Málstofa í stærðfræði

### Fyrirlesari: Ahmed Zeriahi, Université Toulouse III, Paul Sabatier

### Titill: Pluripotential Kähler-Ricci flows

Staðsetning: VR-II 258

Tími: Fimmtudagur 28. mars kl. 11.40

### Ágrip:

We will first give an introduction to the Kähler-Ricci flow

on Kähler compact manifolds.

Next, we will review a recent joint work with Chinh H. Lu and Vincent

Guedj on a weak version of this theory

motivated by the study of the Kähler-Ricci flow on complex projective

varieties with mild singularities.

This study requires the development of a Parabolic Pluripotential Theory

on compact Kähler manifolds.

It bowls down to define and study weak solutions for some parabolic

complex Monge-Ampère equations,

extending the celebrated Bedford-Taylor theory in the degenerate

elliptic case.

arXiv:1810.02121 https://arxiv.org/abs/1810.02121

Málstofa í stærðfræði

### Fyrirlesari: Nick Poovuttikul, Háskóli Íslands

### Titill: Is hydrodynamics a theory of series expansion ?

Staðsetning: VR-II 258

Tími: Fimmtudagur 21. mars kl. 11.40

### Ágrip:

Hydrodynamics is one of the most successful theories in physics which describe dynamics across various length scales: from a few micrometers to the scale of galaxies. (some) Physicists tried to come up with an explanation why such a simple set of equations works so well. One of the most accepted explanations is based on the theorem by Nother which related the existence of divergence free quantities to the continuous global symmetries of the system. According to this, hydrodynamics is the gradient expansion of these quantities.

There are, however, many problems with this ‘explanation’, loosely speaking due to the lack of proper definitions of this gradient expansion scheme. I will go through a few scenarios where sometimes the procedure gives a non-sensible prediction such as the water is unstable, sometimes the gradient expansions is non-analytical (which can be observed experimentally), sometimes it gives a signal that travel faster than the speed of light or doesn’t even give the same collective excitations that were observed in the real systems, even in the regime where the hydrodynamic should be applicable.

Unfortunately, I have no mathematically satisfying answer to this question. So this overview talk will be a list of personal questions and puzzles I found while trying to understand what hydrodynamics really means.

Málstofa í stærðfræði

### Fyrirlesari: Ögmundur Eiríksson, Max-Planck-Institute for Mathematics, Bonn

### Titill: Orbits in varieties of quiver representations

Staðsetning: VR-II 258

Tími: Fimmtudagur 14. mars kl. 11.40

### Ágrip:

We recall the basics of representations of quivers.

On one hand we consider categories of representations and the varieties of such representations on the other. These varieties have a natural action by products of general linear groups, and the orbits correspond to isomorphism classes in the category of representations.

Next we consider the adjoint action of parabolic subgroups of general linear groups on their nilradical.

There is a description by Brüstle-Hille-Ringel-Röhrle of the orbits of this group action via the standard-filtered modules over the Auslander algebra of a truncated polynomial ring. This is one of many examples where orbits correspond to isomorphism classes of standard filtered modules over a quasi-hereditary algebra.

Recent joint work with J.Sauter considers the action of parabolic subgroups of general linear groups on compatible closed subvarieties of representation varieties for quivers. We show a connection to standard filtered modules that generalises that of BHRR.

Málstofa í stærðfræði

### Fyrirlesari: Long Li, Háskóli Íslands

### Titill: On the constant scalar curvature Kaehler metrics with cone like singularities along divisors

Staðsetning: V-258, VR-II

Tími: Fimmtudagur 14. febrúar kl. 11.40

### Ágrip:

In this talk, we will discuss the uniqueness and the existence problems for the constant scalar curvature Kaehler(cscK) metrics with conic singularities along divisors, on a compact complex Kaehler manifold. The uniqueness of these conic cscK metrics follows from the convexity of the so called conic-Mabuchi functional along geodesics in the space of all conic Kaehler metrics. The proof of the existence traces back to a conic version of Chen’s continuity path for cscK metrics, and we will establish all the sufficient „a prior estimates“ to prove the closedness of this continuity path.

