**Welcome to Mathematical Kaleidoscope II (2021)**

**Description:**

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

- Jakob Gulddahl Rasmussen: Temporal point processes and the conditional intensity function
- Horia Cornean: On some fundamental spectral properties of finite dimensional stochastic matrices
- Olav Geil: Applications of Gröbner basis theory to problems in information theory
- Martin Raussen: What are manifolds, what are they good for, and some surprising results
- Anton Evgrafov: Finite element analysis of elliptic problems
- Poul Svante Eriksen: Introduction to genetic algorithms

- Times of events, such as earthquakes, can be modelled by temporal point processes. When modelling temporal point processes, the so-called conditional intensity function is a particularly useful tool. Roughly speaking, the conditional intensity function tells us if an event is going to happen now given we know the times of events in the past. Likelihood functions, simulation algorithms and model checking procedures can be constructed from the conditional intensity function. We will look at some of the theory behind temporal point processes and conditional intensity functions, and apply this theory to data.
- We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for 3-dimensional time-reversal symmetric topological insulators. This problem is translated in the study of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We provide an algorithm which constructs a "multi-step" logarithm that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.
- Algebraic methods play a fundamental role in the theory of error-correcting codes. Different languages are applied such as algebraic geometry, function field theory, pure algebraic methods, and recently also Gröbner basis theory. This course gives an introduction to the use of Gröbner basis theory in algebraic coding theory and other information theoretical applications.
For most of history, geometry was the study of objects in Euclidean space. Motivated by physical considerations, B. Riemann and H. Poincaré were among the first suggesting and studying alternatives: Manifolds that are locally but not globally similar to Euclidean space (in various dimensions). They come up naturally, e.g. as solutions to one or several equations (if polynomial, then in algebraic geometry) or as configuration spaces, in physics, robotics, control theory etc. There are several mathematical areas – differential and algebraic geometry and topology and also mathematical physics, in which the study of geometric and algebraic properties of manifolds and their classification is central. Several modern results were utterly surprising and unexpected.

- Most mathematical models of physical phenomenon are based on partial differential equations (PDE), and there is rich theory on existence and uniqueness properties of such problems. However, in practice, in order to simulate solutions to such problems, it is necessary to discretize the problems into a finite dimensional setting where the before mentioned theory does not directly apply. One commonly used approach for discretization is the socalled Finite Element Method (FEM). We will for second order elliptic PDEs consider how to apply an FEM for computation of approximative solutions. Furthermore, we will go through the theory required to show that the approximations converge (and how fast they converge), in an appropriate norm, to the true solution when the discretization is refined.
- Genetic algorithms are designed to seek for solutions of a given optimality problem. The basic idea is inspired by population biology, where we consider a population of potential solutions. We randomly create new generations and adapt the darwinistic principle: survival of the fittest. Concepts like mutation, recombination and migration are included to allow drift against different solutions.

**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. In order to pass the course, at least the exercises related to four of the five topics should be satisfactory evaluated.

**Organizer:**Professor Jesper Møller - jm@math.aau.dk

**Lecturers:**Associate Professor Jakob Gulddahl Rasmussen, Professor Horia Cornean, Professor Olav Geil, Professor Martin Raussen, Associate Professor Anton Evgrafov, Associate Professor Poul Svante Eriksen.

**ECTS:**5

**Time:**January 27-29 all days 8:15-12:00 (Jakob Gulddahl Rasmussen);

February 1, 8:15-12:00 and February 2, 8.15-16.15 (Horia Cornean);

February 4, 8:15-12.00 and February 5, 8:15-16:15 (Olav Geil);

February 8, 12:30-16.15 and February 9, 8:15-16:15 (Martin Raussen);

February 11, 8:15-12 and February 12, 8:15-16:15 (Anton Evgrafov);

February 15, 8:15-12:00 and February 16, 8:15-16:15 (Poul Svante Eriksen)

**Place:**

**ONLINE**

Zip code:

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**City:**

**Number of seats:**12

**Deadline:**6 January 2021

**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 3.000 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 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: Svante Eriksen
- Teacher: Anton Evgrafov
- Teacher: Olav Geil
- Teacher: Jesper Møller
- Teacher: Jakob Gulddahl Rasmussen
- Teacher: Martin Raussen