### Mathematical Kaleidoscope III (2023)

Mathematical Kaleidoscope III.

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

1. J. Eduardo Vera-Valdés: Missing data and multiple imputation methods.

2. Jesper Møller: Auxiliary variable methods for distributions with intractable normalizing constants, with a view to simulation-based Bayesian inference.

3. Morten Nielsen: Bases in Banach spaces and their approximation properties.

4. Aysegül Kivilcim: Stability of hybrid systems.

5. Lisbeth Fajstrup: Algebraic topology - geometry as reflected in algebra.

6. Giovanna Marcelli: Adiabatic perturbation theory in quantum dynamics.

1. Missing data is a common problem in all branches of data science. They can occur due to human error in capturing the data or equipment fail, for example. To address the issue, we will analyze several imputation methods. Of particular interest will be the Multivariate Imputation by Chained Equations (MICE) method. The multiple imputations account for the statistical uncertainty. In addition, the chained equations approach can handle complexities such as measurement bounds due to old equipment.

2. Probability distributions with intractable normalizing constants appear in many cases of modern statistics, particularly in spatial statistics and statistical physics when studying Gibbs distributions.

This complicates likelihood based inference and simulation procedures such as the Metropolis-Hastings algorithm, where various auxiliary variable methods have proven to be successful. I will start by considering the celebrated Ising model, where the auxiliary variable method known as the Swensen-Wang algorithm applies. Secondly, I discuss a recent auxiliary variable method in a Bayesian setting which solves the problem with an intractable normalizing constant appearing in the likelihood term of the posterior density. This method will be illustrated in connection to the statistical analysis of a point pattern data set.

3. I will give an introduction to the rich theory of bases in Banach spaces. The focus will be on so-called Schauder bases and unconditional bases, and I will give some of the well-known results on such bases.

The second part is concerned with approximation properties of bases in Banach spaces. The notion of best m-term approximation in a Banach space is introduced, and we study the related (and quite recent) concept of a greedy basis. Examples from wavelet theory will be considered.

4. I will start with some preliminaries for hybrid systems. Then, I will mention existence and uniqueness, and continuation of solutions of hybrid systems. Then, some results will be provided on the stability of hybrid systems. Finally, we will analyze stability of some basic mechanical problems.

5. The geometric objects studied in algebraic topology are topological spaces. The aim of algebraic topology is to study topological spaces via combinatorial and algebraic methods, e.g. to break a space up into smaller pieces, to build spaces by gluing pieces together, taking products etc. The focus will be on spaces built from simplices – higher dimensional analogues of triangles and tetrahedra.

Topological invariants are algebraic structures associated to topological spaces and hence to the combinatorial representation of the space. They reveal properties of the space which are robust to deformation.

We will study spaces and invariants from the combinatorial (and hence computational) point of view. In particular, we will define and work with homology groups. Within the last 10 years, such topological invariants have become part of data analysis, image analysis and many other areas. We will have a look at the general ideas.

6. First, I will explain the basic notions of quantum mechanics, dealing with bounded Hamiltonians. Then, I will gently introduce the fundamental concepts of adiabatic perturbation theory. This will be done in the framework of the time-adiabatic theorem, which is the prototype of adiabatic theorems in quantum mechanics. Moreover, I will explain its generalization to super-adiabatic approximation. The spectral gap condition of the reference Hamiltonian is crucial in this analysis (remarks on absence of this property and inclusion of disorder will be pointed out as well). Physical applications, such as linear response for gapped systems of noninteracting fermions at zero temperature, will be discussed.

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 five of the six topics should be satisfactory evaluated.

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

Lecturers: Associate Professor Lisbeth Fajstrup, Assistant Professor Aysegül Kivilcim, Postdoc Giovanna Marcelli, Professor Jesper Møller, Professor Morten Nielsen, Associate Professor J. Eduardo Vera-Valdés.
Time: October 19, 8:15-12:00 and October 20, 8.15-16.15 (J. Eduardo Vera-Valdés);

October 23, 8:15-12:00 and October 24, 8.15-16.15  (Jesper Møller);

October 26, 8:15-12:00 and October 27, 8.15-16.15  (Morten Nielsen);

October 30, 8:15-12:00 and October 31, 8.15-16.15 (Aysegül Kivilcim);

November 2, 8:15-12:00 and November 3, 8.15-16.15 (Lisbeth Fajstrup);

November 6, 8:15-12:00 and November 7, 8.15-16.15 (Giovanna Marcelli).

ECTS: 5.0.

Place: Aalborg University, Department of Mathematical Sciences, Skjernvej 4 A, Room 2.120.

Number of seats:  TBA