Málstofa í stærðfræði

### Fyrirlesari: Tony Guttmann, The University of Melbourne

### Titill: On the number of Av(1324) permutations

Staðsetning: V-147 (VR-II)

Tími: Mánudagur 28. maí kl. 10:50

### Ágrip:

We give an improved algorithm for counting the number of 1324-avoiding permutations, resulting in 14 further terms of the generating function, which is now known to length 50.

We re-analyse the generating function and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of 1324-avoiding permutations of length n behaves as \(B\cdot \mu^n \cdot \mu_1^{\sqrt{n}} \cdot n^g\).

We estimate \(\mu = 11.600 \pm 0.003.\) The presence of the stretched exponential term \(\mu_1^{\sqrt{n}}\) is an unexpected feature of the conjectured solution, but we show that such a term is present in a number of other combinatorial problems.

(A.J. Guttmann with A.R. Conway and P. Zinn-Justin)

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.

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.

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.

Málstofa í stærðfræði

### Fyrirlesari: Bjarki Ágúst Guðmundsson, University of Iceland

### Titill: Enumerating permutations sortable by k passes through a pop-stack

Staðsetning: VRII-158

Tími: Mánudagur 18. september kl. 15:00

### Ágrip:

In an exercise in the first volume of his famous series of books, Knuth considered sorting permutations by passing them through a stack. He noted that, out of the \(n!\) permutations on \(n\) elements, \(C_n\) of them can be sorted by a single pass through a stack, where \(C_n\) is the \(n\)-th Catalan number. Many variations of this exercise have since been considered, including allowing multiple passes through the stack and using different data structures. West classified the permutations that are sortable by 2 passes through a stack, and a formula for the enumeration was later proved by Zeilberger. The permutations sortable by 3 passes through a stack, however, have yet to be enumerated. We consider a variation of this exercise using pop-stacks. For any fixed \(k\), we give an algorithm to derive a generating function for the permutations sortable by \(k\) passes through a pop-stack. Recently the generating function for \(k=2\) was given by Pudwell and Smith (the case \(k=1\) being trivial). Running our algorithm on a computer cluster we derive the generating functions for \(k\) at most 6. We also show that, for any \(k\), the generating function is rational.

Málstofa í stærðfræði

### Fyrirlesari: Nuno Romao, IHES

### Titill: Vortex moduli and the physics of gauged sigma-models

Staðsetning: Naustið, Endurmenntun

Tími: Föstudagur 15. september kl. 13:30

### Ágrip:

Vortices appear as static and stable solutions in field theories known as gauged sigma-models; these are defined on surfaces and can have both linear and nonlinear targets. I will give an overview of recent results concerning the underlying moduli spaces (parametrizing all vortex configurations up to gauge equivalence with fixed topology) and explain their physical significance. My talk will focus on the case where the target is a two-sphere with circle action; in this simple nonlinear model, many important questions can be answered at least in particular examples.