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 Waagepetersen