Type: Seminar

Distribution: World

Expiry: 22 May 2012

CalTitle1: Group Actions Seminar: Kalka -- Conjugacy in braid groups with applications in non-commutative and non-associative cryptography

Calendar2: 22 May 2012 1505-1555

CalLoc2: AGR Carslaw 829

CalTitle2: Group Actions Seminar: Reid -- Local Sylow theory of totally disconnected, locally compact groups

Auth: athomas(.pmstaff;2039.2002)@p615.pc.maths.usyd.edu.au

The next Group Actions Seminar will be on Tuesday 22 May at the University of Sydney. Our speakers will be Arkadius Kalka (Queensland) at 12 noon and Colin Reid (Louvain) at 3pm. ---------------------------------------------------------------------------- Date: Tuesday 22 May Time: 12 noon Location: Access Grid Room, Room 829, Carslaw Building, University of Sydney Speaker: Arkadius Kalka (Queensland) Title: Conjugacy in braid groups with applications in non-commutative and non-associative cryptography Abstract: We review braid and Garside groups and the history of conjugacy in this groups. Then we also consider the friends of the conjugacy problem like the subgroup (subCP), shifted (ShCP), simultaneous conjugacy (SCP) and the double coset problem (DCP). In particular we improved invariants for the SCP, developed the first deterministic algorithms for ShCP, and for subCP and DCP for parabolic subgroups of braid groups. This is based on joint work with several coauthors from Bar-Ilan University, Ramat Gan, Israel. Further motivation for these problems comes from non-commutative public key cryptography, and we discuss basic key agreement protocols. Dehornoy’s shifted conjugacy leads us to left-selfdistributive (LD) systems, multi-LD systems, and our new idea of non-associative cryptography. ---------------------------------------------------------------- Date: Tuesday 22 May Time: 3pm Location: Access Grid Room, Room 829, Carslaw Building, University of Sydney Speaker: Colin Reid (Université catholique de Louvain) Title: Local Sylow theory of totally disconnected, locally compact groups Abstract: Totally disconnected, locally compact (t.d.l.c.) groups are a class of topological groups that occur naturally as automorphism groups of locally finite combinatorial structures, such as graphs or simplicial complexes. Compact totally disconnected groups are known as profinite groups, which can also be characterised as inverse limits of finite groups, and some familiar concepts from finite group theory generalise directly to profinite groups. The inverse limits of finite p-groups are known as pro-p groups, and for these we have a generalisation of Sylow’s theorem: given a profinite group G, every pro-p subgroup of G is contained in a maximal pro-p subgroup (a ’p-Sylow subgroup’), and all p-Sylow subgroups of G are conjugate. At the same time, profinite groups play a key role in the general theory of t.d.l.c. groups, because every t.d.l.c. group has an open profinite subgroup, and all such subgroups are commensurable. Thus we can develop a ’local Sylow theory’ for t.d.l.c. groups, based on the Sylow subgroups of their open compact subgroups. Starting from an arbitrary t.d.l.c. group G, we produce a new t.d.l.c. group, the ’p-localisation’ of G: this is naturally determined by G up to isomorphism, embeds in G with dense image, and has an open pro-p subgroup corresponding to a local Sylow subgroup of G. I will describe the construction and some properties of the p-localisation, illustrating the concepts with the example of the automorphism group of a regular tree of finite degree. ----------------------------------------------------------------