Welcome to Mathematical Kaleidoscope I

The five subjects to be covered by the course are described below.

- Jesper Møller: Perfect simulation
- Morten Nielsen: Sparse representation of data
- Olav Geil: Secret sharing
- Horia Cornean: Pseudospectra of matrices
- Rasmus Waagepetersen: Estimating functions and spatial point processes

1. Perfect simulation methods extend conventional Markov chain Monte Carlo methods by ensuring that the sample is not only approximately, but exactly from the stationary distribution. We review various such algorithms, including the Propp-Wilson algorithm (for simulation of e.g. the celebrated Ising model), Fill's algorithm, and dominated coupling from the past.

2. Sparse approximation techniques have been at the core of a rapidly evolving and very active area of research since the 1990s. Their most visible technological success has certainly been in the compression of high-dimensional data with wavelets. However, approximating a signal or an image with a sparse linear expansion from a possibly overcomplete dictionary of basis functions (called atoms) has turned out to be an extremely useful tool to solve many other signal processing problems. In this talk, I will discuss some of the mathematical and computational aspects of sparse representations using redundant dictionaries in a Hilbert space. Our main focus will be on sparse representations using 'coherent' dictionaries in a finite dimensional space, but we will also mention some very recent results on infinite dimensional time-frequency dictionaries that have clusters of coherent atoms.

3. In secret sharing a group of n participants share a secret in such a way that whenever any group of size at most t1 join forces, they are not able to gain any information. All groups of size at least t2 on the other hand can reconstruct the secret in full. Every linear secret sharing scheme corresponds to a residue class construction involving two error-correcting codes. In this course we shall study the coding parameters: relative generalized Hamming weights. These parameters give a full picture of the information leakage in the system, that is, they explain how much information a group of size t can gain when t1 < t < t2. We will study codes defined by means of algebra. This will allow us to estimate or sometimes even find the relative generalized Hamming weights.

4. The pseudospectrum of a matrix is a tool that can be used to investigate the intermediate dynamics of linear systems. In particular, the techniques can show that a linear system, while in theory stable, may be highly unstable. A number of methods and results will be presented and illustrated with numerical computations.

5. The score function given by the derivative of the log likelihood function is just one example of an estimating function. For some statistical models it is hard to compute the score function and the simpler estimating functions may become useful. We will review general theory regarding statistical inference using estimating functions and consider specific examples of estimating functions for spatial point processes.

**Organiser: **Professor Jesper Møller, *e-mail: jm@math.aau.dk*

**Lecturers: **Professor Horia Cornean, Professsor Olav Geil, Professor Jesper Møller, Professor Morten Nielsen, and Professor Rasmus P. Waagepetersen, all from Department of Mathematical Sciences, AAU.

**ECTS: **2.0.

**Time:** 30 January and 1, 6, 8, February 2017 each day at 12.30-16.15, and 13 February 8.15-12.00.

**Place: **Fredrik Bajers Vej 7G, room G5-112.

** Number of seats: **30

**Deadlin****e**: 9 January 2017

**Important information concerning PhD courses**** **We have over some time experienced problems with no-show for both project and general courses. It has now reached a point where we are forced to take action. Therefore, the Doctoral School has decided to introduce a no-show fee of DKK 5,000 for each course where the student does not show up. Cancellations are accepted no later than 2 weeks before start of the course. Registered illness is of course an acceptable reason for not showing up on those days. Furthermore, all courses open for registration approximately three months before start. This can hopefully also provide new students a chance to register for courses during the year. We look forward to your registrations.

- Teacher: Horia Cornean
- Teacher: Olav Geil
- Teacher: Jesper Møller
- Teacher: Morten Nielsen
- Teacher: Rasmus Plenge Waagepetersen

Welcome to Numerial Physics

**Description**: In this PhD course, we study various topics in mathematics and physics that require numerical tools. We will discuss problems, solution strategies, programming and solve several practical problems. The topics selected range from quantum physics over nanooptics to electronic device simulations.

