Weekly Bulletin

Bhatt-Scholze's prismatic cohomology is a p-adic cohomology theory for p-adic formal schemes, recovering the other known cohomology theories at p. I will explain why considerations from p-adic non-abelian Hodge theory as well as from the p-adic Langlands program make it desirable to extend the picture from formal schemes to rigid-analytic varieties and what we know about how to do it at the moment. Based on joint work with Johannes Anschütz, Juan Esteban Rodriguez Camargo and Peter Scholze.

One of the most surprising aspects of quantum theory is that it tells us that we live in a nonlocal universe in which random correlations seem to appear instantaneously between arbitrarily distant locations. This idea was completely abhorrent to Einstein, who dismissed it as "spooky action at a distance", yet the 2022 Nobel Prize in Physics was awarded for experimental demonstrations half a century ago of this phenomenon. It is even said that so-​called loophole-​free experiments confirmed nonlocality beyond any reasonable doubt. But have they really? I shall argue that no experiment whose purpose is to confirm the predictions of quantum theory can possibly be used as an argument in favour of nonlocality because any theory of physics that does not allow instantaneous signalling to occur and has reversible dynamics (such as unitary quantum theory) can be explained in a purely local and realistic universe. What if Einstein was right after all?... Once again!

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar/Gilles-Brassard.htmlcall_made

L-functions are one of the central objects of study in number theory. There are many beautiful theorems and many more open conjectures linking their values to arithmetic problems. The most famous example is the conjecture of Birch and Swinnerton-Dyer, which is one of the Clay Millenium Prize Problems. I will discuss some recent progress on these conjectures using tools called `Euler systems’. I will also revisit the question of using Euler systems in the proof of Fermat’s last theorem.

Let X be the homogeneous space Gamma \ G, where G is the semidirect product of SL(2,R) and a direct sum of k copies of R^2, and where Gamma is the subgroup of integer elements in G. I will present a result giving effective equidistribution of 1-dimensional unipotent orbits in the space X. The proof makes use of the delta method in the form developed by Heath-Brown. Joint work with Anders Södergren and Pankaj Vishe.

The Hilbert scheme of points in affine n-dimensional space parametrizes finite subschemes of a given length. It is smooth and irreducible if n is at most 2, singular and reducible if n is at least 3. Understanding its irreducible components, their singularities and birational geometry, has long been an inaccessible problem. In this talk, I will describe substantial progress on this problem achieved in recent years. In particular, I will focus on the discovery of elementary components, Murphy’s law up to retraction, and the problem of rationality of components. This is based on works of Joachim Jelisiejew and on a joint work of Gavril Farkas, Rahul Pandharipande, and myself.

In the last few decades, Lee-metric codes gained a lot of attention especially with their promising application to code-based cryptography as well as their connection to lattices. Even though, begin a rather old metric considered in coding theory, there are still many open questions regarding Lee metric codes regarding both the algebraic structure of codes in the Lee metric as well as efficient decoding algorithms. In this talk we present some selected topics on Lee-metric codes. We start by introducing generalized Lee distances in order to derive improved upper bounds on the minimum Lee distance. We discuss the derivation of the bound as well as the density of codes achieving it. In a next step we turn the attention to channel coding where we introduce two channel models in the Lee metric. At this point we specifically focus on a static channel model, introducing an error of fixed weight. We derive its marginal distribution using typical sequences. Additionally, this channel can be viewed as a design parameter to code-based cryptography and hence, the knowledge of the marginal distribution shows some implication in the field of cryptography too. Lastly, we introduce classical random low-density parity-check (LDPC) codes over a finite integer residue ring endowed with the Lee metric, and we present their expected weight enumerator.

We will start with friendly introduction to contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. We will look into recent techniques developed to study these by understanding special knots(legendrian knots) in overtwisted 3 manifolds. We will look at what more classification results can we hope to get using the current techniques and what is far-fetched.

Scientific computing is routinely assisting in the design of systems or components, which have potentially very complex shapes. In these situations, it is often underestimated that the mesh generation process takes the overwhelming portion of the overall analysis and design cycle. If high order discretizations are sought, the situation is even more critical. Methods that could ease these limitations are of great importance, since they could more effectively interface with meta-algorithms from Optimization, Uncertainty Quantification, Reduced Order Modeling, Machine Learning, and Artificial Neural Networks, in large-scale applications. Recently, immersed/embedded/unfitted boundary finite element methods (cutFEM, Finite Cell Method, Immerso-Geometric Analysis, etc.) have been proposed for this purpose, since they obviate the burden of body-fitted meshing. Unfortunately, most unfitted finite element methods are also difficult to implement due to: (a) the need to perform complex cell cutting operations at boundaries, (b) the necessity of specialized quadrature formulas on cut elements, and (c) the consequences that these operations may have on the overall conditioning/stability of the ensuing algebraic problems. This talk introduces a simple, stable, and accurate unfitted boundary method, named “Shifted Boundary Method” (SBM), which eliminates the need to perform cell cutting operations. Boundary conditions are imposed on the boundary of a “surrogate” discrete computational domain, specifically constructed to avoid cut elements. Appropriate field extension operators are then constructed by way of Taylor expansions (or similar operators), with the purpose of preserving accuracy when imposing boundary conditions. An extension of the SBM to higher order discretizations will also be presented, together with a summary of the numerical analysis results. The SBM belongs to the broader class of Approximate Boundary Methods, a less explored or somewhat forgotten class of algorithms, which however might have an important role in the future of scientific computing. The performance of the SBM is tested on large-scale problems selected from linear and nonlinear elasticity, fluid mechanics, shallow water flows, thermos-mechanics, porous media flow, and fracture mechanics.

Let $K$ be a number field. The Grunwald problem for a finite group (scheme) G/K asks what is the closure of the image of $H^1(K,G) \to \prod_{v \in M_K} H^1(K_v,G)$. For a general $G$, there is a Brauer—Manin obstruction to the problem, and this is conjectured to be the only one. In 2017, Harpaz and Wittenberg introduced a technique that managed to give a positive answer (BMO is the only one) for supersolvable groups. I will present a new fibration theorem over quasi-trivial tori that, combined with the approach of Harpaz and Wittenberg, gives a positive answer for all solvable groups. Partial results were also obtained independently by Harpaz and Wittenberg.

As we are all too aware, organizations accumulate vast amounts of data from a variety of sources nearly continuously. Big data and data science advocates promise the moon and the stars as you harvest the potential of all these data. And now, AI threatens our jobs and perhaps our very existence. There is certainly a lot of hype. There’s no doubt that some savvy organizations are fueling their strategic decision making with insights from big data, but what are the challenges? Much can wrong in the data science process, even for trained professionals. In this talk I'll discuss a wide variety of case studies from a range of industries to illustrate the potential dangers and mistakes that can frustrate problem solving and discovery -- and that can unnecessarily waste resources. My goal is that by seeing some of the mistakes I (and others) have made, you will learn how to better take advantage of data insights without committing the "Seven Deadly Sins."

In the 80's Mumford and his collaborators developed the theory of compactifications of bounded symmetric domains, which included moduli spaces of K3 and abelian varieties. We will go through that theory and explain how some functorial properties of these compactifications can be applied to some modern problems, like tautological projections of Shimura varieties, or obtaining relations in the Chow ring of certain period domains for weight 2 VHS.