Weekly Bulletin

I will report on our (ongoing; partially completed) classification, joint with Sue Tolman, of Hamiltonian torus actions of complexity one.

<p><span data-olk-copy-source="MessageBody">This is a report of work in progress. Annala and Elmanto </span><span data-olk-copy-source="MessageBody">constructed motivic Steenrod operations in positive characteristic using a motivic refinement of Niziol’s theorem. In this talk, I will show that Ayoub's unipotent nearby cycles functor sends MZ to MZ, which yields another construction of motivic Steenrod operations in positive characteristic. The proof requires the motivic six-functor formalism for log schemes.</span></p>

<div dir="auto" style="font-family: Aptos, Aptos_MSFontService, -apple-system, Roboto, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(33, 33, 33) !important;" data-ogsc="rgb(33, 33, 33)" data-olk-copy-source="MessageBody">In k-clustering problems, a collection of points in a metric space is given, and the goal is to partition them into k clusters, so as to minimize some objective. The most well-known objectives in that family are k-Median and k-Means, where one must minimize the sum of the distances (resp. squared distances) between each point and the center of its cluster.</div> <div dir="auto" style="font-family: Aptos, Aptos_MSFontService, -apple-system, Roboto, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(33, 33, 33) !important;" data-ogsc="rgb(33, 33, 33)">In this seminar, we will go over the techniques used to design approximations for these problems. The first difficulty is the hard constraint of at most k clusters: we will cover Lagrangian Multiplier Preserving (LMP) approximations. Then, we will present how to exploit the linear relaxation via Dual Fitting. Finally, we will go into the specificities of these two clustering problems, discussing recent advancements.</div>

The human genome comprises over 20,000 genes that bring about life and sustain it over decades. However, genes are silent and must be given voice through a fundamental biological process called gene expression. The first step in gene expression is transcription, in which genetic information encoded in DNA is copied into RNA. The lecture explores the molecular mechanisms of transcription and its regulation to explain how genes are switched on during embryonic development, for cellular function, and in human health and disease.

More information: https://pauli-lectures.ethz.ch/lectures26.htmlcall_made

Kepler discovered that the planets move in elliptical orbits; Newton showed, by means of a tour de force of Euclidean geometry, that this corresponds to an inverse-square law of attraction towards the sun. Since the time of Newton, mathematicians have returned again and again to this ancient problem, solving and re-solving it with ever-changing tools and concepts. I will talk about both the mathematics, and what this process of rewriting suggests about the past and future of our discipline.

Moduli spaces often admit several natural compactifications. The comparison of different compactifications is referred to as wall-crossing. Some compactifications are simpler than others, making wall-crossing one of the most powerful established techniques, for example, in the study of moduli spaces of sheaves. I will discuss wall-crossing phenomena for moduli spaces of maps (Gromov–Witten theory) and other related spaces. The first glimpses of wall-crossing in Gromov–Witten theory go back to its beginnings, namely to Givental’s proof of mirror symmetry for the quintic in 1990s. Later, independently of Givental’s work, Manin and Bayer were the first to explore the idea of wall-crossing in the setting of maps, extending Hassett’s ideas on weighted marked curves. Only with the work of Ciocan-Fontanine, Kim, and Maulik from 2010s, which in turn builds on the work of many others, did it become possible to exploit different compactifications of moduli spaces of maps with considerable flexibility for GIT quotients, thereby revealing Givental’s proof as a wall-crossing calculation. Zhou then made the next major step by proving the Ciocan-Fontanine–Kim wall-crossing formulas for GIT quotients in full generality. The goal is to explain how the ideas of Ciocan-Fontanine, Kim, and Zhou can be applied in three different settings: configuration spaces of points, maps to GIT quotients, and Hurwitz theory. Based on these examples, I will try to formulate some general principles underlying such wall-crossing phenomena. The final goal is to combine all three settings to provide a wall-crossing between spaces of maps and Hilbert schemes associated to threefolds of the form Surface × Curve.

Mixed models for repeated measures (MMRM) analysis has been extensively used to analyze longitudinal datasets. SAS has been the gold standard for this analysis in the past, and previously R packages have fallen short for one of the following reasons: model convergence issues, unavailability of covariance structures or adjusted degrees of freedom, or numerical results being far from SAS. To fill in this important gap in the open-source statistical software landscape, a cross-company workstream of openstatsware.org has developed the {mmrm} R package. A critical advantage of {mmrm} over existing implementations is that it is faster and converges more reliably. It also provides a comprehensive set of features: users can specify a variety of covariance matrices, weight observations, fit models with restricted or standard maximum likelihood inference, perform hypothesis testing with Satterthwaite or Kenward-Roger adjusted degrees of freedom, extract the least square means estimates using the emmeans package, and use tidymodels for easy model fitting. We introduce the modeling framework, the implementation strategy and discuss open source collaboration as a critical ingredient to success.

