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Part I: Geometric TopologyExotic Topology in Geometry and DynamicsF. Thomas Farrell, Andrey Gogolev, and Pedro Ontaneda1 The space of negatively curved metrics2 The Teichmiiller and the moduli spaces of negatively curved metrics3 Bundles with negatively curved fibers4 Exotic topology and hyperbolic dynamical systemsReferencesA Survey of Surface Braid Groups and the Lower Algebraic K-Theory of Their Group RingsJohn Guaschi and Daniel Juan-Pineda1 Introduction2 Basic definitions of surface braid groups3 Some properties of surface braid groups4 Braid groups of the sphere and the projective plane5 K-theory of surface braid groupsAcknowledgmentsReferencesThe Isomorphism Conjecture for Groups with Generalized Free Product StructureSayed K. Roushon1 Introduction2 Lower K-theory and surgery L-theory of groups acting on trees3 The isomorphism conjecture for groups acting on treesAcknowledgmentsReferencesGroup Actions on Banach SpacesPiotr W. Nowak1 Introduction2 Preliminaries3 Fixed points4 Metrically proper actions5 Final remarksAcknowledgmentsReferencesTopological Spherical Space FormsIan Hambleton1 Introduction2 First steps: pre-19603 Swan complexes: 1960-19704 Semicharacteristic obstructions and 3-manifolds5 Poincare complexes6 Surgery theory and stable existence7 The finiteness obstruction8 Space forms in the period dimension9 Topological Euclidean space forms10 Further topicsReferencesPart II: Representations and DeformationsDynamics on Character Varieties: A SurveyRichard D. Canary1 Overview2 Basic definitions3 Teichmuller space4 Convex cocompact representations5 Hitchin representations6 Anosov representations7 PSL(2, C)-character varieties8 AppendixReferencesHiggs Bundles and Representations of Complex Hyperbolic LatticesJulien Maubon1 Introduction2 The symmetric space SV(p,q)/S(V(p) �� U(q))3 Representations, flat bundles, Higgs bundles4 Proof of the Milnor-Wood type inequality5 Maximal representationsReferencesGeometric Limits of Kleinian Groups and Their ApplicationsKen'ichi Ohshika1 Introduction2 Preliminaries3 Examples of geometric limits4 Classification of geometric limits5 Boundaries of deformation spacesReferencesDeformations of Surfaces Embedded in 4-dimensional Manifolds and Their Mapping Class GroupsSusumu Hirose1 Introduction2 Preliminaries3 Surfaces standardly embedded in S44 Flexible surfaces in 4-manifolds5 Knotted embeddings in S46 ProblemsAcknowledgmentsReferencesOn the Geography of Lefschetz FibrationsNaoyuki Monden1 Introduction2 Lefschetz fibrations3 Examples of Lefschetz fibrations4 Signatures of Lefschetz fibrations5 Fundamental groups of Lefschetz fibrations6 Some methods to construct Lefschetz fibrations7 Sections of Lefschetz fibrations8 The geography problems of symplectic 4-manifolds and Lefschetz fibrations9 Holomorphic and non-holomorphic Lefschetz fibrations10 The slopes of Lefschetz fibrations with b+ 2= 1AcknowledgmentsReferencesPart III: Geometric GroupsThe Modular Group and Its ActionsA. Muhammed Uludag1 Introduction2 Rise of the modular group3 The bipartite Farey tree F and its boundary4 The projective general linear group over Z5 Action on the upper half plane H6 Modular graphs and dessins7 The action on binary quadratic forms8 Et ceteraAppendix: Gosper's algorithm for continued fraction arithmeticAcknowledgmentsReferencesGroups with Structures and Homotopy GroupsJie Wu1 Introduction2 Simplicial groups3 Simplicial structure on geometric groupsAcknowledgmentsReferencesAbstract Homomorphisms of Algebraic Groups and ApplicationsIgor A. Rapinchuk1 Introduction2 On algebraic rings3 K-theoretic background4 Rigidity results over commutative rings5 Rigidity results over noncommutative rings6 Applications to character varieties and deformations of representationsAcknowledgmentsReferencesPart IV: Geometric Invariants and Growth of Discrete GroupsSome Topics in Geometric Invariant Theory over Non-algebraically Closed Fields1 Introduction2 An overview of geometric invariant theory. Observability and related notions3 Stability in geometric invariant theory over non-algebraically closed fields4 Topology of relative orbits for actions of algebraic groups over completely valued fieldsAcknowledgmentsReferencesComputation of the Spherical Growth Series of Finitely Generated Groups and Monoids by Using AutomataMichihiko Fujii1 Introduction2 Finite-state automata and the subset construction3 Unique-geodesic word acceptors for groups and monoids, and growth series4 Examples of rational growth series and irrational growth series5 The spherical growth series of Artin groups and Artin monoids6 The pure Artin group PI2(k) and the monoid P+ I2(k) of dihedral type7 Automata for geodesic representatives of P+ I2 (k)8 Unique-geodesic word acceptors constructed by subset construction9 The spherical growth series of P+ I2 (k)10 The spherical growth series of P+ I2 (k)AcknowledgmentsReferencesPart V: Music and Group ActionsMathematics and Group Theory in MusicAthanase Papadopoulos1 Introduction2 A brief overview of the interaction between mathematics and music3 The music of Olivier Messiaen4 Rhythm5 Counterpoint6 Modes of limited transpositionReferences