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Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.

This chapter focuses on notion of adjoint functor, which applies everything that has been learned so far to unify and subsume all the different universal mapping properties encountered, from free groups to limits to exponentials. It also captures an important mathematical phenomenon that is invisible without the use of the lens of category theory. It is argued that adjointness is a concept of fundamental logical and mathematical importance not captured elsewhere in mathematics. Topics discussed include hom-set definition, examples of adjoints, order adjoints, quantifiers as adjoints, RAPL 197, locally cartesian closed categories, and the adjoint functor theorem.

This chapter presents a third characterization of adjunctions. This one has the virtue of being entirely equational. Topics discussed include the triangle identities, monads and adjoints, algebras for a monad, comonads and coalgebras, and algebras for endofunctors. Exercises are given in the last part of the chapter.

The deformation theoretic techniques of Wiles-Taylor were introduced for elliptic modular forms in the introductory Chapter 1, and are generalized to Hilbert modular forms (following Fujiwara's treatment) in this chapter. In particular, Fujiwara's ‘R=T’ theorem (the identification of the Galois deformation ring and the corresponding Hecke algebra) is proven in the minimal case. In addition to the Taylor-Wiles methods, an explicit formula of the L-invariant (of the adjoint L-functions) as well as an integral solution to Eichler's basis problem are presented for Hilbert modular forms.

This chapter proves the torsion of the anticyclotomic Iwasawa module of a (p-ordinary) CM field, and presents an explicit formula of the L-invariant of the CM field, which is a natural generalization of the formula by Ferrero-Greenberg and Gross-Koblitz from the 1970s for imaginary quadratic fields. These results are proven through the comparison theorem (the ‘R=T’ theorem) between the Iwasawa-theoretic version of the universal deformation ring and the universal nearly p-ordinary Hecke algebra over the (infinite) cyclotomic Iwasawa tower. These combined results enable us to compute the adjoint L-invariant of a CM theta family in terms of the U(p)-eigenvalue of the theta family.

In practice, it is often the case that only partial information on eigenvalues and eigenvectors is available. In many cases, just a few eigenpairs can determine much of the desirable reconstruction. This chapter illustrates this point by concentrating on the Toeplitz structure and the self-adjoint quadratic pencils. The possibility of model updating or tuning applications is discussed.

This chapter drops the assumption of symmetry, or at least it does not assume the dissipativity in a classical sense. The search for a necessary condition for maximal estimates (strong well-posedness in Kreiss' sense) yields the so-called uniform Kreiss-Lopatinskii condition. The chapter investigates the case of a characteristic boundary. It gives practical devices to check the K.-L. condition, including the construction of a Lopatinskii determinant. It shows that the adjoint BVP shares with the direct one the K.-L. condition, a fact exploited in the duality method employed in the existence theory. The latter is carried out for the BVP in weighted (in time) spaces under the assumption that a Kreiss' dissipative boundary symmetrizer exists. Its existence is stated for a constantly hyperbolic operator, but the proof will be seen in the next chapter. The evolutionary property is shown with the use of the Paley-Wiener Theorem. Rauch's Theorem tells what the solution at a given time T can be estimated to; this extends the well-posedness to the full IBVP.

This chapter provides a detailed and self-contained exposition of Hacon and McKernan's construction of pl flips in dimension n assuming minimal models with scaling in dimension n-1. The construction is based on the key notion of an ‘adjoint algebra’. The chapter contains an introduction to multiplier ideals, and the celebrated lifting lemma is developed from first principles. Key ideas of the minimal model program for real pairs are also developed from the ground up.

This chapter describes sensitivity analysis and automatic differentiation (AD). These include the theory, then an object oriented library for AD by operator overloading, and finally the authors' experience with AD systems using code generation operating in both direct and reverse modes. The chapter describes the different possibilities and through simple programs, gives a comprehensive survey of direct AD by operator overloading and for the reverse mode, the adjoint code method. Several elementary and more advanced examples help the understanding of this central concept.

