhering to this law are healthier trading venues in terms of sensitivity of liquidity and agents’ activity. Regulators and practitioners could benefit from our model by developing related pool health monitoring tools.

2:15-3:00 Žan Žurič, CDT student, Imperial College London

A Random Neural Network Approach to Pricing SPDEs for Rough Volatility

We propose a novel machine learning-based scheme for solving partial differential equations (PDEs) and backward stochastic partial differential equations (BSPDE) stemming from option pricing equations of Markovian and non-Markovian models respectively. The use of the so-called random weighted neural networks (RWNN) allows us to formulate the optimisation problem as linear regression, thus immensely speeding up the training process. Furthermore, we analyse the convergence of the RWNN scheme and are able to specify error estimates in terms of the number of hidden nodes. The performance of the scheme is tested on Black-Scholes and rBergomi models and shown to have superior training times with accuracy comparable to existing deep learning approaches.

I will present a model for an optimal portfolio allocation and consumption problem for a portfolio composed of a risk-free bond and two illiquid assets. Two forms of illiquidity are presented, both illiquidities based on Lévy processes. The goal of the investor is to maximise a certain utility function, and the optimal utility is found as a solution of a nonlinear PIDE of the Hamilton-Jacobi-Bellman kind.

Néron models are mathematical objects which play a very important role in contemporary arithmetic geometry. However, they usually behave badly, particularly in respect of exact sequences and base change, which makes most problems regarding their behaviour very delicate. Chai introduced the base change conductor, a rational number associated with a semiabelian variety $G$ which measures the failure of the Néron model of $G$ to commute with (ramified) base change. Moreover, Chai conjectured that this invariant is additive in certain exact sequences. We shall introduce a new method to study the Néron models of Jacobians of proper (possibly singular) curves, and sketch a proof of Chai's conjecture for semiabelian varieties which are also Jacobians. 

Let K be a number field and let L/K be an abelian extension. The genus field of L/K is the largest extension of L which is unramified at all places of L and abelian as an extension of K. The genus group is its Galois group over L, which is a quotient of the class group of L, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions L/K with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran.