Bjarki Ágúst Guðmundsson (18/09/17)

Anders Claesson, September 13, 2017

Math Colloquium

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.

Nuno Romao (15/09/17)

Anders Claesson, September 11, 2017

Math Colloquium

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.

Uwe Leck and Ian Roberts (01/09/17)

Anders Claesson, August 29, 2017

Math Colloquium

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.

Marston Conder (13/07/17)

Anders Claesson, July 12, 2017

Math Colloquium

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.

Daniel Fernandez Moreno (07/07/17)

Anders Claesson, July 4, 2017

Math Colloquium

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.

Christopher Kellett (06/07/17)

Anders Claesson, July 4, 2017

Math Colloquium

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.

Peter J. Olver (09/06/17)

Anders Claesson, June 7, 2017

Math Colloquium

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.

Örn Arnaldsson (06/06/17)

Anders Claesson, June 2, 2017

PhD thesis defense rehearsal

Title: Involutive moving frames

Location: Tg-227 (Tæknigarður, 2. hæð)
Time: Tuesday 6 June at 10:50

Abstract:

Cartan’s equivalence method and the method of the equivariant moving frames are the two best known methods for solving equivalence problems in differential geometry, differential equations, calculus of variations and control theory. My thesis demonstrates how the two methods are really two sides of the same coin and combines them in a powerful hybrid method that increases the computational efficiency of both of its progenitors. This novel viewpoint provides effortless proofs to some previously hard-to-prove results, such as the Lie-Tresse theorem. Furthermore, we obtain proof of a long standing conjecture of Cartan on the termination of his equivalence method.

Guðmundur Helgason (01/06/17)

Benedikt Magnússon, June 1, 2017

Thesis presentation in Master of Applied Statistics (MAS)

Guðmundur Helgason Titill: Hversu lengi þarf ég að bíða? Forspárlíkön fyrir biðtíma í þjónustuveri CCP

Location: V-157, VRII
Time: Thursday  1. June at14:00.

Abstrat:

Í þessari rannsókn, með hjálp ýmissa tölfræðiaðferða, spáum við fyrir um biðtíma eftir svari við tölvupósti með gögnum frá þjónustuveri CCP, framleiðanda  tölvuleiksins EVE Online. Að mestu leyti er notast við tvíkosta tölfræðilíkön þar sem spáð er fyrir um hvort að svar sé gefið fyrir ákveðinn tímapunkt eða ekki. Samfelldar aðferðir eru þó einnig notaðar, bæði til að spá fyrir um biðtíma í sjálfu sér og hvort svar sé gefið fyrir ákveðinn tíma eða ekki. Auk greiningarlegra aðferða til forspáar er einnig notast við einfaldari empírískar aðferðir til að meta dreifingu biðtíma og líkindi á svari eftir ákveðinn tíma. Tiltækar rannsóknir á sviði þjónustuvera, gæða í þjónustu, áhrifa þess að bíða eftir þjónustu og aðferða sem notast hefur verið við til að spá fyrir um biðtíma eru skoðaðar. Aðferðirnar sem notast var við til biðtíma forspáar eru bornar saman, kostir þeirra og gallar ræddir, auk hugsanlegra hagnýtra eiginleika.

Leiðbeinendur:
Anna Helga Jónsdóttir
Matthías Kormáksson

Prófdómari: Thor Aspelund

Stella Kristín Hallgrímsdóttir

Benedikt Magnússon, May 29, 2017

Thesis presentation in Master of Applied Statistics (MAS)

Stella Kristín Hallgrímsdóttir Title: Samband veðurs og komufjölda á bráðamóttökur Landspítala

Location: V-157, VRII
Time: Monday 29. May at 14:00.

Abstract:

The objective of this project is to study the seasonal and weekly fluctuations in number of arrivals to the emergency departments of the University Hospital of Iceland and also to assess the influence of weather on the number of arrivals. Four emergency departments were examined; the Emergency Department in Fossvogur, the Emergency Unit in the Children‘s Hospital Department, Hjartagátt which is the emergency department for people with suspected acute heart problems, and the Psychiatry Emergency Department. The weather variables that were mostly looked into are temperature, wind speed, precipitation and cloudiness. Seasonal fluctuations were modeled with sine and cosine waves and with the help of linear regression a new variable was made that describes the seasonal fluctuations and linear increase in the number of arrivals. A few ARIMA models were built to predict the number of arrivals in the Emergency Department in Fossvogur and in the Children‘s Emergency Department. The models were compared to find the best prediction model for each department. To assess whether weather affects the number of arrivals in the emergency departments, the weather variables were added one by one to the best prediction model for each department to see if the model‘s prediction root-mean squared error (RMSE) decreases when information about weather is added to the model. Principal components analysis was also used to combine the weather variables into fewer new variables. The new variables were then added to the ARIMA models to assess their effect on the goodness of the models. The results show that adding the weather information slightly decreases prediction RMSE in the Emergency Department in Fossvogur but increases it for the Children‘s Emergency Department. That both applies to when each weather variable was looked into separately and when the principal components were used. Therefore, it can be concluded that weather does not affect the number of arrivals to the Children‘s Emergency Department but it has a minor effect on the number of arrivals to the Emergency Department in Fossvogur. Furthermore, the results show that a good prediction model for the number of arrivals to the emergency departments can be developed only using calendar variables.

Advisors: Dr. Sigrún Helga Lund and Dr. Tryggvi Helgason
Examiner: Dr. Ólafur Pétur Pálsson