Weekly Bulletin

A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, an operad, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. Kaledin classes were introduced as an obstruction theory fully characterizing the formality of associative algebras over a characteristic zero field. In this talk, I will present a generalization of Kaledin classes to any coefficients ring, to other algebraic structures (encoded by operads, possibly colored, or by properads), and to address a more general problem: the existence of homotopy equivalences between algebraic structures. I will prove new formality criteria based on this obstruction theory, presenting applications in several domains such as algebraic geometry, representation theory and mathematical physics.

A contact homeomorphism is defined as a C⁰-limit of a sequence of contact diffeomorphisms. In this talk, we will discuss C⁰-flexibility and C⁰-rigidity properties of various classes of submanifolds in the contact setting. Our primary focus will be on Legendrian submanifolds, which exhibit C⁰-rigidity in the closed case and C⁰-flexibility in the open case. More precisely, we will focus on the following question: if a contact homeomorphism maps a Legendrian submanifold to a smooth submanifold, must the image necessarily be Legendrian?

Fixing a closed surface and a number \(k\), we ask: How many crossings do any \(k\) non-homotopic simple closed curves on the surface necessarily create? In this talk, we focus on small minimising systems on a surface of genus 2 and relate them to optimal curve systems appearing in other contexts. If time allows, we discuss more general results concerning higher genus surfaces and asymptotic behaviour.

For a smooth expanding map of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers along all periodic orbits. This set is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In the talk, I will present a length spectral rigidity result for expanding circle maps. Namely, I will show that a smooth expanding circle map, under certain assumptions on the sparsity of its length spectrum, cannot be perturbed with an arbitrarily small smooth perturbation (depending on the map) so that the length spectrum stays the same. The proof uses the Whitney extension theorem, a quantitative Livcis-type theorem, and a novel iterative scheme. This is joint work with Vadim Kaloshin.

After a general introduction to applications of metric graph theory in geometric group theory, focused on quasi-median graphs, I will explain how one can deduce from such a perspective new quasi-isometric invariants for right-angled Artin groups. This is joint work with Carolyn Abbott and Eduardo Martinez-Pedrosa.

The Plateau problem asks: which are the surfaces of least m-dimensional area spanning a given (m-1)-dimensional boundary? To guarantee existence of minimizers and desirable compactness properties for sequences of surfaces, one must consider a weak notion of surface, thus allowing for area-minimizing "surfaces” to have singularities. Two particularly natural frameworks for this problem are integral currents and mod(q) currents, which have both been studied in great depth since the 1950s, pioneered by works of De Giorgi, Federer & Fleming, Almgren, Taylor and White, and built upon by many others. The former framework allows for surfaces to have integer multiplicities, while the latter allows for multiplicities modulo a fixed integer q. I will explain the history of the problem and some recent breakthroughs in the regularity theory for each framework, as well as regularity (and failure thereof) for almost area-minimizers.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar/Anna-Skorobogatova.htmlcall_made

I will describe recent work joint with Olga Balkanova and Dmitry Frolenkov on a restricted divisor function and its associated divisor problem. This problem shares properties with the usual Dirichlet divisor problem and the Hardy-Littlewood problem to count lattice points in a right triangle. The latter depends deeply on the arithmentic nature of the slope of the triangle the restricted divisor problem has a similar property. For the H-L problem, Hecke showed how to go much further when the slope comes from a real quadratic field. We apply Hecke’s idea to our problem, which introduces a number of interesting new difficulties,

In the last 10 years colossal cloud infrastructure investments behind the rise of near-ubiquitous global mobile technologies have trickled down to scientific research through innovative infrastructure including cloud compute and storage, I/O tools, data analysis and modeling frameworks, which in turn have generated broad and expanding communities of users and supporters. Arguably, the recent success of Large Language Models were catalyzed by the resulting technological innovations of 1) open and accessible massive data, and 2) re-executable discovery pipelines for model estimation and prediction. These changes are deeply disruptive to the research community since they open new paths to knowledge creation that were previously inaccessible and largely culturally unknown. The scientific community is faced with the challenge of responding to changes in research modalities due to these technological innovations. Research is now conducted as an “Olympics” of benchmarked competitions between Machine Learning models leveraged by the opaque results of Large Language Models, access to massive data, and redeployment of complex scientific discovery workflows. In this seminar I provide a roadmap of challenges and responses by various stakeholders in the research community to ensure that scientific results remain reliable and reproducible, and secure within a position of trust in the broader society.

I will present a formula of the S_n-equivariant Euler characteristics of genus one stable maps to the projective space. It enriches the ordinary Euler characteristics, which are previously unkown, and continues the results of Getzler - Pandharipande on genus zero stable maps. The approach connects torus localization, composition structure on the Grothendieck ring of varieties, and graph enumeration techniques via wreath symmetric functions. Joint work with Siddarth Kannan.