L4

Advances in manufacturing technologies, most prominently in additive manufacturing or 3d printing, are making it possible to fabricate highly optimised products with increasing geometric and hierarchical complexity. This talk will introduce our ongoing work on design optimisation that combines CAD-compatible geometry representations, multiresolution geometry processing techniques and immersed finite elements with classical shape and topology calculus. As example applications,the shape optimisation of mechanical structures and electromechanical components, and the topology optimisation of lattice-skin structures will be discussed.

The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed.

However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.

After giving a brief general introduction to reduced order models, we discuss developments in two different directions. In the first part, we discuss recent developments of reduced methods that conserve chosen invariants for nonlinear time-dependent problems. We pay particular attention to the development of reduced models for Hamiltonian problems and propose a greedy approach to build the basis. As we shall demonstrate, attention to the construction of the basis must be paid not only to ensure accuracy but also to ensure stability of the reduced model. Time permitting, we shall also briefly discuss how to extend the approach to include more general dissipative problems through the notion of port-Hamiltonians, resulting in reduced models that remain stable even in the limit of vanishing viscosity and also touch on extensions to Euler and Navier-Stokes equations.

The second part of the talk discusses the combination of reduced order modeling for nonlinear problems with the use of neural networks to overcome known problems of on-line efficiency for general nonlinear problems. We discuss the general idea in which training of the neural network becomes part of the offline part and demonstrate its potential through a number of examples, including for the incompressible Navier-Stokes equations with geometric variations.

This work has been done with in collaboration with B.F. Afkram (EPFL, CH), N. Ripamonti EPFL, CH) and S. Ubbiali (USI, CH).

In this talk we will introduce and analyse a class of robust numerical methods for nonlocal possibly nonlinear diffusion and convection-diffusion equations. Diffusion and convection-diffusion models are popular in Physics, Chemistry, Engineering, and Economics, and in many models the diffusion is anomalous or nonlocal. This means that the underlying “particle" distributions are not Gaussian, but rather follow more general Levy distributions, distributions that need not have second moments and can satisfy (generalised) central limit theorems. We will focus on models with nonlinear possibly degenerate diffusions like fractional Porous Medium Equations, Fast Diffusion Equations, and Stefan (phase transition) Problems, with or without convection. The solutions of these problems can be very irregular and even possess shock discontinuities. The combination of nonlinear problems and irregular solutions makes these problems challenging to solve numerically.
The methods we will discuss are monotone finite difference quadrature methods that are robust in the sense that they “always” converge. By that we mean that under very weak assumptions, they converge to the correct generalised possibly discontinuous generalised solution. In some cases we can also obtain error estimates. The plan of the talk is: 1. to give a short introduction to the models, 2. explain the numerical methods, 3. give results and elements of the analysis for pure diffusion equations, and 4. give results and ideas of the analysis for convection-diffusion equations. 
 

A very common problem in Science is that we have some Data Observations and we are interested in either approximating the function underlying the data or computing some quantity of interest about this function.  This talk will discuss what are best algorithms for such tasks and how we can evaluate the performance of any such algorithm.
 

A C^k-extension of a smooth and connected Lorentzian manifold (M,g) is an isometric embedding of M into a proper subset of a connected Lorentzian manifold (N,h) of the same dimension, where the Lorentzian metric h is C^k regular. If no such extension exists, then we say that (M,g) is C^k-inextendible. The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity.

The Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a C^2 regular metric. In this talk I will describe how one
proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric.

Let $\mathfrak g$ be a semisimple Lie algebra.  A $\mathfrak g$-algebra is an associative algebra $R$ on which $\mathfrak g$ acts by derivations.  There are several significant examples.  Let $V$ a finite dimensional $\mathfrak g$ module and take  $R=\mathrm{End} V$ or $R=D(V)$ being the ring of derivations on  $V$ . Again take $R=U(\mathfrak g)$.   In all these cases  $ S=U(\mathfrak g)\otimes R$ is again a $\mathfrak g$-algebra.  Finally let $T$ denote the subalgebra of invariants of $S$.
 
For the first choice of $R$ above the representation theory of $T$ can be rather explicitly described in terms of Kazhdan-Lusztig polynomials.  In the second case the simple $T$ modules can be described in terms of the simple $D(V)$ modules.  In the third case it is shown that all simple $T$ modules are finite dimensional, despite the fact that $T$ is not a PI ring,  except for the case $\mathfrak  g =\mathfrak {sl}(2)$.

After outlining the principles of Algebraic Quantum Field Theory (AQFT) I will describe the generalization of Hochschild cohomology that is relevant to describing deformations in AQFT. An interaction is described by a cohomology class.

I will discuss a classical six-dimensional superconformal field theory containing a non-abelian tensor multiplet which we recently constructed in arXiv:1712.06623.

This theory satisfies many of the properties of the mysterious (2,0)-theory: non-abelian 2-form potentials, ADE-type gauge structure, reduction to Yang-Mills theory and reduction to M2-brane models. There are still some crucial differences to the (2,0)-theory, but our action seems to be a key stepping stone towards a potential classical formulation of the (2,0)-theory.

I will review in detail the underlying mathematics of categorified gauge algebras and categorified connections, which make our constructions possible.

Algebraic quantum field theories (AQFTs) are traditionally described as functors that assign algebras (of observables) to spacetime regions. These functors are required to satisfy a list of physically motivated axioms such as commutativity of the multiplication for spacelike separated regions. In this talk we will show that AQFTs can be described as algebras over a colored operad. This operad turns out to be interesting as it describes an interpolation between non-commutative and commutative algebraic structures. We analyze our operad from a homotopy theoretical perspective and determine a suitable resolution that describes the commutative behavior up to coherent homotopies. We present two concrete constructions of toy-models of algebras over the resolved operad in terms of (i) forming cochains on diagrams of simplicial sets (or stacks) and (ii) orbifoldization of equivariant AQFTs.