Weekly Bulletin

I will show that every cycle on the moduli space of abelian varieties decomposes canonically as a sum of a tautological and non-tautological class. The key input is the vanishing of the top Chern class of the Hodge bundle when restricted to the boundary of any toroidal compactification. This is joint work with Molcho, Oprea, and Pandharipande.

''Knot theory is the study of knots, that is embedded circles into R^3 (or S^3) considered up to isotopy. A generic intersection of a knot with a 3-ball is called a tangle. In this talk, I will focus on tangles of a particular kind, called rational tangles. These tangles are in 1:1-correspondence with rational numbers and share many of their properties. After introducing the relevant ingredients and results, I will present the proper rational unknotting number, an invariant of knots defined in terms of rational tangles.

How can we explain the patterns of genetic variation in the world around us? This question lies at the heart of theoretical population genetics. The genetic composition of a population can be influenced by its spatial structure, natural selection, mutation, random mating, and other genetic, ecological and evolutionary mechanisms. In trying to understand the ways in which these factors interact with one another, and their relative importance, we are led to mathematical models. In this lecture we'll discuss some of the ways in which we can distill our understanding of biological processes into workable mathematical models of surprisingly broad applicability.

More information: https://math.ethz.ch/news-and-events/events/lecture-series/alice-roth-lectures.htmlcall_made

In hyperbolic dynamical systems, one can often prove the spatial central limit theorem (CLT), where the starting point is randomized with respect to the SRB measures. In zero-entropy systems such as irrational rotations, the spatial CLT often fails due to lack of mixing properties. However, using coding and Markov chains, Bromberg and Ulcigrai showed that a temporal CLT holds for bounded type irrational rotations with step functions whose jump point lies in a full Hausdorff dimension set. Here "temporal" means that we randomise time while fixing the starting point. In an ongoing joint work with Bromberg and Ulcigrai, we extend this result from full Hausdorff dimension to full Lebesgue measure.

In the moduli space of principally polarized abelian varieties, it is natural to consider the loci determined by those varieties that are not simple. I will give a description of the irreducible components of these loci, and explain how to obtain their projection to the tautological ring of A_g (as recently developed by Canning, Molcho, Oprea and Pandharipande), using the theory of toroidal compactifications of Siegel domains. Then I will discuss some connections to the enumerative geometry of curves.

In this talk, we discuss coding schemes for data storage systems, including flash memories, racetrack memories, and DNA-based systems. In particular, we present several constrained codes avoiding specific patterns and focus on codes correcting synchronization errors, such as deletions and sticky-insertions. Firstly, we show that constrained codes can be useful for avoiding some potential errors. Especially, we present ICI-free codes for flash memories and locally constrained codes for DNA-based storages. Then, we present several new families codes correcting synchronization errors, including q-ary codes, permutation codes, multi-dimensional codes, and symbol-pair codes and their applications in flash memories, and racetrack memories. Finally, we study several constrained codes avoiding specific set of patterns that can be used to correct synchronization errors in racetrack memories and DNA-based storage systems. A few open questions will be discussed in the end of the talk. This talk is based on joint works with Yeow Meng Chee, Tuvi Etzion, Kees A. Schouhamer Immink, Manabu Hagiwara, Hanmao Kiah, Alexander Vardy, and Eitan Yaakobi.

Norbert Wiener defined “Cybernetics” as “the science of control and communication in the animal and the machine”, anticipating some of the goals and the future development of Artificial Intelligence. The traditional Applied Mathematics program, combining modelling, analysis, numerical approximation, and scientific computing, when facing practical applications, must often be complemented by additional efforts to address control issues, to better understand how dynamics changes when varying free parameters. This frequently leads to new complex and fascinating analytical and computational challenges that require significant unexpected further developments. We will lecture on some recent success stories that arise when facing, for instance, source identification problems, and the regulation of collective dynamics. We shall also discuss the issue of the optimal placement of sensors and actuators, which plays a key role when designing efficient control mechanisms. Control techniques also play an unexpected relevant role in other contexts such as the large time asymptotics for partially dissipative systems in fluid mechanics. We will describe the links between these problems and their analytical and numerical treatment, as one further manifestation of the unity and interconnections of all mathematical disciplines. We shall conclude pointing towards some perspective for future research in connection with Machine Learning. We will begin by briefly discussing the origins of mathematical control theory and machine learning, emphasizing their intimate analogies and links. We will then recall some basic results on the control of linear finite-dimensional systems and the Universal Approximation Theorem. Later we will address the problem of supervised learning, formulated as a simultaneous or ensemble control problem for the so-called neural differential equations, driven by Lipschitz nonlinearities, the activation functions in the neural network ansatz for learning. We will present an iterative and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies. The very role that the nonlinear nature of the activation functions plays will be emphasized. Unnecessary technical difficulties will be avoided. Several open problems and perspectives for future research will be formulated.

