This week

Many popular Markov chain Monte Carlo samplers for statistical mechanics models drastically slow down in the vicinity of a phase transition. This slowdown is deeply tied to the geometry of the model. I will discuss some aspects of this phenomenon and show in particular that, on certain geometries, it is possible to identify the cause(s) of the slowdown. This can then be used to build a physically reasonable dynamics, which samples from the model from which slow objects have been removed in a sense I will make precise, and for which fast sampling guarantees can be obtained beyond the phase transition. This is a joint work with Roland Bauerschmidt and Thierry Bodineau based on the following paper: https://link.springer.com/article/10.1007/s00440-024-01326-9 .

In this paper, a new approach for solving the problems of pricing and hedging derivatives is introduced in a general frictionless market setting. The method is applicable even in cases where an equivalent local martingale measure fails to exist. Our main results include a new superhedging duality for American options when wealth processes can be negative and trading strategies are subject to a cone constraint. This answers one of the questions raised by Fernholz, Karatzas and Kardaras.

This is joint with Miklos Rasonyi.

The differential geometry of spheres has long been a source of central problems in mathematics, and Sasaki–Einstein metrics—odd-dimensional analogues of Kähler–Einstein metrics—offer a particularly rich perspective, with significance both in geometry and in theoretical physics. In joint work with Yuchen Liu and Taro Sano, we construct infinitely many Sasaki–Einstein metrics on odd-dimensional spheres that bound parallelizable manifolds, thereby confirming conjectures of Boyer–Galicki–Kollár and Collins–Székelyhidi. Our approach is based on establishing the K-stability of certain Fano weighted hypersurfaces.

The Consumer Price Index (CPI) is a key economic statistic that informs various stakeholders, such as citizens, governments, about the recent past and future direction of inflation. Accurate inflation assessment and good forecasts are vital to understand, inform (e.g. internet charges or train fares that are tied to inflation indices) and control the economy, such as interest rate setting in central banks. The CPI averages various volume-weighted price indices of a basket of underlying component items taken from across the economy and is separately computed in very similar ways by the statistical offices of many countries. In the UK, the Bank of England has come in for criticism for its inflation forecasts, but the recent Bernanke 2024 review noted that `the forecast errors made by the Bank and those made by external forecasters are barely distinguishable.' Recently, it has been shown several times in the literature that generalised network autoregressive (GNAR) time series models have proved to be powerful forecasters in a variety of applied situations. We develop a technique (RaGNAR) to forecast monthly UK CPI using GNARs models by averaging forecasts obtained over many randomised networks and models, focussing on CPI as the objective. RaGNAR significantly outperforms both traditional benchmark and pre-trained probabilistic time series forecasting models across all horizons, and delivers materially more accurate forecasts than the Bank of England across four- to six-month horizons. This is somewhat surprising given the time-consuming and complex nature of the Bank’s forecasting process and the speed and efficiency of RaGNAR. Our methods also permit us to identify CPI components that most strongly influence the CPI during different periods.

Optimal control problems often involve the solution of high dimensional nonlinear PDEs, which is a key computational bottleneck. In this talk we will consider how neural networks can be used as a computational tool for these problems, how simple test cases can work deceptively well, and how fine details of the approach can lead to different results.
Based on joint work with Justin Sirignano, Deqing Jiang and Jackson Hebner.

This paper explores the application of Monte Carlo (MC), Quasi-Monte Carlo (QMC), and Randomized Quasi-Monte Carlo (RQMC) methods in the context of option pricing and risk analysis under the time-homogeneous hyperbolic local volatility (HLV) model. While standard MC methods suffer from slow convergence, QMC techniques leverage low-discrepancy sequences to achieve superior convergence rates, particularly for problems with low effective dimension. However, the deterministic nature of QMC prevents reliable error estimation, a limitation overcome by RQMC through randomized sequences such as Owen’s nested scrambling. The study incorporates variance reduction techniques such as Brownian Bridge (BB) and Principal Component Analysis (PCA) to reduce effective dimension and enhance convergence. Numerical experiments on Asian options demonstrate significant accuracy gains using RQMC over MC and QMC, especially when PCA is used. The paper also analyzes the convergence behavior and effective dimensions of price and Greeks (Delta, Gamma), confirming that RQMC-PCA offers the best performance in high dimensional settings. Joint work with Sergei Kucherenko.

Nonpairwise "higher-order" network interactions can shape the collective
network dynamics of interacting nodes. But what is the right
"higher-order" object to encode these interactions? For examples of
synchronization and heteroclinic dynamics, we start with the dynamical
phenomenon and give conditions what type of higher-order interactions
are required for the dynamics to arise. This highlights limitations of
using classical hypergraphs to model higher-order interactions.

Title: Newform congruences of local origin for classical and Hilbert modular forms

Abstract: The theory of Eisenstein congruences dates back to Ramanujan’s surprising discovery that the Fourier coefficients of the discriminant function are congruent to the 11th power divisor sum modulo 691. This observation can be explained via the congruence of two modular forms of weight 12 and level 1; the discriminant function and the Eisenstein series, E_{12}. Eisenstein congruences were later used by Ribet in his proof of the converse to Herbrand's theorem.

The full details can be found: https://www.londonmathfinance.org.uk/lgs-phd-day