Málstofa í stærðfræði

### Fyrirlesari: Michael Melgaard, University of Sussex

### Titill: Rigorous mathematical results on nonlinear PDEs arising in Quantum Chemistry

Staðsetning: V-147 (VR-II)

Tími: Föstudagur 13. apríl kl. 13:30

### Ágrip:

An introduction to electronic structure models is given and rigorous results are discussed on the existence of solutions (ground states and excited states) to weakly coupled, semi-linear elliptic PDEs with nonlocal operators arising in Hartree-Fock, Kohn-Sham and multiconfigurative many-particle models in quantum chemistry, in particular for systems with relativistic effects and external magnetic fields.

Math Phys seminar

### Fyrirlesari: Matteo Baggioli, Autonomous University of Barcelona

### Titill: Holographic quantum phase transitions: time to get dirty!

Staðsetning: HB-5 (Háskólabíó)

Tími: Mánudagur 9. apríl kl. 10:50

### Ágrip:

We study the effects of quenched one-dimensional disorder on the holographic Weyl semimetal quantum phase transition (QPT). We observe the smearing of the sharp QPT linked to the appearance of rare regions at the horizon where the local order parameter is non-zero. We discuss the role of the disorder correlation and we compare our results to the weakly coupled expectations from condensed matter theory and simulations. We analyze also the interplay of finite temperature and disorder and we find preliminary indications for the presence of log-oscillatory structures in the order parameter.

Málstofa í stærðfræði

### Fyrirlesari: Anthony Thomas Lyons, Waterford Institute of Technology

### Titill: The dressing method for the Camassa-Holm equation

Staðsetning: HB-5 (Háskólabíó)

Tími: Mánudagur 26. mars kl. 10:50

### Ágrip:

The Camassa-Holm equation is a nonlinear shallow water model which has been the focus of a great deal of mathematical research in hydrodynamics for the past two decades. This interest is in part due to the versatility of the system, being relevant as a fluid model possessing solutions which display wave-breaking along with global solutions in the form of soliton, peakon and cuspon solutions.

The inverse scattering transform has been successfully implemented to construct numerous global solutions of this system, and in this talk we present a recently developed variation of this method for the Camassa-Holm equation, known as the dressing method. This efficient implementation allows one to integrate several nonlinear hydrodynamical models, and in particular we shall outline the details of this new dressing method and use it to construct the one and two-soliton solutions of the Camassa-Holm equation.

Málstofa í stærðfræði

### Fyrirlesari: Phillip Wesolek, Binghamton University

### Titill: Totally disconnected locally compact groups: from examples to general theory

Staðsetning: HB-5 (Háskólabíó)

Tími: Mánudagur 5. mars kl. 10:50

### Ágrip:

Locally compact groups arise in many areas of mathematics as well as in physics. The study of locally compact groups splits into two cases: the connected groups and the totally disconnected groups. There is a rich and deep theory for the connected groups, which was developed over the last century. On the other hand, the study of totally disconnected locally compact groups groups only seriously began in the last 30 years, and moreover, these groups today appear to admit an equally rich and deep theory. In this talk, we will explore in details a wide variety of examples of totally disconnected locally compact groups. In particular, we discuss Lie groups over over the p-adic numbers, Galois groups, and automorphism groups of locally finite trees. We will then survey some recent results in the theory of totally disconnected locally compact groups.

### Fyrirlesari: Alexander Wendland, University of Warwick

### Titill: Facially restricted graph colouring’s

Staðsetning: HB-5 (Háskólabíó)

Tími: Mánudagur 19. febrúar kl. 10:50

### Ágrip:

Arguably one of the best known theorems from combinatorics is the four colour theorem, stating that every planar graph can be coloured using at most four colours such that no edge connects two vertices of the same colour. In this talk I will discus variants on this theorem in particular list colouring’s and facial restriction’s on the colouring. In this, I present the method of discharging in Graph Theory, used to finally prove the four colour theorem nearly 140 years after it was first stated, which has been used to prove theorems elsewhere in Mathematics.

Applications are invited for a postdoctoral position at the University of Iceland financed by The Icelandic Research Fund. The research project is called:

„Scaling limits of random enriched trees“

and is in the field of probabilistic combinatorics. The project includes studying scaling limits of random graphs, statistical mechanical models on random planar maps and related subjects. The application deadline is March 12, however applications will continue to be accepted until the position is filled.

