### Mathematical Kaleidoscope 3 (2016)

Welcome to Mathematical Kaleidoscope 3

Description:

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

1. Leif Kjær Jørgensen: Girth and diameter of graphs. (February 9)

2. Jesper Møller: Auxiliary variable methods for distributions with intractable normalizing constants, with a view to simulation-based Bayesian inference. (February 10)

3. Lisbeth Fajstrup: Algebraic topology - geometry as reflected in algebra. (February 5 - date changed)

4. Arne Jensen: Interpolation and numerical differentiation. (February 17)

5. Morten Nielsen: Bases in Banach spaces and their approximation properties. (February 23)

1. In a communication network (graph) we want communication between any two vertices to pass as few edges as possible. The largest number of edges in a shortest path is the diameter of the graph. We want to construct graphs with diameter D where every vertex has degree (at most) k. The number of vertices in such graph is at most the Moore bound M(k,D). The Moore bound is also a lower bound on the number of vertices in a graph with girth 2D+1 where every vertex has degree (at least) k. The girth is the length of a shortest cycle. I will consider construction and proof of non-existence of graphs of order close to the Moore bound and either girth 2D+1 or diameter D. Methods of construction includes combinatorial, algebraic (from finite geometry) and probabilistic. Non-existence proofs are either combinatorial or based on eigenvalues. I will also consider computer enumeration. I will show how an orderly search algorithm is used to find all graphs with a given number of vertices and with the required properties.

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. The geometric objects studied in algebraic topology are topological spaces. To study continuity and convergence on a space, one needs to know the open sets, i.e., the topology.  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.

Topological invariants are algebraic structures associated to topological spaces. The association of invariants is required to be functorial:  Given X a topological space, i(X) an invariant, say this invariant is always a group. Suppose f:X ->Y is continuous, then there should be a map i(f): i(X) -> i(Y), which is  a homomorphism of the groups. Given another space Z, and a map g:Y ->Z, we require i(g o f)=i(g)oi(f). This very abstract definition is extremely efficient, as we will see.

We will study one family of such invariants, the fundamental group and higher order homotopy groups. These measure "holes" in spaces and we will see some applications, among them the Brouwer fixed point theorem. Within the last 10 years, topological invariants have become part of data analysis, image analysis and many other areas. The tools for this are more intricate invariants, but we will have a look at the general ideas.

4. I will give well known results on interpolation with polynomials, and also new results on fast algorithms for computation of interpolating polynomials. The special role of the Chebyshev points (the points cos(j*pi/N), j=0,...,N in the interval [-1,1]) will be emphasized. The second part is concerned with numerical differentiation. Polynomial interpolation will be used to give a unified treatment of the finite difference methods and the spectral differentiation methods. The methods will be illustrated with approximate eigenvalue determination of some differential equation boundary value problems. For this type of problem we will show that the spectral methods are vastly superior to the finite difference methods.

5. 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.

Organizer: Professor Jesper Møller, e-mail: jm@math.aau.dk
Lecturers: Associate Professor Leif Kjær Jørgensen, Associate Professor Lisbeth Fajstrup, Professor Arne Jensen, Professor Jesper Møller, Professor Morten Nielsen.

ECTS: 2.0

Time: February 5, 9, 10, 16, 17, 23. Each day 12.30-16.15

Place: G5-110, Fredrik Bajers Vej 7G

February 5: G5-109

Zip code: 9220

City: Aalborg East Number of seats: 30