Goal-oriented adaptive FEMs with optimal computational complexity
Tuesday, 1.2.22, 14:15-15:15, https://uni-freiburg.zoom.us/j/62025301433?pwd=TURsRWs3NHN6T0lCNGZsekFKMGNtQT09
We consider a linear symmetric and elliptic PDE and a linear goal functional. We design \na goal-oriented adaptive finite element method (GOAFEM), which steers the adaptive \nmesh-refinement as well as the approximate solution of the arising linear systems by \nmeans of a contractive iterative solver like the optimally preconditioned conjugate gradient \nmethod (PCG). We prove linear convergence of the proposed adaptive algorithm with optimal \nalgebraic rates with respect to the number of degrees of freedom as well as the computational cost.
Der Satz von Lindström
Tuesday, 1.2.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Der Satz von Lindström charakterisiert die Prädikatenlogik erster Stufe\nals stärkste Logik, in welcher der Kompaktheitssatz und der Satz von\nLöwenheim-Skolem abwärts gelten. Um den Satz sauber formulieren zu\nkönnen, muss zunächst geklärt werden, was unter einer Logik zu\nverstehen ist. Hierzu wird in diesem Vortrag der von Chang und Keisler\neingeführte Begriff einer abstrakten Logik diskutiert. Anschließend\nkann der Satz von Lindström präzise formuliert und bewiesen werden. Im\nBeweis, der sich ebenfalls am Vorgehen von Chang und Keisler\norientiert, wird die Charakterisierung von elementarer Äquivalenz\nmithilfe von Back-and-Forth Systemen eine wichtige Rolle spielen.\n
Thursday, 3.2.22, 17:00-18:00, Hörsaal II, Albertstr. 23b
Higgs bundles twisted by a vector bundle
Friday, 4.2.22, 10:30-11:30, Talk on BBB
Graviton scattering and differential equations in automorphic forms
Monday, 7.2.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
Green, Russo, and Vanhove have shown that the scattering amplitude for gravitons (hypothetical particles of gravity represented by massless string states) is closely related to automorphic forms through differential equations. Green, Miller, Russo, Vanhove, Pioline, and K-L have used a variety of methods to solve eigenvalue problems for the invariant Laplacian on different moduli spaces to compute the coefficients of the scattering amplitude of four gravitons. We will examine two methods for solving the most complicated of these differential equations on \(SL_2(\bmathbb{Z})\bbackslash\bmathfrak{H}\). We will also discuss recent work with S. Miller to improve upon his original method for solving this equation.
Neural network approximations for high-dimensional PDEs
Tuesday, 8.2.22, 14:15-15:15, https://uni-freiburg.zoom.us/j/64146027041?pwd=dCtxbTBqbjI4MjlOcFV2WDJ5ODI2dz09
Most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality (CoD) in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision \(\bvarepsilon > 0\) grows exponentially in the PDE dimension and/or the reciprocal of \(\bvarepsilon\). Recently, certain deep learning based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the CoD in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension \(d \bin \bN\) and the reciprocal of the prescribed approximation accuracy \(\bvarepsilon > 0\). In this talk we show that solutions of suitable Kolmogorov PDEs can be approximated by DNNs without the CoD.
Small groups of finite Morley rank with a supertight automorphism
Tuesday, 8.2.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
The famous Cherlin-Zilber Algebraicity conjecture proposes that any infinite simple \(\baleph_1\)-categorical group is isomorphic to a simple algebraic group over an algebraically closed field.\n\n\nIn my talk, I will first explain the current state of the Cherlin-Zilber conjecture. I will then introduce a recent approach towards this conjecture, which is based on the notion of a supertight automorphism. I will discuss a result, proven jointly with P. Ugurlu, on “small” infinite simple groups of finite Morley rank with a supertight automorphism whose fixed-point subgroup is pseudofinite.
How much of a covariance is causal?
Friday, 11.2.22, 12:00-13:00, online: Zoom
Can we quantify how much of the covariance between two variables is due to the causal effects of one variable on the other? I will introduce new approaches to this problem, drawing on recent advances in the theory of causal inference. As an application, I consider the relationships between an individual’s traits and their fitness in the context of evolutionary biology. By analysing such relationships casually, we can explain why certain traits evolve over time.
Three-dimensional, homogenized PDE/ODE model for bone fracture healing
Tuesday, 15.2.22, 11:10-12:10, Raum 226, Hermann-Herder-Str. 10 & und Online auf Zoom: https://un1-freiburg.zoom.us/j/64612030148?pwd=eUF6WTk4bVVVb1g4YnhucFhOUS9jZz09
We present a three-dimensional, homogenized PDE/ODE model for bone\nfracture healing in the presence of a porous, bio-resorbable scaffold\nand an associated PDE constrained optimization problem concerning the\noptimal scaffold density distribution for an ideal healing environment.\nThe model is analyzed mathematically and a well-posedness result is\nprovided. Concerning the optimization problem, we show the existence of\nan optimal scaffold design. We touch delicate regularity results for\nelliptic and parabolic equations with mixed boundary conditions which\nare crucial for the analysis of the optimal control problem, extending\nresults from the literature. Numerical simulations for the PDE/ODE\nsystem and the PDE constrained optimization problem are presented,\nillustrating the effect of stress-shielding on optimal scaffold design.