## Lecture SeriesMladen Bestvina (University of Utah)The geometry of mapping class groups and Out(F_n)I will review the basic theory, due to Masur-Minsky, of the geometry of mapping class groups, including the hyperbolicity of the curve complex and the subsurface projections. We will then consider the situation for Out(F_n), which is much less understood and where there is a lot of current research. Dave Witte Morris (University of Lethbridge)Arithmetic subgroups of SL(n,R)SL(2,Z) is an "arithmetic" subgroup of SL(2,R). The other arithmetic subgroups are not as obvious, but they can be constructed by using quaternion algebras. Replacing the quaternion algebras with larger division algebras yields many arithmetic subgroups of SL(n,R), with n > 2. In fact, a calculation of group cohomology shows that the only other way to construct arithmetic subgroups of SL(n,R) is by using unitary groups. |
Joan Porti (Universitat Autònoma de Barcelona)Dynamics at infinity of symmetric spacesThis talk discusses the dynamical behaviour at infinity of discrete groups of isometries of symmetric spaces. The situation in higher rank is very different from rank one (e.g. hyperbolic spaces). I plan to start with a very brief introduction to higher rank symmetric spaces and to provide examples. This is joint work with M. Kapovich and B. Leeb. |

## Research TalksDelaram Kahrobaei (Graduate Center, CUNY)Polycyclic groups: A Secure Platform for Ko-Lee ProtocolThe Ko-Lee protocol was proposed in 2000 and published in CRYPTO. It generalizes the Diffie-Hellman protocol-which is based on the difficulty of the discrete log problem in a finite abelian group- to non-commutative version which uses the hardness of search conjugacy problem in certain infinite non-abelian groups. Particularly braid groups were proposed as the platform for this protocol by Ko-Lee. In 2002, there was an attack to this cryptosystem, namely the length-based attack. This attack was proposed mainly to break the Anshel-Anshel-Goldfeld commutator key exchange protocol (1999) using braid groups. In 2004 together with Bettina Eick (Germany) we proposed infinite, non-virtually nilpotent, polycyclic groups with high Hirsch length as a new platform for Ko-Lee protocol. Such groups are finitely presented, finding normal form is efficiently solvable, they have exponential growth rate and the search conjugacy problem has conjectured to be solved in exponential time using proper experiments. Recently together with my PhD student Ha T. Lam and David Garber (Israel) we showed that polycyclic groups with high enough Hirsch length are secure against the length-based attacks. In this talk I will survey on a couple of non-commutative public keys and digital signatures that use conjugacy search problem. Then I will talk about the length based attack and how our algorithms work for polycyclic groups. Youngju Kim (KIAS) Quasiconformal deformations of Schottky groups in complex hyperbolic spaceWe generalize a Schottky group construction to complex hyperbolic space and study its quasiconformal deformation in a complex hyperbolic quasi-Fuchsian space. In particular, we construct a fundamental domain whose sides consist of disjoint non-asymptotic packs for the action of the Schottky group acting on complex hyperbolic space. Then we prove that a smooth deformation of such a Schottky group is quasiconformally stable. Thilo Kuessner (KIAS)Proportionality principle for simplicial volumeThe proportionality principle for the simplicial volume states that the quotient of simplicial and geometric volume of a Riemannian manifold depends only on the isometry class of its universal covering. In joint work with Sungwoon Kim we extend the proportionality principle to noncompact manifolds of pinched negative curvature. |
Jung Hoon Lee (Chonbuk National University)Topologically minimal surfacesTopologically minimal surfaces are defined by Bachman as topological analogues of geometrically minimal surfaces. We introduce known results on topologically minimal surfaces and discuss about topologically minimal surfaces in the 3-sphere. Sangyop Lee (Chung-Ang University)Twisted torus knotsTwisted torus knots are obtained by adding full twists to some parallel strands of torus knots. We will discuss some properties of these knots. Seonhee Lim (Seoul National University)Subword complexity and Sturmian colorings of treesWe define subword complexity of colorings of trees as a generalization of subword complexity of words. We characterize those of bounded subword complexity. We also investigate colorings of minimal unbounded subword complexity, which we call Sturmian colorings. This is a joint work with Dong Han Kim. Ken'ichi Ohshika (Osaka U)Geometric limits and deformation spaces of Kleinian groupsI shall discuss how geometric limits help to understand the defamation spaces of Kleinian groups. In particular, I shall introduce a new kind of compactification of Teichmuller space using geometric limits. Catherine Pfaff (Universite d'Aix-Marseille)Stratifying the set of fully Irreducible elements of Out(F_r)By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmuller flow invariant stratification of the space of quadratic di fferentials. We give a first step to an Out(F_r) analog of the Masur-Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher give a strictly finer invariant in the analogous Out(F_r) setting of a fully irreducible outer automorphism, we determined which of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in Out(F_3). |

designed by Sang-hyun Kim