# Fakultätskolloquium im Sommersemester 2016

**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, Daniel Matthes.

Vorträge im vergangenen Semester:

Wintersemester 2015/16.

## Vorträge im Sommersemester 2016

Datum und Zeit |
Sprecher |
Titel |

11.05.2016 um 14:30 |
Mark Iwen (Michigan State) |
Sparse Fourier Transforms: A General Framework with Extensions |

11.05.2016 um 16:00 |
Joachim Weickert (U Saarland) |
Image Compression with Differential Equations |

25.05.2016 um 14:30 |
Gary Froyland (UNSW+TUM/JvN) |
Dynamics, mixing, and coherence |

08.06.2016 um 14:30 |
Bruno Nachtergaele (UC Davis+TUM/JvN) |
Quantum Many-Body Systems: Particles and Spectra |

08.06.2016 um 16:00 |
Johanna Neslehova (McGill+TUM/JvN) |
Assessing dependence between extreme risks |

29.06.2016 um 14:30 |
Robert König (TUM/M5) |
Fault-tolerance in quantum information processing |

29.06.2016 um 16:00 |
Nicole Megow (TUM/M9) |
Resource Minimization under Uncertainty |

#### Mark Iwen — Sparse Fourier Transforms: A General Framework with Extensions

Compressive sensing in its most practical form aims to recover a function that exhibits sparsity in a given basis from as few function samples as possible. One of the fundamental results of compressive sensing tells us that $O(s \log^4 N)$ samples suffice in order to robustly and efficiently recover any function that is a linear combination of $s$ arbitrary elements from a given bounded orthonormal set of size $N > s$. Furthermore, the associated recovery algorithms (e.g., Basis Pursuit via convex optimization methods) are efficient in practice, running in just polynomial-in-$N$ time. However, when $N$ is very large (e.g., if the domain of the given function is high-dimensional), even these runtimes may become infeasible.

If the orthonormal basis above is Fourier, then the sparse recovery problem above can also be solved using Sparse Fourier Transform (SFT) techniques. Though these methods aim to solve the same problem, they have a different focus. Principally, they aim to reduce the runtime of the recovery algorithm as much as absolutely possible, and are willing to sample the function a bit more often than a compressive sensing method might in order to achieve that objective. By doing so, one can indeed achieve similar recovery guarantees to Basis Pursuit, but with radically reduced runtimes that depend only logarithmically on $N$. However, SFTs are highly adapted to the special properties of the Fourier basis, making their extension to other orthonormal bases difficult.

In this talk we will present a general framework that can be used in order to construct a highly efficient SFT algorithm. The framework abstracts many of the components required for SFT design in an attempt to simplify the application of SFT ideas to other basis choices. Extension of arbitrary SFTs to the Chebyshev and Legendre polynomial bases will also be discussed.

#### Joachim Weickert — Image Compression with Differential Equations

Partial differential equations (PDEs) are widely used
to model phenomena in nature. In this talk we will see
that they also have a high potential to compress digital
images.

The idea sounds temptingly simple: We keep only a small
amount of the pixels and reconstruct the remaining data
with PDE-based interpolation. This gives rise to three
interdependent questions:
1. Which data should be kept?
2. What are the most useful PDEs?
3. How can the selected data be encoded efficiently?
Solving these problems requires to combine ideas from
different mathematical disciplines such as mathematical
modelling, optimisation, interpolation and approximation,
and numerical methods for PDEs.

Since the talk is intended for a broad audience, we focus
on intuitive ideas, and no specific knowledge in image
processing is required.

#### Gary Froyland — Dynamics, mixing, and coherence

Coherent structures in geophysical flows play fundamental roles by organising fluid flow and obstructing transport. For example, in the ocean, coherence manifests itself at global scales down to scales of at least tens of kilometres, and strongly influences the transportation of heat, salt, nutrients, phytoplankton, pollution, and garbage. I will describe some recent mathematical constructions, ranging across dynamical systems, probability, and geometry, which enable the accurate identification and tracking of such structures, and the quantification of associated mixing and transport properties. I will present case studies from a variety of geophysical settings.

#### Bruno Nachtergaele — Quantum Many-Body Systems: Particles and Spectra

The energy spectrum of a quantum many-body system and its physical properties at low temperature can often be understood in terms of a particle description of the eigenvectors of the Hamiltonian of the system. These (quasi-)particles derive some of their most important properties from the local structure of the interactions and the physical space in which the system resides. I will review old and new results in mathematical physics that describe and exploit this particle structure in order to understand physical properties of the system, including recent results and open problems regarding topologically ordered phases of matter.

#### Johanna Neslehova — Assessing dependence between extreme risks

Dependence between rare events is of prime concern in risk management. For example, extreme comovements of prices or huge operational losses in different business lines represent a substantial risk for financial institutions. Severe losses or insurance claims can also result from the simultaneous occurrence of violent storms, fires, earthquakes or floods. Modelling such dependencies is one of the objectives of extreme-value theory, an area of research that stands at the crossroads between analysis, statistics, and probability. In this talk, I will show how dependencies between extreme risks can be quantified using copula-based models and illustrate some techniques of inference relevant to this issue.

#### Robert König — Fault-tolerance in quantum information processing

Classical information theory provides quantitative answers to basic questions about communication and computation. In the presence of quantum effects, its basic tenets need to be reassessed as fundamentally novel information-processing primitives become possible. Their potential appears promising, but their realization hinges on our ability to construct mechanisms protecting information against unwanted noise.

In this talk, I will consider two problems associated with communication and computation. First, I will discuss the additivity problem for classical capacities. I will review some of the more recent results in this direction: for continuous-variable channels, quantum generalizations of certain geometric inequalities yield operationally relevant statements. Second, I will discuss the problem of performing gates on information encoded in an error-correcting code, and explain how this relates to automorphisms of the latter.

#### Nicole Megow — Resource Minimization under Uncertainty

Uncertainty in the input data is an omnipresent issue when solving real-world optimization problems: resources may become unavailable, material arrives late, jobs may take more or less time than originally estimated, weather conditions may cause severe delays, etc. Uncertain data is typically modeled through stochastic parameters or as online information that is incrementally revealed. In this talk, I will discuss different models and solution methods for optimization under uncertainty. As a main example I show recent results on an online machine minimization problem, and I will mention some results and intriguing open questions on stochastic scheduling.