Málstofa í stærðfræði

### Fyrirlesari: Thomas Weigel, Università di Milano-Bicocca

### Titill: The capitulation kernel and Hilbert’s theorem 94

Staðsetning: VR-II, 158

Tími: Miðvikudagur 11. desember kl. 11.00

### Ágrip:

One of the central theorems in Algebraic Number theory

is the finiteness of The capitulation kernel and Hilbert’s theorem 94.

One of the central theorems in Algebraic Number theory

is the finiteness of the Ideal class group of a number field.

The capitulation kernel k(R/O) is the subgroup of ideal classes which

become principal under an extension of Dedekind domains R/O.

Hilbert’s theorem 94 states that for a finite cyclic Galois extension

L/K of number fields of prime power degree, the order of k(R/O) is divisible

by |L:K|. This fact motivated D. Hilbert to formulate his

Principal ideal conjecture which was proved by P. Furtwängler 30 years later.

In this seminar we show a strong version of Hilbert’s theorem 94, which is based

on an abstract version of Hilbert’s theorem 90.f the Ideal class group of a number field.

The capitulation kernel k(R/O) is the subgroup of ideal classes which

become principal under an extension of Dedekind domains R/O.

Hilbert’s theorem 94 states that for a finite cyclic Galois extension

L/K of number fields of prime power degree, the order of k(R/O) is divisible

by |L:K|. This fact motivated D. Hilbert to formulate his

Principal ideal conjecture which was proved by P. Furtwängler 30 years later.

In this seminar we show a strong version of Hilbert’s theorem 94, which is based

on an abstract version of Hilbert’s theorem 90.

### Speaker: Ragnar Sigurðsson, University of Iceland

### Title: Siciak’s extremal functions and Helgason’s support theorem

Location: VR-II 157

Time: Friday December 7 at 11.40

### Abstract:

We prove that a function, which is defined on a union

of lines $\C E$ through the origin in $\C^n$ with direction

vectors in $E\subset \C^n$ and is holomorphic

of fixed finite order and finite type along each line,

extends to an entire holomorphic function on $\C^n$

of the same order and finite type, provided that $E$ has

positive homogeneous capacity in the sense of Siciak and all

directional derivatives along the lines satisfy a necessary

compatibility condition at the origin.

We are able to estimate the indicator function of

the extension in terms of Siciak’s weighted

homogeneous extremal function, where the weight

is the type of the given function on each given line.

As an application we prove a generalization of

Helgason’s support theorem by showing how the support

of a continuous function with rapid decrease at infinity

can be located from partial information on the support

of its Radon transform.

This is a joint work with Jöran Bergh at Chalmers University of

Technology and University of Gothenburg.

Math Colloquium

### Speaker: Hermann Þórisson, University of Iceland

### Title: What is typical?

Location: Naustið (Endurmenntun)

Time: Friday November 30 at 11.40

### Abstract:

The word “typical” is often used in a loose sense for events in a stationary random process (such as occurrences of heads in repeated coin tosses). This concept can be made precise using so-called Palm version of the process. It is however not well known that there are in fact two Palm versions and that it is the less known version that captures the typicality property. So using the well known version is flawed except in the special case when the two versions coincide.

In this talk the elementary example of repeated coin tosses (indexed by the integers) will be used to make transparent what the issue is.

In the latter half of the talk we shift drastically to random measures on a rather general class of Abelian groups (in the coin tossing example the group is the integers under addition and the random measure is formed by mass points of size one at the heads). After giving the formal definition of the two Palm versions we present a theorem motivating the claim that it is the less known Palm version that captures the typicality property. If time allows we skim through the proof which relies on the concepts of shift-coupling and mass-stationarity.

Math colloquium

### Speaker: Watse Sybesma, University of Iceland

### Title: Black holes and good vibrations

Location: Naustið (Endurmenntun)

Time: Friday November 23 at 11.40

### Abstract:

Matter falling into a black hole is a dynamical process that can be described by a complicated wave equation, which has to be disentangled from a system of PDEs. Computing the eigenvalues of such a wave equation allows one to obtain the characteristic time it takes for this process to take place, which physically is an interesting quantity. However, in general it is very hard to solve the wave equation or even to disentangle the initial system of PDEs. In this talk I will introduce a series of ways one can approximate and solve these types of problems.

Math colloquium

### Speaker: Hjalti Þór Ísleifsson, University of Iceland

### Title: Weak Topologies in Banach Spaces

Location: Naustið (Endurmenntun)

Time: Friday November 16 at 11.40

### Abstract:

We begin by defining the weak topology on normed spaces and the weak* topology on their dual spaces. The fundamental properties of these topologies will be discussed quite thoroughly but we will focus primarily on compactness. We will prove the Banach-Alaoglu theorem which states that the closed unit ball in the dual space of a normed space is compact in the weak* topology. Then we will discuss reflexive Banach spaces and their basic properties, discuss the Milman-Pettis theorem which gives a sufficient geometric condition for the reflexivity of a Banach space. We will prove the theorem of Kakutani which states that the closed unit ball of a Banach space is compact in the weak topology if and only if the space is reflexive. Finally, we will discuss the Eberlein-Smulian theorem which states that compactness, sequential compactness and limit point compactness are equivalent for subsets of normed spaces endowed with the weak topology.

