**Welcome to Mathematical Kaleidoscope I.**

**Description:** The six subjects to be covered by the course are described below.

- Rasmus Waagepetersen: Estimating functions and spatial point processes
- Jesper Møller: Perfect simulation
- Morten Nielsen: Sparse representation of data
- Oliver Wilhelm Gnilke: Applications of error-correcting codes
- Horia Cornean: Three ODE iteration methods, with rigorous error bounds
- Orimar Sauri: Estimation methods of SDEs with jumps.

1. 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.

2. 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.

3. 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.

4. Codes are nowadays used for more than just error correction. We will give a short introduction into the theory of linear block codes, with special focus on generalized Reed-Solomon codes. Then we will present a handful of applications:

- Secret Sharing, are protocols that allows a message to be shared amongst a group of N people such that any subset of K-1 of them learn nothing about the message, but any set of K users can decode the message.

- Private Information Retrieval, are schemes for retrieving data from a database without the database learning which entry was accessed.

- Distributed Computation, are ways of outsourcing calculating a given function on sensitive data, using helper nodes, who cannot learn anything about the data.

We will discuss how the use of error correcting codes enables and enhances each of these applications. A solid knowledge of linear algebra, and a basic understanding of finite fields is encouraged, and we will provide material covering these essentials.

5. First, we review three basic numerical integration rules: trapezoidal, midpoint and Simp-

son. Second, we use these rules to derive three approximation schemes for solving initial value problems for first order ordinary differential equations: Euler's first order, Heun's second order, plus another scheme of order three. Third, we control how the local errors accumulate on a finite interval.

6. Stochastic differential equations (SDEs) play a crucial role in modeling complex systems affected by random processes, making estimation techniques essential for various scientific areas including finance, engineering, physics, and biology. Throughout this part, we will study classical estimation methods, such as the method of moments and maximum likelihood estimation, as well as high-frequency techniques, in the context of SDEs. Furthermore, we will emphasize the practical applications of SDEs in e.g. risk management and the stochastic modeling of particle motion.

**Prerequisites**: Basic knowledge in mathematics and statistics.

**Evaluation:** The participants are evaluated during each lecture by solving exercises which provide and prove an understanding of the presented theory. To pass the course, at least the exercises related to five of the six topics should be satisfactory evaluated.

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

**Lecturers:** Professor Horia Cornean, Assistant Professor Oliver Wilhelm Gnilke, Professor Jesper Møller, Professor Morten Nielsen, Associate Professor Orimar Sauri, Professor Rasmus P. Waagepetersen.

**Time:**

April 16, 8:15-16:00 and April 17, 8.15-12.00 (Rasmus Waagepetersen)

April 29, 8.15-16.15 and April 30, 8:15-12:00 (Orimar Sauri);

May 1, 8:15-12:00 and May 2, 8.15-16.15 (Jesper Møller);

May 6, 8:15-12:00 and May 7, 8.15-16.15 (Morten Nielsen);

May 9, 8:15-12:00 and May 10, 8.15-16.15 (Oliver Wilhelm Gnilke);

May 13, 8:15-12:00 and May 14, 8.15-16.15 (Horia Cornean).

**ECTS**: 5.0.

**Place:** Aalborg University

**Number of seats:**

**Deadline:** 29 March 2024

**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 3000 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 four 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: Oliver Wilhelm Gnilke
- Teacher: Jesper Møller
- Teacher: Morten Nielsen
- Teacher: Orimar Sauri Arregui
- Teacher: Rasmus Waagepetersen