Math Phys Seminar

### Titill: Holographic Complexity of Two-Dimensional Black Holes

### Fyrirlesari: Lukas Schneiderbauer, HÍ

Staðsetning: VR-II 158

Tími: Mánudaginn 29. apríl, kl. 10:00

### Ágrip:

Interest in holographic complexity arose when it was proposed as the dual quantity of the complexity of a quantum state in the context of AdS/CFT. However, it was realized that this quantity behaves reasonably even for geometries which are asymptotically flat and therefore do not fall under the AdS/CFT umbrella.

Motivated by this, we adapt and evaluate proposals for holographic complexity in black hole solutions of the CGHS model and variants thereof. These solutions describe 1+1 dimensional black hole geometries which are asymptotically flat.

The advantage of working with these models is that they allow for analytic treatment even in time-dependent backgrounds such as gravitational collapse or even black hole evaporation.

Málstofa í stærðfræði

### Titill: Properties of solution trajectories for a metrically regular generalized equation

### Fyrirlesari: Iman Mehrabi Nezhad, HÍ

Staðsetning: VR-II 258

Tími: Fimmtudagur 11. apríl kl. 10.00

### Ágrip:

The presentation starts from a tangible example, analysis of electrical circuits. Using the circuit theory laws, and considering *set-valued maps* to model the i-v characteristics of semiconductors like diode, and transistor, a *generalized equation* is obtained. The main concern of the talk is to investigate how perturbing the input signal will affect the output variables. The problem is studied in two cases: the static case, where the input signal is a DC source; and the dynamic case, where there exists an AC source in the circuit. We will review the electronic part very briefly as we are more interested in the mathematical model.In the static case, the problem can be reduced to the existence or absence of *local stability* properties of the solution map, or *metric regularity* for the inverse map. In the dynamic case, using methods of variational analysis and strong metric regularity property of an auxiliary map, we are able to prove the regularity properties of the solution trajectories inherited by the input signal. Furthermore, we establish the existence of continuous solution trajectories for the perturbed problem.

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: Carolina Canales Gonzalez, Pontificia Universidad Católica de Chile

### Titill: Looking for a complex Poincaré-Bendixson theorem

Staðsetning: V-258, VR-II

Tími: Fimmtudagur 28. febrúar kl. 11.40

### Ágrip:

The goal of this talk is to introduce the study of holomorphic foliations.

First, we will recall some things about differential equations on $\mathbb{R}^2$, like the Poincaré-Bendixson theorem and Hilbert’s 16th problem and then we will pass to the complex context where we will introduce holomorphic foliations.

The idea is to talk about the problem of the exceptional minimal set, who is an analog of Poincaré-Bendixson theorem in this new context, and about the developments in complex geometry, analysis and dynamics related with the study of this problem.

Málstofa í stærðfræði

### Fyrirlesari: Carlos Argaez Garcia, Háskóli Íslands

### Titill: Numerical methods for dynamic systems: Analysis of stability

Staðsetning: V-258, VR-II

Tími: Fimmtudagur 21. febrúar kl. 11.40

### Ágrip:

Dynamic systems describe the evolution over time of quantities governed by

differential equations. Therefore they are a powerful descriptive tool of

phenomena originated in applied disciplines.

A complete Lyapunov function describes the dynamic behavior of such

systems without requiring the explicit solution of the differential

equations. However, these have the disadvantage of being difficult to

obtain.

The algorithms proposed here, reduce the effort to obtain such functions

and even more, are able to isolate the regions whose dynamics have a

periodic behavior.

Throughout this talk, simple examples of applications in both two- and

three-dimensional systems will be given.

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.