Math Colloquium

### Speaker: Þórður Jónsson, University of Iceland

### Title: The structure of the spatial slices of 3-dimensional causal triangulations

Location: Naustið (Endurmenntun)

Time: Friday November 2 at 11.40

### Abstract:

We show that there is a bijection between the spatial slices of 3-dimensional causal triangulations and a class of two-dimensional cell complexes satisfying some simple conditions. The talk will be preceded by a short introduction to the subject.

Math Colloquium

### Speaker: Eggert Briem, University of Iceland

### Title: Gelfand Theory for Real Banach Algebras

Location: Naustið (Endurmenntun)

Time: Friday October 19 at 11.40

### Abstract:

A real Banach algebra is a Banach algebra over the reals. We will only consider commutative Banach algebras with unit. An example is the algebra of continuous functions, f , on the unit disc, analytic in the interior of the disc, satisfying f (z) = f (z). The norm on the algebra is the sup-norm.

Another example is the algebra of continuously differentiable real-valued functions on the unit interval with the norm given by

∥f∥ = ∥f∥∞ +∥f′∥∞

According to Gelfand theory, a commutative Banach algebra A with unit, over the complex numbers, can be represented as an algebra of continuous complex valued functions on a compact Hausdorff space X, with

sup_{x ∈ X}|ã(x)| = r(a) := lim∥a^n∥^1/n

for a ∈ A. Here X is the space of multiplicative linear functionals on A, equipped with the w∗-topology, and ã(x) = x(a) for x ∈ X.

This result also holds for real Banach algebras. Furthermore, the representati- on consists of real valued functions if and only if

r(a^2) ≤ r(a^2 +b^2) a,b ∈ A.

We will prove this using only real Banach space theory. If there is time we

will also talk about the general case where there is no condition on A.

Math Colloquium

### Speaker: Thomas Selig, University of Iceland

### Title: The Abelian sandpile model on permutation graphs

Location: Naustið (Endurmenntun)

Time: Friday October 12 at 11.40

### Abstract:

A permutation graph is a graph whose edges are given by the inversions of a permutation. The Abelian sandpile model (ASM) is a Markov chain on the set of so-called configurations of a graph. Of particular interest are the recurrent configurations, i.e. those that appear infinitely often in the long-time running of the model. We exhibit a bijection between the set of recurrent configurations for the ASM on permutation graphs and the set of tiered trees, introduced by Duggan et al. This provides a new bijective proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We also show a link between the minimal recurrent configurations and the set of complete non-ambiguous binary trees, introduced by Aval et al.

Math Colloquium

### Speaker: Þorsteinn Jónsson, University of Guelph

### Title: Variational Techniques for Learning Distributions of Data

Location: V-147 (VR-II)

Time: Monday 2 July at 10:30

### Abstract:

In this talk I will give an overview of a set of techniques that allow us to define latent variable models of data generating distributions.

To this end we introduce neural networks as an efficient and a surprisingly effective way of finding variational parameters that best fit some specified objective.

I will discuss a couple of different approaches that can be taken to define this objective and show you some interesting results.

Math Colloquium

### Speaker: Thomas Selig, University of Strathclyde

### Title: EW-tableaux, permutations and recurrent configurations of the sandpile model on Ferrers graphs.

Location: VRII, V-147

Time: Wednesday 27 June at 10:30

### Abstract:

The Abelian sandpile model (ASM) is a dynamic process on a graph. More specifically, it is a Markov chain on the set of configurations on that graph. Of particular interest are the recurrent configurations, i.e. those that appear infinitely often in the long-time running of the model. We study the ASM on Ferrers graphs, a class of bipartite graphs in one-to-one correspondence with Ferrers diagrams. We show that minimal recurrent configurations are in one-to-one correspondence with a set of certain 0/1 fillings of the Ferrers diagrams introduced by Ehrenborg and van Willigensburg. We refer to these fillings as EW-tableaux, and establish a bijection between the set of EW-tableaux of a given Ferrers diagram and a set of permutations whose descent bottoms are given by the shape of the Ferrers diagram. This induces a bijection between these permutations and minimal recurrent configurations of the ASM. We enrich this bijection to encode all recurrent configurations, via a decoration of the corresponding permutation. We also show that the set of recurrent configurations over all Ferrers graphs of a given size are in bijection with the set of alternating trees of that size.