...This chapter gives explicit descriptions of various moduli stacks in the case d = 1, relating them to moduli stacks of Weil group representations. It describes some of key constructions explicitly in the case where d = 1. It also gives explicit descriptions...

... rejects the notion that individual and group interests are inherently conflicting. The chapter challenges traditional views in evolutionary biology that assume conflict between individual and group interests, suggesting that this premise hinders the investigation of collective behavior evolution...

...This chapter introduces the reader to Artin's classical braid groups Bₙ. The group Bₙ is isomorphic to the mapping class group of a disk with n marked points. Since disks are planar, the braid groups lend themselves to special pictorial...

...This chapter specializes the theory back to the case of local Shimura varieties, and explains the relation with Rapoport-Zink spaces. It begins with a local Shimura datum. A local Shimura datum is a triple (G, b, µ) consisting of a reductive group...

... isocrystals and Higgs crystals, cohomology of Higgs isocrystals, and representations of the fundamental group, the chapter presents the main result: the construction of a fully faithful functor from the category of Higgs (iso)crystals satisfying an overconvergence condition to that of small generalized...

... representations” of the étale fundamental group is constructed for curves over a p-adic field. The definition of “generalized representations” uses p-adic Hodge theory and almost étale coverings, and it includes usual representations which form a full subcategory. The equivalence...

...This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction. Let G be a topological group and M a left G-space. The Borel construction mixes...

... the module structure of equivariant cohomology. Suppose a topological group G acts on the left on a topological space M. Let EG → BG be a universal G-bundle. The homotopy quotient M_G fits...

...This chapter studies vector-valued forms. Ordinary differential forms have values in the field of real numbers. This chapter allows differential forms to take values in a vector space. When the vector space has a multiplication, for example, if it is a Lie algebra or a matrix group, the vector...

...This chapter illustrates the Maurer-Cartan form. On every Lie group G with Lie algebra g, there is a unique canonically defined left-invariant g-valued 1-form called the Maurer-Cartan form. The chapter describes the Maurer-Cartan form...

...This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex...

...This chapter looks at the Cartan model. Specifically, it generalizes the Cartan model from a circle action to a connected Lie group action. The chapter assumes the Lie group to be connected, because the condition that LX α = 0 is sufficient for a differential form α on M...

...This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular...

...This chapter deals with the semi-group on the space Β⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components before discussing the null-space of the adjoint under the hypothesis...

...This chapter deals with central extensions and groups locally of minimal type. It begins with a discussion of the general lemma on the behavior of the scheme-theoretic center with respect to the formation of central quotient maps between pseudo-reductive groups; this lemma generalizes a familiar...

...This chapter describes the construction of canonical central extensions that are analogues for perfect smooth connected affine k-groups of the simply connected central cover of a connected semisimple k-group. A commutative affine k-group scheme...

...This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root system Bn over ks...

...This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification...

.... It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative...

...This chapter presents the proof for the Fundamental Theorem of Descent in buildings: that if Γ is a descent group, the set of residues of a building Δ that are stabilized by a subgroup Γ of Aut(Γ) forms a thick building. It begins with the hypothesis: Let Π be an arbitrary Coxeter diagram, let...