Vorlesung

Dozent: Volker Diekert

Übungen: Armin Weiß

Zeit	Raum	Termine
Mo 17:20–18:50	0.124	am 29.04. in Raum 0.463
Do 17:30–19:00	0.108

Inhalt

Bereits 1911 formulierte Max Dehn drei fundamentale algorithmische Probleme in der (kombinatorischen) Gruppentheorie:

Wortproblem: Ist ein gegebenes Gruppenelement (als Wort in Erzeugern) das Einselement in der Gruppe?
Konjugationsproblem: Sind zwei Elemente konjugiert?
Isomorphieproblem: Definieren zwei gegebene Darstellungen isomorphe Gruppen?

Im Allgemeinen sind alle diese Fragen unentscheidbar, also kann man positive Antworten nur in Spezialfällen erhalten. Die weitreichensten Ergebnisse liegen für das Wortproblem vor. Hier gibt es eine große Klasse von Gruppen, die in der Praxis auftreten und für die man sehr gute Algorithmen kennt.

Die Vorlesung wiederholt zunächst die grundlegenden Techniken der algorithmischen Gruppentheorie. Hierzu gehört insbesondere die Theorie der konvergenten Ersetzungssysteme. Diese werden benutzt, um die Bass-Serre-Theorie herzuleiten, die heute ein Fundament der kombinatorische Gruppentheorie bildet. Aufbauend auf die Bass-Serre Theorie wird die Theorie kontext-freier bzw. virtuell freier Gruppen entwickelt. Die beleuchtet insbesondere eine enge Verflechtung von Konzepten der Grupppentheorie, Graphentheorie sowie Formalen Sprachen.

Folien und Skript zur Vorlesung

Aktuelle Folien (Stand: 30.04.2019)
Folien Algorithmische Gruppentheorie vom Wintersemester 2018/19 (Stand: 01.02.2019)

Übungsblätter

Blatt 1 (Besprechung am 15.04.)
Blatt 2 (Besprechung am 06.05.)
Blatt 3 (Besprechung am 04.06.)

Scheinbedingungen

Scheinbedingung ist eine aktive Teilnahme an den Übungen, insbesondere sollte mindestens einmal vorgerechnet werden. Die Erfüllung der Scheinkriterien ist eine notwendige Voraussetzung um zur Prüfung zugelassen zu werden.

Literatur

Volker Diekert, Manfred Kufleitner, Gerhard Rosenberger: Diskrete algebraische Methoden, Walter de Gruyter, 2013.
Diekert, Weiß: Context-Free Groups and Bass-Serre Theory. In Algorithmic and Geometric Topics Around Free Groups and Automorphisms, Advanced Courses in Mathematics, Birkhäuser, 2017.
Lyndon, Schupp: Combinatorial Group Theory, Springer, 1977.
Serre: Trees, Springer, 1980.

News

[Jun’23] The paper “Parallel algorithms for power circuits and the word problem of the Baumslag group” by Caroline Mattes and Armin Weiß has been accepted at Computational Complexity.

[Oct’22] The paper “Lower Bounds for Sorting 16, 17, and 18 Elements” by Florian Stober and Armin Weiß has been accepted at ALENEX 2023.

[Sep’22] The paper “Conelikes and Ranker Comparisons” by Viktor Henriksson and Manfred Kufleitner has been accepted at LATIN 2022.

[Sep’22] The paper “Improved Parallel Algorithms for Generalized Baumslag Groups” by Caroline Mattes and Armin Weiß has been accepted at LATIN 2022.

[Apr’22] The paper “Reachability Games and Parity Games” by Volker Diekert and Manfred Kufleitner has been accepted at ICTAC 2022.

[Apr’22] The paper “Satisfiability Problems for Finite Groups” by Pawel M. Idziak, Piotr Kawalek, Jacek Krzaczkowski and Armin Weiß has been accepted at ICALP 2022.

All news

[Mar’22] The paper “The Power Word Problem in Graph Products” by Florian Stober and Armin Weiß was accepted at DLT 2022.

[Nov’20] Volker Diekert is Partner Investigator in the Australian ARC grant “Geodetic groups: foundational problems in algebra and computer science” at University of Technology Sydney.

[Apr’20] The paper “Groups with ALOGTIME-hard word problems and PSPACE-complete circuit value problems” by Laurent Bartholdi, Michael Figelius, Markus Lohrey and Armin Weiß has been accepted at CCC 2020.

[Apr’20] The paper “Hardness of equations over finite solvable groups under the exponential time hypothesis” by Armin Weiß has been accepted at ICALP 2020.

[Dec’19] The paper “An Automaton Group with PSPACE-Complete Word Problem” by Jan Philipp Wächter and Armin Weiß has been accepted at STACS 2020.

[Nov’19] Carlos Camino was awarded the stuvus Special Prize for exceptional commitment in teaching.

[Jun’19] The paper “The power word problem” by Markus Lohrey and Armin Weiß has been accepted at MFCS 2019.

[May’19] The paper “On the Average Case of MergeInsertion” by Florian Stober and Armin Weiß has been accepted at IWOCA 2019.

[Oct’18] The paper “Worst-Case Efficient Sorting with QuickMergesort” by Stefan Edelkamp and Armin Weiß has been accepted at ALENEX 2019.

[Jun’18] At CCC 2018, Lukas Fleischer received a Best Student Paper Award for his submission “On the Complexity of the Cayley Semigroup Membership Problem”.

[Jun’18] The paper “Testing Simon’s congruence” by Lukas Fleischer and Manfred Kufleitner was accepted at MFCS 2018.

[Jun’18] The paper “The Intersection Problem for Finite Semigroups” by Lukas Fleischer was accepted at DLT 2018.

[Apr’18] The paper “The isomorphism problem for finite extensions of free groups is in PSPACE” by Géraud Sénizergues and Armin Weiß was accepted at ICALP 2018.

[Apr’18] The paper “On the Complexity of the Cayley Semigroup Membership Problem” by Lukas Fleischer was accepted at CCC 2018.

[Jan’18] On March 24-29, 2019 Volker Diekert, Markus Lohrey, Olga Kharlampovich and Alexei Miasnikov will organize the Schloss Dagstuhl Seminar “Algorithmic Problems in Group Theory”.

[Dec’17] The paper “The Intersection Problem for Finite Monoids” by Lukas Fleischer and Manfred Kufleitner was accepted at STACS 2018.

[Jun’17] At the 12th International Computer Science Symposium in Russia (CSR), Lukas Fleischer and Manfred Kufleitner received a Best Paper Award for their publication “Green’s Relations in Finite Transformation Semigroups”, and Armin Weiss received a Best Paper Award for “The conjugacy problem in free solvable groups and wreath product of abelian groups is in $\text{TC}^0$ “ which is joint work with Alexei Miasnikov and Svetla Vassileva.