We are looking for a candidate who has completed a PhD within the last 5 years or is close to defending a PhD thesis. Her/his specialization and interests should be in this area.

Applications should be sent directly by e-mail to sigurdur[at]hi.is, including a CV, list of publications or an abstract of a planned PhD thesis, a research statement and names and e-mail addresses of two referees, who have agreed to provide recommendation.

The appointment is temporary for two years from the 1st of August 2018, or otherwise according to agreement, with a possibility of an extension of one year. All applications will be answered.

For further information please contact:

Ass. Prof. Sigurdur Orn Stefansson (e-mail: sigurdur[at]hi.is)

Applications are invited for a three year PhD position in mathematics at the University of Iceland with a starting date in Fall 2018. The position is funded by a grant from the Icelandic Research Fund.

The successful candidate will work in the area of probabilistic combinatorics with emphasis on scaling limits of random graphs, statistical mechanical models on random planar maps and related subjects. A master degree, or equivalent, in mathematics is required. The application deadline is March 12, however applications will continue to be accepted until the position is filled.

Applications should be sent directly by e-mail to sigurdur[at]hi.is, including a CV, transcripts from undergraduate and master studies, a short description of research interests and names and e-mail addresses of two referees, who have agreed to provide recommendation.

For further information please contact:

Ass. Prof. Sigurdur Orn Stefansson (e-mail: sigurdur[at]hi.is)

Málstofa í stærðfræði

### Fyrirlesari: Hjörtur Björnsson, University of Iceland

### Titill: Lyapunov functions for almost sure exponential stability

Staðsetning: VRII-158

Tími: Mánudagur 27. nóvember kl. 15:00

### Ágrip:

We present a generalization of results obtained by X. Mao in his book „Stochastic Differential Equations and Applications“ (2008). When studying what Mao calls „almost sure exponential stability“, essentially a negative upper bound on the almost sure Lyapunov exponents, he works with Lyapunov functions that are twice continuously differentiable in the spatial variable and continuously differentiable in time. Mao gives sufficient conditions in terms of such a Lyapunov function for a solution of a stochastic differential equation to be almost surely exponentially stable. Further, he gives sufficient conditions of a similar kind for the solution to be almost surely exponentially unstable. Unfortunately this class of Lyapunov functions is too restrictive. Indeed, R. Khasminskii showed in his book „Stochastic Stability of Differential Equations“ (1979/2012) that even for an autonomous stochastic differential equation with constant coefficients, of which the solution is stochastically stable and such that the deterministic part has an unstable equilibrium, there cannot exists a Lyapunov function that is differentiable at the origin. These restrictions are inherited by Mao’s Lyapunov functions. We therefore consider Lyapunov functions that are not necessarily differentiable at the origin and we show that the sufficiency conditions Mao proves can be generalized to Lyapunov functions of this form.

Málstofa í stærðfræði

### Fyrirlesari: Sigurður Freyr Hafstein, University of Iceland

### Titill: Dynamical Systems and Lyapunov functions

Staðsetning: VRII-158

Tími: Mánudagur 13. nóvember kl. 15:00

### Ágrip:

We discuss dynamical systems and the theory of Lyapunov functions and complete Lyapunov functions. Further, we discuss several different numerical methods for the computation of Lyapunov functions and the corresponding estimation of basins of attraction.

Málstofa í stærðfræði

### Fyrirlesari: Sigurður Örn Stefánsson, University of Iceland

### Titill: The phase structure of random outerplanar maps

Staðsetning: VRII-158

Tími: Mánudagur 30. október kl. 15:00

### Ágrip:

An outerplanar map is a drawing of a planar graph in the sphere which has the property that there is a face in the map such that all the vertices lie on the boundary of that face. We study the phase diagram of random outerplanar maps sampled according to non-negative weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its boundary when its number of vertices tends to infinity. The outerplanar maps are then shown to converge in the Gromov-Hausdorff sense towards the α-stable looptree introduced by Curien and Kortchemski (2014), with the parameter α depending on the specific weight-sequence. This allows us to describe the transition of the asymptotic geometric shape from a deterministic circle to the Brownian tree.

Based on arXiv:1710.04460 with Benedikt Stufler.