This week

In the first part of the seminar I will review the main features of the Hopfield model, providing as well an overview on its numerous and broad applications. Then, inspired by biological information-processing, I will enrich the framework and make the network able to successfully and cheaply handle structured datasets. In particular, I will recast reinforcement and remotion mechanisms occurring in mammal’s brain during sleep into suitable machine-learning hyperparameters. Finally, I will show that this framework is also effective to face a classical inference problem, that is, matrix factorisation. Results presented are both analytical and computational.

There is a recent debate on whether sustainable investing necesarily impact portfolio performance negatively. We model the financial impact of portfolio constraints by attributing the performance of dynamic portfolios to contributions from individual constraints. We consider a mean-variance portfolio problem with unknown asset returns. Investors impose a dynamic constraint based on a firm characteristic that contains information about returns, such as the environmental, social, and governance (ESG) score. We characterize the optimal investment strategy through two stochastic Riccati equations. Using this framework, we demonstrate that, depending on the correlation between returns and firm characteristics, incorporating the constraint can, in certain cases, enhance portfolio performance compared to a passive benchmark that disregards the information embedded in these constraints. Our results shed light on the role of implicit information contained in constraints in determining the performance of a constrained portfolio.

This project is joint work with Ruixun Zhang (Peking University), Yufei Zhang (Imperial College London) and Xunyu Zhou (Columbia University).

Having nearly finished our second year of teaching MATH0039, an ancillary course on differential and integral calculus, we would like to share our experience and what we have learnt. We will discuss some of the abstract principles that guided us when writing lecture notes, problem sheets and exams, before sharing some tips on how to give a good lecture and in particular, the role that comedic techniques and structures can play. We hope that many of the ideas are applicable in the much broader context of mathematics communication, including outreach workshops, tutorials, and talks at seminars or reading groups.

olymers with associative motifs—chains bearing specific “sticker” groups capable of forming reversible, finite-lifetime bonds—exhibit phase behaviour governed by a competition between connectivity, transient crosslinking, and polymer fluctuations. The classical reversible-gelation theory of Semenov and Rubinstein provides the fundamental thermodynamic description of these systems, showing how sticker–sticker associations generate a sol–gel transitions. Subsequent sequence-dependent sticker–spacer models, including those developed by Pappu and collaborators, demonstrated that the spatial arrangement and density of associative motifs modulate condensation behaviour in biomolecular polymers. At a more universal level, scale-free models of droplet formation by Maritan and coworkers revealed nucleation and coarsening dynamics that emerge independently of molecular details.
In this seminar, I will present a field-theoretic framework that aims to unify these descriptions within a single microscopic model. Starting from a polymer Hamiltonian with explicit sticker variables—treated consistently in both quenched and annealed ensembles—we derive a saddle-point theory that yields the bonding free energy directly, without relying on combinatorial assumptions. A preliminary analysis of the resulting effective action further shows how condensation behaviour naturally emerges from the interplay between reversible crosslinking and density fluctuations.

In this talk we introduce the notion of generalized/categorical families of QFTs. These families of theories have a structure which is analogous to the categorical symmetries in Lagrangian QFTs and may be thought of as the family structure that arises when a categorical symmetry is explicitly broken by deformations of the action. After describing this structure, we will explain the anomalies of these families and their implications. We will then use this structure to study some higher dimensional deformations of 4d N=1 SUSY gauge theories.

TBD

The Metropolis-Adjusted Langevin Algorithm (MALA) is a widely used Markov Chain Monte Carlo (MCMC) method for sampling from high-dimensional distributions. However, MALA relies on differentiability assumptions that restrict its applicability. In this paper, we introduce the Metropolis-Adjusted Subdifferential Langevin Algorithm (MASLA), a generalization of MALA that extends its applicability to distributions whose log-densities are locally Lipschitz, generally non-differentiable, and non-convex. We evaluate the performance of MASLA by comparing it with other sampling algorithms in settings where they are applicable. Our results demonstrate the effectiveness of MASLA in handling a broader class of distributions while maintaining computational efficiency.

TBD