PhD Courses in Denmark

Welcome to Mathematical Kaleidoscope II

Description: The five 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

Anton Evgrafov: Finite element analysis of elliptic problems

Poul Svante Eriksen: Introduction to genetic algorithms

1.       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.

2.       We will start by investigating the Jordan normal form of any square matrix by using elementary methods from complex function theory. We will then focus on left stochastic matrices (square matrices with non-negative entries whose columns sum to one), by proving classical spectral results like the Gershgorin and Perron-Frobenius Theorems.  As an application we will study the approach to equilibrium of non-homogeneous Markov chains in finite state spaces.

3.       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.

4.       Finite Element Method (FEM) is objectively the most widely used numerical approach for finding approximate solutions of partial differential equations.

Starting from the simplest case of conforming approximations to problems defined by symmetric, coercive, and bounded bilinear forms we will explore the rich theory and practice of FEM arising, when the underlying assumptions are successively relaxed.

5.       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.

Learning objectives: 

Form of evaluation:

Active participation is required (including a show up of minimum 80% corresponding to 4 topics) together with a successful evaluation of the solutions for a selected part of the exercises (as agreed with the lecturer) for at least 3 topics (selected by each student).

Organizer: Jesper Møller, e-mail: jm@math.aau.dk

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

ECTS: 5

Time: 

January 30 2025, 8:15-16:15 (Jakob Gulddahl Rasmussen);

February 3 2025, 8.15-16.15 (Horia Cornean);

February 6 2025, 8:15-16:15 (Olav Geil);

February 10 2025, 8:15-16:15 (Anton Evgrafov);

February 13 2025, 8:15-16:15 (Poul Svante Eriksen).

Place: Thomas Manns Vej 23, Room: 1.344

Zip code: 9220

City: Aalborg

Maximal number of participants: XX

Deadline: 9. januar 2025

Important information concerning PhD courses: 

There is 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 the 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 of the course.

We cannot ensure any seats before the deadline for enrolment, all participants will be informed after the deadline, approximately 3 weeks before the start of the course.

To attend courses at the Doctoral School in Medicine, Biomedical Science and Technology you must be enrolled as a PhD student.

For inquiries regarding registration, cancellation or waiting list, please contact the PhD administration at aauphd@adm.aau.dk When contacting us please state the course title and course period. Thank you.