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This chapter explores Shou-Wu Zhang's minicourse on Faltings heights and L-functions. It essentially consists of three parts. The first part discusses conjectures and results in the literature which give bounds, or formulae in terms of L-functions, for “Faltings heights.” The authors also mention various applications of such conjectures and results. The second part is devoted to the work of Yuan–Zhang in which they proved the averaged Colmez conjecture. Here, the authors detail the main ideas and concepts used in their proof. The third part focuses on the work of Yuan–Zhang in the function field world. Therein they compute special values of higher derivatives of certain automorphic L-functions in terms of self-intersection numbers of Drinfeld–Heegner cycles on the moduli stack of shtukas. The result of Yuan–Zhang might be viewed as a higher Gross–Zagier/Chowla–Selberg formula in the function field setting. The authors then motivate and explain the philosophy that Chowla–Selberg type formulae are special cases of Gross–Zagier type formulae coming from identities between geometric and analytic kernels.

In quantum electrodynamics, the matter degrees of freedom are coupled to the radiation field through a local, gauge-invariant interaction density. Due to the linear structure of this coupling, the problem of constructing a complete formal representation of the reduced matter dynamics can be solved exactly if the electromagnetic radiation field is initially in a Gaussian state. This chapter combines field theoretical methods with superoperator and influence functional techniques to derive an exact, relativistic representation for the reduced density matrix of the matter degrees of freedom which completely describes the influence of the electromagnetic radiation field on the matter dynamics. Several applications are treated, such as the suppression of the quantum coherence of charged particles caused by the emission of bremsstrahlung and the decoherence of many-particle states.

This introductory chapter sets out the book's purpose, which is to study Weil's conjecture over function fields: that is, fields K which arise as rational functions on an algebraic curve X over a finite field F_q. It reformulates Weil's conjecture as a mass formula, which counts the number of principal G-bundles over the algebraic curve X. An essential feature of the function field setting is that the objects that we want to count (in this case, principal G-bundles) admit a “geometric” parametrization: they can be identified with Fq-valued points of an algebraic stack Bun_G(X). This observation is used to reformulate Weil's conjecture yet again: it essentially reduces to a statement about the ℓ-adic cohomolog of Bun_G(X), reflecting the heuristic idea that it should admit a “continuous Künneth decomposition”.

The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.

Chapter 11 employs variational differential techniques and the Schwinger Action Principle to derive coupled-field Green’s function equations for a multi-component system, modeled as an interacting electron-hole-phonon system. The coupled Fermion Green’s function equations involve five interactions (electron-electron, hole-hole, electron-hole, electron-phonon, and hole-phonon). Starting with quantum Hamilton equations of motion for the various electron/hole creation/annihilation operators and their nonequilibrium average/expectation values, variational differentiation with respect to particle sources leads to a chain of coupled Green’s function equations involving differing species of Green’s functions. For example, the 1-electron Green’s function equation is coupled to the 2-electron Green’s function (as earlier), also to the 1-electron/1-hole Green’s function, and to the Green’s function for 1-electron propagation influenced by a nontrivial phonon field. Similar remarks apply to the 1-hole Green’s function equation, and all others. Higher order Green’s function equations are derived by further variational differentiation with respect to sources, yielding additional couplings. Chapter 11 also introduces the 1-phonon Green’s function, emphasizing the role of electron coupling in phonon propagation, leading to dynamic, nonlocal electron screening of the phonon spectrum and hybridization of the ion and electron plasmons, a Bohm-Staver phonon mode, and the Kohn anomaly. Furthermore, the single-electron Green’s function with only phonon coupling can be rewritten, as usual, coupled to the 2-electron Green’s function with an effective time-dependent electron-electron interaction potential mediated by the 1-phonon Green’s function, leading to the polaron as an electron propagating jointly with its induced lattice polarization. An alternative formulation of the coupled Green’s function equations for the electron-hole-phonon model is applied in the development of a generalized shielded potential approximation, analysing its inverse dielectric screening response function and associated hybridized collective modes. A brief discussion of the (theoretical) origin of the exciton-plasmon interaction follows.

Chapter 12 introduces Graphene, which is a two-dimensional “Dirac-like” material in the sense that its energy spectrum resembles that of a relativistic electron/positron (hole) described by the Dirac equation (having zero mass in this case). Its device-friendly properties of high electron mobility and excellent sensitivity as a sensor have attracted a huge world-wide research effort since its discovery about ten years ago. Here, the associated retarded Graphene Green’s function is treated and the dynamic, non-local dielectric function is discussed in the degenerate limit. The effects of a quantizing magnetic field on the Green’s function of a Graphene sheet and on its energy spectrum are derived in detail: Also the magnetic-field Green’s function and energy spectrum of a Graphene sheet with a quantum dot (modelled by a 2D Dirac delta-function potential) are thoroughly examined. Furthermore, Chapter 12 similarly addresses the problem of a Graphene anti-dot lattice in a magnetic field, discussing the Green’s function for propagation along the lattice axis, with a formulation of the associated eigen-energy dispersion relation. Finally, magnetic Landau quantization effects on the statistical thermodynamics of Graphene, including its Free Energy and magnetic moment, are also treated in Chapter 12 and are seen to exhibit magnetic oscillatory features.

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category D_ℓ(X). This chapter attempts to remedy the situation by introducing a mathematical object Shv_ℓ (X), which refines the triangulated category D_ℓ (X). This object is not itself a category but instead is an example of an ∞-category, which is referred to as the ∞-category of ℓ-adic sheaves on X. The triangulated category D_ℓ (X) can be identified with the homotopy category of Shv_ℓ (X); in particular, the objects of D_ℓ (X) and Shv_ℓ (X) are the same. However, there is a large difference between commutative algebra objects of D_ℓ (X) and commutative algebra objects of the ∞-category Shv_ℓ (X). We can achieve (b') by viewing the complex B as a commutative algebra of the latter sort.