# Fakultätskolloquium im Wintersemester 2017/18

**Termin:** Das Kolloquium findet in loser Folge mittwochs statt, mit jeweils zwei Vorträgen von 14:30 – 15:30 Uhr und von 16:00 – 17:00 Uhr im Hörsaal 3 (MI 00.06.011).

In der Pause ist für Getränke und Brez'n in der Magistrale gesorgt.

**Ansprechpartner:** Felix Krahmer, Robert König.

Vorträge im vergangenen Semester:

Sommersemester 17.

## Vorträge im Wintersemester 2017/18

#### Ralf Hiptmair — Shape Differentiation: New Perspectives

The presentation examines the "derivative'' of solutions of second-order boundary value problems and of output functionals based on them with respect to the shape of the domain. A rigorous approach relies on encoding shape variation by means of deformation vector fields, which will supply the direction for taking shape derivatives. These derivatives and methods to compute them numerically are key tools for studying shape sensitivity, performing gradient based shape optimization, and small-variation shape uncertainty quantification.

A unifying view of second-order elliptic boundary value problems recasts them in the language of differential forms (exterior calculus). Fittingly, the shape deformation through vector fields matches the concept of Lie derivative in exterior calculus. This paves the way for a unified treatment of shape differentiation in the framework of exterior calculus. The obtained formulas can be employed in the so-called adjoint approach to derive shape gradients of concrete output functionals. The resulting expressions allow different reformulations. Though equivalent for exact solutions of the involved boundary value problems, they deliver vastly different accuracies in the context of finite element approximation, as confirmed by a rigorous asymptotic a priori convergence analysis for a number of important cases.

This is joint work with J.-Z. Li (SUSTC, Shenzen), A. Paganini (Mathematical Institute, University of Oxford) and S. Sargheini (Seminar for Applied Mathematics, ETH Zürich).

#### Gilad Lerman — Non-convex Robust Subspace Recovery

The setting of robust subspace recovery assumes datasets composed of inliers drawn around a low-dimensional subspace and of outliers that do not lie nearby this subspace. The goal is to robustly determine the underlying subspace of such datasets, while having low computational complexity. We present a mathematical analysis of a non-convex energy landscape for robust subspace recovery. We prove that an underlying subspace is the only stationary point and minimizer in a large neighborhood if a generic condition holds for a dataset. We further show that if the generic condition is satisfied, a geodesic gradient descent method over the Grassmannian manifold can exactly recover the underlying subspace with proper initialization. The condition is shown to hold with high probability for a certain model of data. We also discuss guarantees of another nonconvex strategy and open problems in the area.

#### Fabrizio Catanese — The (never ending?) story of Algebraic geometry: history of the origins and modern classification theory

The history of algebraic geometry starts with Pythagoras' theorem, and even more with the theorem of Pappus (300 A.D.) , which I shall illustrate as a motivating case for the introduction of projective geometry. There was a long lapse of time between Pappus' theorem and Pascal's theorem (1640 ca). Pascal did not prove his theorem, and similarly occurred for the theorem of Bezout in 1750. The appropriate tools were developed only in the 19th century, through use of the projective coordinates (extending the barycentric coordinates of Moebius) and through the theory of resultants. The Bezout theorem allows powerful generalizations of the theorem of Pappus-Pascal, which I will illustrate. But the greatest breakthrough came in the theory of algebraic surfaces: for instance the discovery of the Lines on a cubic surface, due to Cayley, and conceptually understood through the plane model due to Cremona. The latter brought to the development of birational geometry, where essential was the contribution of the Italian school, not only on the projective side (Del Pezzo, Fano), but especially with the birational classification of algebraic surfaces, due to Castelnuovo and Enriques (1895-1914).

I will especially explain the connections to topology, the genus of a curve, to finally illustrate the \(P_{12}\) theorem which gives the "rough" classification of algebraic surfaces through the 12th plurigenus \(P_{12}\) and the linear genus. Time permitting, I shall talk about later and recent developments (transcendental methods, the fine classification of surfaces, moduli spaces) and open questions.

#### Daniel Peterseim — Numerical homogenization beyond periodicity and scale separation

Physical processes in micro-heterogeneous media can often be modelled by partial differential equations (PDEs) with oscillatory coefficients that represent complex material microstructures. Given the multiscale nature of these processes, the construction of computable macroscopic (homogenized or effective) models is crucial for the efficient and reliable numerical simulation in this context. Numerical homogenization is a multiscale method for the derivation and simulation of such macroscopic models. At the example of a prototypical linear elliptic model diffusion problem, this lecture illustrates such an approach. It is based on the computation of operator-dependent subspaces with a quasi-local basis and approximation properties independent of oscillations and roughness of the diffusion coefficient. While constructive approaches in the mathematical theory of homogenization require strong structural assumptions such as (local) periodicity and scale separation of the coefficients, the new subspace decomposition approach to homogenization is applicable, reliable and accurate beyond these restrictions. This added value of the approach is demonstrated by theoretical and experimental results.