Málstofa í stærðfræði

### Fyrirlesari: Dagur Tómas Ásgeirsson

### Titill: Palindromes in Finite Groups

Staðsetning: TG-227 Tæknistofan, (Tæknigarður)

Tími: Miðvikudagur 6. febrúar kl. 11.00

### Ágrip:

A subset P of a group G is called palindromic if it contains the identity element, and satisfies the property that for all a,b in P, the element aba also belongs to P. The Magnus-Derek game is a two-player game in which one of the players, Magnus, moves a token around a group by specifying a group element while the other player, Derek, decides whether Magnus multiplies the current position of the token by the specified element or its inverse, and moves the token to the resulting element. Magnus’s goal is to maximize the number of group elements the token visits, while Derek’s is to minimize that number. The problem we are interested in is finding f(G), the number of elements visited in the group G assuming optimal play. This problem has previously been solved for abelian groups. In this talk, we give a solution for general groups, in terms of palindromic subsets. Our solution yields a more satisfactory solution, i.e. in terms of subgroups rather than palindromic subsets, for certain classes of groups. Among those are nilpotent groups – a big step forward from the previous solution for abelian groups. After presenting the solution of the game, we consider further properties of palindromic subsets in finite groups. We introduce the notion of a civic group; a group in which every palindromic subset is a subgroup, and prove results about those. For instance, every civic group is the direct product of a cyclic 2-group and a civic group of odd order. We also give the form of minimal non-civic groups of odd order, and prove that the number of palindromes in a group of odd order divides the order.

The talk presented here is based on joint work with Patrick Devlin at Yale University.

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.

Málstofa í stærðfræði

### Fyrirlesari: Ragnar Sigurðsson, University of Iceland

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

Staðsetning: VR-II 157

Tími: Föstudagur 7. desember kl. 11.40

### Ágrip:

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.

Málstofa í stærðfræði

### Fyrirlesari: Hermann Þórisson, University of Iceland

### Titill: What is typical?

Staðsetning: Naustið (Endurmenntun)

Tími: Föstudagur 30. nóvember kl. 11.40

### Ágrip:

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.

Málstofa í stærðfræði

### Fyrirlesari: Watse Sybesma, University of Iceland

### Titill: Black holes and good vibrations

Staðsetning: Naustið (Endurmenntun)

Tími: Föstudagur 23. Nóvember kl. 11.40

### Ágrip:

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.

Málstofa í stærðfræði

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

### Titill: Veik grannmynstur á Banach rúmum.

Staðsetning: Naustið (Endurmenntun)

Tími: Föstudagur 16. Nóvember kl. 11.40

### Ágrip

Við byrjum á að skilgreina veika grannmynstrið á normrúmum og veika* grann- mynstrið á tvírúmum þeirra. Grundavallar eiginleikar þessarra grannmynstra verða kynntir til en sérstök áhersla verður á að skoða þjappleika. Við sönnum setningu Banachs og Alaoglu um að lokaða einingarkúla tvírúms sé þjöppuð í veika* grannmynstrinu. Svo fjöllum við um sjálfhverf Banach rúm, kynnumst grundvallareiginleikum þeirra og skoðum setningu Milman og Pettis sem gefur nægjanlegt rúmfræðilegt skilyrði fyrir því að Banach rúm sé sjálfhverft. Við sönnum setningu Kakutani um að einingarkúla Banach rúms sé þjöppuð í veika grannmynstrinu þá og því aðeins að rúmið sé sjálfhverft. Loks skoðum við setningu Eberlein og Smulian um að þrjár gerðir þjappleika séu jafngildar í normrúmum með veika grannmynstrinu.