Mathematics: We shall deal with various spectral and dynamical aspects of (mostly discrete) Schrodinger operators. Among the topics which will be covered: the Feshbach formula, (almost) invariant subspaces, the adiabatic theorem of quantum mechanics, the limiting absorption principle, asymptotic completeness and the Lippmann-Schwinger equation. Literature: personal notes based on the book "Methods of Modern Mathematical Physics: Analysis of Operators" by M. Reed and B. Simon.

Quantum physics: We will discuss the optical response of two-dimensional semiconductors. In particular, focus will be on excitons including impurity-bound states. Course material includes “Electric, Optical and magnetic Properties of Nanostructures” 2016 version (found on TGP’s homepage).

Nanooptics/photonics: In nanooptics and photonics the theoretical study of propagation and scattering of light by nanostructures and nanophotonics components requires the solution of Maxwells equations. Several methods for solving Maxwells equations exist, and the most appropriate method to use depends on the geometry and what information that should be extracted. In this part of the Ph. D course we will go in depth with one such method, namely the Finite-Difference-Time-Domain method. This will include both the theory of the method and writing of numerical FDTD programs in matlab. We use as course material the relevant chapters in the book: A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, “Numerical Methods in Photonics”, CRC Press, 2015.

Fracture mechanics: The focus will be on fatigue induced fractures in polycrystalline metals – especially thermo-mechanical induced fatigue. Initially we will discuss the concept of fatigue induced fracture mechanics and the problematics of modelling. This will be followed by a case study of the bond wire lift-off failure mechanism observed in high power modules where we will use standard continuum mechanics models in combination with the Paris-Erdogan model to estimate fracture propagation rate. Literature will be specified closer to the start of the course.

The contents are listed below.

- 16/2 The Feshbach formula, limiting absorption principle and the Lippmann-Schwinger formula (Horia Cornean)
- 17/2 Slowly time dependent Schrodinger operators and the adiabatic theorem (Horia Cornean)
- 22/2 The Feshbach reduction for 1D excitons, Wannier model and 2D hydrogen (Horia Cornean/Thomas G. Pedersen)
- 23/2 2D Keldysh model and excitonic optical response (Thomas G. Pedersen)
- 24/2 2D model of impurity-bound excitons (Thomas G. Pedersen)
- 27/2 FDTD (Thomas Søndergaard)
- 28/2 FDTD (Thomas Søndergaard)
- 1/3 FDTD (Thomas Søndergaard)
- 2/3 Fundamental fracture mechanics, paris-erdogan model, J-integral approach (Kristian Bonderup Pedersen)
- 3/3 Case study of bond wire lift-off (Kristian Bonderup Pedersen)

Prerequisites: Basic knowledge in math, optics, quantum physics and electromagnetism.

**Organizer:** Professor Thomas G. Pedersen, email: tgp@nano.aau.dk

**Lecturers:** Professor Horia Cornean, Professor Thomas G. Pedersen, Associate Professor Thomas Søndergaard, and Post doc. Kristian B. Pedersen

**ECTS:** 4

**Time:** 16-17 + 22-24 + 27-28 February 2017 + 1-3 March 2017

**Place**: Skjernvej 4

**Zip code: **9220

**City: **Aalborg East

**Important information concerning PhD courses**** **We have over some time experienced problems with no-show for both project and general courses. It has now reached a point where we are forced to take action. Therefore, the Doctoral School has decided to introduce a no-show fee of DKK 5,000 for each course where the student does not show up. Cancellations are accepted no later than 2 weeks before start of the course. Registered illness is of course an acceptable reason for not showing up on those days. Furthermore, all courses open for registration approximately three months before start. This can hopefully also provide new students a chance to register for courses during the year. We look forward to your registrations.

- Teacher: Horia Cornean
- Teacher: Kristian Bonderup Pedersen
- Teacher: Thomas Garm Pedersen
- Teacher: Thomas Søndergaard