Weekly Bulletin

In Neolithic archaeological sites, there are often large quality of pottery fragments. A foundational work is to establish pottery chronology which provides a relative time-line for different sites in a region. Combining with Carbon-14 dating, one can obtain an absolute time-line. The above traditional method is extremely time consuming and uses less than 10% of the materials. One can easily imagine that the remaining 90% material may contain important information which even changes the conclusion of archaeological study. In the talk, we will present an AI-based technique to date individual pieces of pottery fragments, developed by a team of archaeologist in Sichuan, China in collaborations with mathematician/statistician. The highlight of the project is to build a large dataset consisting of over 200,000 images from more than 20 sites.

More information: https://eth-its.ethz.ch/activities/its-fellows--seminar/yongbin-ruan.htmlcall_made

The main question in quantum chaos is to relate the chaotic properties of a dynamical system (like a billiard in a bounded domain, or the geodesic flow on a closed manifold) to the spectral properties of the corresponding Schrödinger operator in quantum mechanics (the laplacian, in the examples above). This is usually asked for a given dynamical system, but one may try to make the problem more tractable by studying a "random" billiard, or the geodesic flow on a "random manifold". Several years ago, I became interested in the ergodic and spectral properties of random hyperbolic surfaces, in the asymptotic regime where the area of the surface goes to infinity. I will survey the existing techniques and some of the results.

This is joint work with Jon DeWitt. We study skew products over area-preserving Anosov diffeomorphisms on T^2×G, where G is a compact Lie group, given by (x,g)?(f(x),h(x)·g). We establish smooth rigidity; that is, if two such skew products are C^0 conjugate, then they are smoothly conjugate, unless h:T^2->G is cohomologous to a constant and the skew product is, in fact, a product with a translation on G. Interestingly, on twisted principal G-bundles, our approach gives exception-free rigidity.

It is well known that finite element approximations of the Helmholtz equation suffer from the pollution effect for large wavenumbers k>0. This degeneracy can be avoided by the application of high order FEMs, with polynomial degree p chosen proportional to log k. The key ingredient of the respective analysis [1] is a so-called regularity splitting, which decomposes the solution of the Helmholtz equation with a L^2 right hand-side into an analytical part and k-well behaved H^2 part. The generalization of this technique for nonconstant coefficients and other boundary conditions is technical and nontrivial, but has received much attention lately. In this talk I show how the classical Schatz technique can be adapted to circumvent the necessity of any regularity splitting, which significantly simplifies the analysis. In the second part of the talk I discuss the application of this approach to heterogeneous media and Maxwell-impedance problems. [1] M. Melenk and S. Sauter, Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79(272):1871–1914, 2010.

Tensors are fundamental objects in mathematics, physics, statistics, and computer science, and they play an important role in a wide range of applied sciences and engineering disciplines. In this talk, we will focus on concentration inequalities for simple random tensors. We establish sharp dimension-free concentration inequalities and expectation bounds for the deviation of the sum of simple random tensors from its expectation. As part of our analysis, we use generic chaining techniques to obtain a sharp high-probability upper bound on the suprema of multi-product empirical processes. In so doing, we generalize classical results for quadratic and product empirical processes to higher-order settings.