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 Phys/Phys seminar

### Speaker: Jens H. Bárðarson, KTH Stockholm

### Title: On the chiral anomaly in Weyl semimetals

Location: VR-II 158

Time: Tuesday December 4, at 13.30

### Abstract:

I will give an introduction to the physics of Weyl semimetals focussing on Fermi arcs and chiral anomaly. I will then discuss some transport properties of Weyl semimetals including strongly angular-dependent magnetotransport in the presence of long range disorder, and the difference between the longitudinal conductance in the presence of magnetic field and chiral pseudo-magnetic field. The latter points in opposite direction for opposite chiralities and can be induced, for example, by strain. This leads to a discussion of pseudo-landau levels and their connection with Fermi arcs and covariant and consistent chiral anomaly. I will also mention a new axial torsional contribution to the axial anomaly.

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 Phys seminar

### Speaker: Aruna Rajagopal, University of Iceland

### Title: Shock and Rarefaction Waves in Out of Equilibrium Lifshitz Fluids

Location: VR-II 155

Time: Thursday November 22 at 13.30

### Abstract:

Motivated by the recent developments and interest in the emergence of a non equilibrium steady state or NESS, for a relativistic fluid, and its possible holographic dual, we follow a similar framework to study the properties of such a state for a non-relativistic Lifshitz fluid with a general scaling exponent. This is carried out by solving the Riemann problem for the NESS which develops between two semi-infinite heat reservoirs that are brought into contact, and studying the shock and rarefaction waves which emerge as solutions to this problem.

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 Phys seminar

### Speaker: Daniel Areán, IFT Madrid

### Title: Breaking translations in AdS/CMT: Holographic Bad Metals

Location: Naustið (Endurmenntun)

Time: Wednesday October 24, 13.30 – 14.30

### Abstract:

We present holographic models realizing the pseudo-spontaneous breaking of translations. We study the electrical transport of these solutions finding that they reproduce features characteristic of the bad metallic transport observed in the cuprates.

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.