Welcome to Mathematical Kaleidoscope II

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

- Jakob Gulddahl Rasmussen (January 29, 30): Temporal Point Processes and the Conditional Intensity Function
- Horia Cornean (February 5, 7): On the construction of exponentially localized Wannier functions in 3D periodic systems
- Olav Geil (February 8, 9): Applications of Gröbner Basis Theory to Problems in Information Theory
- Poul Svante Eriksen (February 12, 14): Introduction to Genetic Algorithms
- Henrik Garde (February 15, 16): Finite Element Analysis of Elliptic Problems

- 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.
- 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.
- 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.

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, e-mail: jm@math.aau.dk **Lecturers: **Associate Professor Jakob Gulddahl Rasmussen, Professor Horia Cornean, Professor Olav Geil, Associate Professor Poul Svante Eriksen, and Assistant Professor Henrik Garde.**ECTS: **4 **Time: **

Full days (8:15-16:15) on 29. January and 5., 8., 12., and 15. February and

Half days:

(8:15-12:00) on 30 January

(12:30-16:15) on 7, 9, 14, and 16 February

**Place: **Aalborg University, Skjernvej 4A, Room 3.102. **Zip code: **9220

**Aalborg Øst**

City:

City:

**Number of seats:**40

**Deadline:**20. January 2018

**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: Poul Svante Eriksen
- Teacher: Henrik Garde
- Teacher: Olav Geil
- Teacher: Jesper Møller
- Teacher: Jakob Gulddahl Rasmussen

Welcome to Green's Fuction Methods ind PDE and Physics

**Description: **The course has two distinct parts as described below.

**1. Horia Cornean:** “The general mathematical theory behind Green's functions”.

**2. Thomas Søndergaard:** “Green’s function integral equation methods for solving the Helmholtz equation”.

A “Green function” is a generic name given by physicists to the integral kernel of the resolvent of certain PDEs generated by the Laplace, Helmholtz, Maxwell, and D'Alembert operators. In a physics language the Greens function essentially describes the response to a point source.

1. The first part of the course will cover how to find this integral kernel based on solving linear integral equations. We will also discuss several mathematical properties of the integral kernels of resolvents of elliptic partial differential operators acting on (un)bounded domains. In particular, we will be interested in self-adjointness and regularity issues, the Lippmann-Schwinger equation and the Limiting Absorption Principle.

2. The second part of the course covers Green’s function integral equation methods for solving the Helmholtz equation in relation to the type of scattering problems found in electromagnetics / optics. The methods require finding self-consistent solutions to integral equations that express e.g. the total electric field in terms of an incident field and a scattered field, where the latter is given as an overlap integral between a Green’s function and the field itself on or inside a scattering object.

For the purpose of these methods this part of the course will first cover the construction of Green’s functions using eigenmode expansion techniques with radiating (open) or periodic boundary conditions depending on the type of scattering problem. In the case of layered reference structures the Green’s function will be expressed in terms of Sommerfeld integrals, and details of their evaluation in terms of poles of Fresnel reflection and transmission coefficients corresponding to bound modes will be discussed. In addition, details of the handling of singularities in Green’s functions will be covered. The course will cover a range of integral equation methods for theoretical research within the area of nano optics.

**Prerequisites: ** advanced knowledge in mathematical analysis and/or theoretical physics.

**Evaluation:** During each lecture the participants are evaluated through exercises, which provide and prove an understanding of the presented theory. In order to pass the course, at least the exercises related to one of the topics should be satisfactory evaluated.

**Organizers and lecturers: **Professor Horia Cornean, e-mail: cornean@math.aau.dk and Associate Professor Thomas Søndergaard, e-mail: ts@mp.aau.dk

**ECTS: **4 **Time: **First part of the course: Full days (8:15-16:15) on 26 February to 2 March 2018.

Second part of the course: Full days (8:15-16:15) on 5 March to 9 March 2018.

**Place: **Aalborg University, Skjernvej 4A, Room 3.102.**Zip code: **9220

**Aalborg Øst**

City:

City:

**Number of seats:**40

**Deadline:**12 February 2018

**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: Thomas Søndergaard