Veranstaltungen

A very friendly introduction to spectral graph theory and some algorithms this theory gives rise to which are conjectured to be optimal approximation algorithms to NP-hard problems.

More information: https://zucmap.ethz.ch/call_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.

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.

We focus on the Benjamin-Ono equation on the line with a small dispersion parameter. The goal of this talk is to precisely describe the solution at all times when the dispersion parameter is small enough. This solution may exhibit locally rapid oscillations, which are a manifestation of a dispersive shock. The description involves the multivalued solution of the underlying Burgers equation, obtained by using the method of characteristics. This work is in collaboration with Elliot Blackstone, Patrick Gérard, and Peter Miller.

Since distribution shifts are common in real-world applications, there is a pressing need for developing prediction models that are robust against such shifts. Existing frameworks, such as empirical risk minimization or distributionally robust optimization, either lack generalizability for unseen distributions or rely on postulated distance measures. Alternatively, causality offers a data-driven and structural perspective to robust predictions. However, the assumptions necessary for causal inference can be overly stringent, and the robustness offered by such causal models often lacks flexibility. In this paper, we focus on causality-oriented robustness and propose Distributional Robustness via Invariant Gradients (DRIG), a method that exploits general additive interventions in training data for robust predictions against unseen interventions, and naturally interpolates between in-distribution prediction and causality. In a linear setting, we prove that DRIG yields predictions that are robust among a data-dependent class of distribution shifts. We extend our approach to the semi-supervised domain adaptation setting to further improve prediction performance. Finally, we discuss an idea to go beyond specific characteristics but exploit shifts in overall aspects of the distribution, thus leading to potentially more robust predictions. The proposed methods are validated on a single-cell data application.

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.