### Mathematical Kaleidoscope II (2015)

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

1. Jakob Gulddahl Rasmussen (January 30): Temporal Point Processes and the Conditional Intensity Function
2. Jon Johnsen (February 5): Fourier Series and Partial Differential Equations
3. Olav Geil (February 9): Applications of Gröbner Basis Theory to Problems in Information Theory
4. Poul Svante Eriksen (February 16): Introduction to Genetic Algorithms
5. Horia Cornean (February 23): Neogibbsian Thermodynamics and Mechanical Statistics

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. First we shall give a short introduction to Fourier Series, and then proceed to the main subject: how Fourier Series can be applied in the solution of partial differential equations on simple domains. We shall focus on basic examples such as the heat equation, the wave equation and the Laplace equation. In the main we follow Chapter 8 in "Elementary differential equations with boundary value problems" by C. H. Edwards and D. E. Penney, Prentice-Hall 2000 (4th. ed.). We shall emphasize ideas rather than rigour, and therefore we shall combine lectures and exercises.
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. 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.
5. We will introduce thermodynamics in an axiomatic way, showing that all thermodynamical properties of a given system are completely characterized by the knowledge of its entropy. Then we will discuss how the entropy can be computed using the ideas of (classical and quantum) mechanical statistics.

Prerequisites: Basic knowledge in mathematics and statistics.

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

Lecturers: Associate Professor Jon Johnsen, Professor Horia Cornean, Associate Professor Olav Geil, Associate Professor Poul Svante Eriksen and Associate Professor Jakob Gulddahl Rasmussen.

ECTS: 2

Time: 30 January and 5, 9, 16, 23 February, 2015, each day 9:00-15.30

Place: Aalborg University, Fredrik Bajers Vej 7

Room:
30 January: B2-104
5 February: G5-112
9 February: G5-110
16 February: G5-110
23 February: G5-110

Zip code: 9220

City: Aalborg

Number of seats: 40