Gaussian multiplicative chaos (GMC) on the circle is a canonical (random) multifractal measure on the circle which appears in a wide variety of contexts and most recently in relation to Liouville conformal field theory. In this talk, I will present the first results concerning the decay and renormalization of the Fourier coefficients of GMC. In particular, one can show that GMC is a so-called Rajchman measure which means that its Fourier coefficients go to zero when the frequency goes to infinity. Numerous questions remain open. Based on a joint work with C. Garban.

We study partially linear models in settings where observations are arranged in independent groups but may exhibit within-group dependence. Existing approaches estimate linear model parameters through weighted least squares, with optimal weights (given by the inverse covariance of the response, conditional on the covariates) typically estimated by maximising a (restricted) likelihood from random effects modelling or by using generalised estimating equations. We introduce a new ‘sandwich loss’ whose population minimiser coincides with the weights of these approaches when the parametric forms for the conditional covariance are well-specified, but can yield arbitrarily large improvements in linear parameter estimation accuracy when they are not. Under relatively mild conditions, our weighted least squares (within a double machine learning framework) estimated coefficients are asymptotically Gaussian and enjoy minimal variance among estimators with weights restricted to a given class of functions, when user-chosen regression methods are used to estimate nuisance functions. We further expand the class of functional forms for the weights that may be fitted beyond parametric models by leveraging the flexibility of modern machine learning methods within a new gradient boosting scheme for minimising the sandwich loss. We demonstrate the effectiveness of both the sandwich loss and what we call ‘sandwich boosting’ in a variety of settings with simulated and real-world data.

Ever since Osterwalder and Schrader's pioneering work, it has been established that Fermionic systems can be described using Grassmann variables and Grassmann measures. In my talk, I will delve into this perspective, elucidating fundamental stochastic analytical tools, such as Grassmann Brownian motion and non-commutative $L^{p}$ spaces, which allow for a more thorough stochastic analytical characterisation of Grassmann measures. To exemplify the utility of this approach, I will discuss the advancements achieved in describing some prototypical subcritical interacting Grassmann measures. Based on joint work with F. De Vecchi, M. Gordina and M. Gubinelli.

This talk will be a unified overview of some recent contributions in financial mathematics. The financial topics are option pricing with market impact and model calibration. The mathematical tools are fully non-linear partial differential equations and semi-martingale optimal transport. Some new and fun results will be a Black-Scholes-Legendre formula for option pricing with market impact, a Measure Preserving Martingale Sinkhorn algorithm for martingale optimal transport, and a lognormal version of the Bass Martingale.

We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which seeks to approximate a distribution $\pi$ over $\mathbb{R}^d$ by a product measure $\pi^\star$. When $\pi$ is strongly log-concave and log-smooth, we provide (1) approximation rates certifying that $\pi^\star$ is close to the minimizer $\pi^\star_\diamond$ of the KL divergence over a \emph{polyhedral} set $\mathcal{P}_\diamond$, and (2) an algorithm for minimizing $\text{KL}(\cdot\|\pi)$ over $\mathcal{P}_\diamond$ with accelerated complexity $O(\sqrt \kappa \log(\kappa d/\varepsilon^2))$, where $\kappa$ is the condition number of $\pi$. Joint work with Yiheng Jiang and Sinho Chewi.

I will discuss some new results on averages of multiplicative functions over integer sequences from arXiv:2402.08710. We will then give applications to Cohen-Lenstra and Manin's conjecture. Joint work with Chan, Koymans and Pagano.