Multiphysics, multiscale models present significant challenges in terms of computing accurate solutions and for estimating the error in information computed from numerical solutions. In this chapter, we discuss error estimation for a widely used numerical approach for multiphysics, multiscale problems called multiscale operator decomposition. In this approach, a multiphysics model is decomposed into components involving simpler physics over a relatively limited range of scales, and the solution is sought through an iterative procedure involving numerical solutions of the individual components. After describing the ingredients of adjoint-based a posteriori analysis, we describe the extension to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, there are several key ideas underlying a general approach to treat operator decomposition multiscale methods.

This chapter develops a basic working knowledge of operators — objects central to quantum mechanics. The fact that quantum operators are linear determines how they may be manipulated, so linearity forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.

This chapter addresses an inverse problem in glaciology, namely how to infer a climatic scenario (i.e. how to reconstruct past polar temperature) from ice volume records. Ice volume observations are available from oceanic records, giving information about sea level, and therefore about the amount of water stored in ice sheets, ice caps, and glaciers. The link between the climatic scenario and the ice sheet volume is complex. The temperature affects the ice mass budget (accumulating minus melting ice). Then, the ice mass budget is a key factor in ice dynamics, which is quite complex in itself (non-Newtonian fluid and nonlinear rheology). The few previous studies of this subject have used fairly simple inverse methods. The idea of the approach described in this chapter is to explore, with idealized twin experiments, the ability of the adjoint method to solve the inverse problem of reconstructing past temperature given all available observations.

This chapter discusses matrices. Matrices appear in many instances across physics, and it is in this chapter that the background necessary for understanding how to use them in calculations is provided. Although matrices can be a little daunting upon first exposure, they are very handy for a lot of classical physics. This chapter reviews the basics of matrices and their operations. It discusses square matrices, adjoint matrices, cofactor matrices and skew-symmetric matrices. The concepts of matrix multiplication, transpose, inverse, diagonal, identity, Pfaffian and determinant are examined. The chapter also discusses the terms Hermitian, symmetric and antisymmetric, as well as the Levi-Civita symbol and Laplace expansion.

Lie groups and Lie algebras are defined. SU(2), the group whose elements are 2x2 unitary unimodular matrices is described providing an example of a 3-dimensional Lie group. Infinitesimal generators are defined and used to provide a basis for a vector space that leads to the Lie algebra. Structure constants are introduced and shown to provide the so-called adjoint representation. The Cartan metric tensor is defined and Cartan’s criterion for semisimplicity is described. After showing that SU(2) is compact SL(2,R) — the group whose elements are 2x2 real unimodular matrices — is introduced to give an example of a non-compact group. Biographical notes on Euler, Lie and Cartan are given.

Modal logic flourished throughout the twentieth century. Kripke provided a semantics in terms of possible world recognized by mathematicians to be an example of varying sets. This allows a formulation in terms of monads generated by adjunctions. Modal homotopy type theory adds the radical idea that modalities apply to all types, not just propositions, so as to make sense of possible steps and necessary ingredients. The proximity is shown between the structures discovered by modal logicians and common ideas in mathematics of stability under variation. We can then reformulate many ideas in current philosophical metaphysical uses of modal logic, such as rigid designators, counterparts, the de re/de dicto distinction, and so on. Worlds are understood as extended contexts, allowing a formulation of counterfactuals. A form of temporal logic is also easily generated in the same vein.

This chapter examines two advanced constructions in category theory: adjoint functors and topoi. Pairs of adjoint functors, or adjunctions, are formally defined and then illustrated with several concrete examples. Topoi are introduced conceptually by focusing on the role of the subobject classifier within a topos, with examples from set theory and graph theory. The difference between the classical logic of Boolean algebras and the non-classical logic of Heyting algebras is explained in terms of their natural mathematical environments of set theory and topos theory respectively.