Our laboratory uses a combination of structural biology, functional genomics and computational approaches to elucidate the molecular mechanisms and cellular regulation of transcription, the first step in gene expression that governs cell differentiation, embryo development and cancer formation. In my lecture I will provide an overview of our current understanding of the molecular mechanisms underlying gene transcription in eukaryotic cells. I will also provide data that show that transcription is regulated at three key steps, which were revealed by structure-based multiomics analysis.

More information: https://pauli-lectures.ethz.ch/lectures26.htmlcall_made

In Fourier's "Théorie analytique de la chaleur" we find a formula expressing the evolution of temperature in a heated ring. This formula – a "theta function" – soon became a testament to the interconnectedness of mathematics; its symmetries addressed questions raised by Euler, it led Jacobi to fundamental discoveries in number theory, and it was the basis of equally fundamental work in algebraic geometry by Riemann. I will describe some of my favorite parts of this classical theory. The classical theory depends on a positive definite quadratic form. A surprising 20th century discovery (Siegel, Kudla-Millson) is that the theory can be reasonably "continued" to where the form is not positive definite, despite the fact that all the formulas diverge. I will explain a new point of view on this phenomenon, developed with Kenz Kallal and taking inspiration from the 19th century, leading to a quite satisfactory picture.

Despite the availability of vaccines, there remains a need for safe and effective antiviral drugs against coronaviruses to treat vulnerable patients, control transmission, and prepare for future outbreaks. In the spring of 2020, our lab was part of a global race to the 3D structure of the coronavirus SARS-Cov2 RNA-dependent RNA polymerase and provided mechanistic insights into viral RNA replication. We later also used structure-function studies to reveal the mechanisms of antiviral drugs such as remdesivir and molnupiravir, revealing key limitations in their efficacy and safety. Building on this framework, we are identifying novel polymerase inhibitors through large-scale chemical screening, in the hope to uncover leads for the development of next-generation antiviral therapies.

More information: https://pauli-lectures.ethz.ch/lectures26.htmlcall_made

Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based training in which hierarchical feature learning becomes an explicit iterative spectral procedure. In this limit, the dynamics at each layer decouple: given the current representation, the next layer selects directions with maximal accessible low-degree correlation to the label. This yields a tractable surrogate mechanism for deep learning, together with a natural kernel-space interpretation. Neural LoFi provides a mathematically explicit framework for studying multi-layer feature learning beyond the lazy regime. It predicts how representations are selected layer by layer, explains how emergence of concepts arises with given sample complexity, and gives a concrete mechanism by which depth progressively constructs new features from old ones through low-degree compositionality. We complement the theory with mechanistic experiments on fully connected and convolutional architectures, showing that Neural LoFi improves over lazy random-feature baselines, recovers meaningful structured filters, and predicts representations aligned with early gradient-descent feature discovery with real datasets. The talk is based on joint work with Matteo Vilucchio, Luca Arnaboldi, Hugo Tabanelli, and Florent Krzakala (https://arxiv.org/abs/2605.13612).

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

A fundamental question in dynamic optimization problems is to identify the topology under which the associated value functions are continuous. Such stability ensures that optimal decisions remain robust under small perturbations of models. In this talk, we discuss a continuous-time extension of the adapted weak topology, under which a broad class of such optimization problems is stable. While the adapted weak topology has proven useful in applications, it is practically difficult to verify convergence in this topology. In this talk, we aim to address the fundamental, yet challenging question: When does a sequence converge in the adapted weak topology and what are its limit points? We provide a first answer to this question by establishing an Arzela-Ascoli-type characterization of relatively compact families of continuous-time Markovian laws in the adapted weak topology. As a primary application, we establish relative compactness for families of well-posed SDEs, for example uniformly elliptic diffusions, and stability under convergence of the coefficients, allowing in particular for discontinuous drift coefficients and non-equicontinuous diffusion matrices. This talk is based on joint work with Martin Larsson and Johannes Wiesel.

Mathematics abounds with special functions: Bessel, hypergeometric, Legendre, Racah, ... satisfying a huge list of bewildering identities. Some of us fear them, some of us love them, and some (like me) sit hesitantly between the two camps. Many mathematicians have tried to bring order to this unruly topic, and a particularly popular choice for an organizing paradigm is Lie group theory. I will explain this and then give it a modern (Langlands-colored) coat of paint, talking about my own attempts, jointly with Kevin Kwan and Griffin Wang, to connect a collection of classical identities with invariant theory. Finally, I will ask: what makes a special function special?