Math Colloquium

### Speaker: Hjörtur Björnsson, University of Iceland

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

Location: VRII-158

Time: Monday 27 November at 15:00

### Abstract:

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.

Math Colloquium

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

### Title: Dynamical Systems and Lyapunov functions

Location: VRII-158

Time: Monday 13 November at 15:00

### Abstract:

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.

Math Colloquium

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

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

Location: VRII-158

Time: Monday 30 October at 15:00

### Abstract:

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.

Math Colloquium

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

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

Location: VRII-158

Time: Monday 18 September at 15:00

### Abstract:

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.

Math Colloquium

### Speaker: Nuno Romao, IHES

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

Location: Naustið, Endurmenntun

Time: Friday 15 September at 13:30

### Abstract:

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.

Math Colloquium

### Speaker: Uwe Leck and Ian Roberts, University of Flensburg and Darwin

### Title: Extremal problems for finite sets related to antichains

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 1 September at 13:30

### Abstract:

This talk requires little more than mathematical maturity as it relates to problems concerning finite sets. The basic ideas are simple, and the problems are easy to state, but the problems range from simple to very hard.

An antichain (or Sperner family) in the Boolean lattice \(B_n\) is a collection \(A\) of subsets of \([n] = \{1,2,…,n\}\) such that no set in \(A\) is a subset of another. By Sperners famous theorem, the largest possible cardinality of an antichain in \(B_n\) is \({n \choose\lfloor n/2\rfloor}\). Antichains are fundamental in extremal set theory; but also with applications in other areas such as Search Theory.

We will discuss several extremal problems and results involving antichains such as: minimizing the union-closure of uniform antichains of a given size; finding the number of different antichains in \(B_n\); determining the possible cardinalities of maximal antichains; and others.

Some of the problems are solved and some provide tantalising unsolved problems.

Math Colloquium

### Speaker: Marston Conder, University of Auckland

### Title: Experimental Algebra and Combinatorics

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Thursday 13 July at 11:00

### Abstract:

Some 40 years after the computer-based proof of the 4-Colour Theorem by Appel and Haken, there is still a degree of healthy skepticism about the use of computers to prove nice theorems in mathematics. But there is a distinction between proofs that are highly dependent on computation (verifiable or otherwise), and the use of computer-based experimentation to analyse and construct examples, to produce data that might exhibit patterns from which conjectures can be drawn and tested, or to investigate a range of possible scenarios — subsequently leading to theorems that can be proved by hand.

In this talk I will describe a range of instances of experimental computations involving finite and infinite groups that have led to unexpected but theoretically provable discoveries about discrete objects possessing a high degree of symmetry. These include discoveries about the genus spectra of particular classes of regular maps on surfaces, the smallest regular and chiral polytopes, and various kinds of edge-transitive graphs. Such examples highlight the value of experimental computation, and the surprising outcomes it can often produce.

Math Colloquium

### Speaker: Daniel Fernandez Moreno, University of Iceland

### Title: Entanglement entropy at non-equilibrium in holography

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 7 July at 13:20

### Abstract:

In recent years, holographic models have proved to be successful at studying far-from-equilibrium physics. This provides a new approach to studying quantum quenches in strongly coupled systems. In this talk, based on ArXiv:1705.04696, I will focus on the local quench-like time evolution obtained by joining two 1+1 dimensional heat baths at different temperatures. I will present results for the entanglement entropy of different entangling regions obtained by adapting the time-dependent Hubeny-Rangamani-Takayanagi prescription. The interest of this study relies on the presence of emergent collective behavior, which can provide insight into the interplay between quantum effects and out of equilibrium physics.

Math Colloquium

### Speaker: Christopher Kellett, University of Newcastle (Australia)

### Title: Back and Forth in Lyapunov’s Second Method: Non-Uniform Subtleties

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Thursday 6 July at 13:20

### Abstract:

Lyapunov’s second or direct method provides an easy-to-check sufficient condition for stability properties of equilibria. The converse question – given a stability property, does there exist an appropriate Lyapunov function? – has been fundamental in differentiating and classifying different stability properties, particularly with regards to “uniform” stability.

In this talk, I will review the usual textbook definitions for Lyapunov functions for time-varying systems and describe where they are deficient. Some interesting new sufficient (and probably necessary) conditions pop up along the way.

Math Colloquium

### Speaker: Peter J. Olver, University of Minnesota

### Title: Equivalence, Invariants, Puzzles, and Cancer

Location: Tg-227 (Tæknigarður, 2. hæð)

Time: Friday 9 June at 16:00

### Abstract:

A fundamental issue in computer vision is recognizing when two objects in an image are the “same”. The underlying mathematical apparatus for studying such equivalence problems is transformation group (or, more generally, groupoid) theory. Cartan’s solution to the equivalence and symmetry problem for submanifolds relies on the associated geometric invariants, through what is now known as the differential invariant signature. Furthermore, the new equivariant approach to the method of moving frames provides a systematic and algorithmic approach that can be applied to very general Lie group and even Lie pseudo-group actions. The talk will conclude with recent applications to automated assembly of broken objects, such as jigsaw puzzles, and cancer detection.