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Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soulé arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Arithmetical algebraic geometry. --- Shimura varieties. --- Varieties, Shimura --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Arithmetical algebraic geometry --- Number theory --- Abelian group. --- Addition. --- Adjunction formula. --- Algebraic number theory. --- Arakelov theory. --- Arithmetic. --- Automorphism. --- Bijection. --- Borel subgroup. --- Calculation. --- Chow group. --- Coefficient. --- Cohomology. --- Combinatorics. --- Compact Riemann surface. --- Complex multiplication. --- Complex number. --- Cup product. --- Deformation theory. --- Derivative. --- Dimension. --- Disjoint union. --- Divisor. --- Dual pair. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elliptic curve. --- Endomorphism. --- Equation. --- Explicit formulae (L-function). --- Fields Institute. --- Formal group. --- Fourier series. --- Fundamental matrix (linear differential equation). --- Galois group. --- Generating function. --- Green's function. --- Group action. --- Induced representation. --- Intersection (set theory). --- Intersection number. --- Irreducible component. --- Isomorphism class. --- L-function. --- Laurent series. --- Level structure. --- Line bundle. --- Local ring. --- Mathematical sciences. --- Mathematics. --- Metaplectic group. --- Modular curve. --- Modular form. --- Modularity (networks). --- Moduli space. --- Multiple integral. --- Number theory. --- Numerical integration. --- Orbifold. --- Orthogonal complement. --- P-adic number. --- Pairing. --- Prime factor. --- Prime number. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Quadratic form. --- Quadratic residue. --- Quantity. --- Quaternion algebra. --- Quaternion. --- Quotient stack. --- Rational number. --- Real number. --- Residue field. --- Riemann zeta function. --- Ring of integers. --- SL2(R). --- Scientific notation. --- Shimura variety. --- Siegel Eisenstein series. --- Siegel modular form. --- Special case. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Test vector. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Trace (linear algebra). --- Triangular matrix. --- Two-dimensional space. --- Uniformization. --- Valuative criterion. --- Whittaker function.

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Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors.An essential primer for undergraduates making the leap to graduate work, the book begins with free groups-actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples.Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Geometric group theory. --- "ient. --- 4-valent tree. --- Cantor set. --- Cayley 2-complex. --- Cayley graph. --- Coxeter group. --- DSV method. --- Dehn function. --- Dehn twist. --- Euclidean space. --- Farey complex. --- Farey graph. --- Farey tree. --- Gromov hyperbolicity. --- Klein's criterion. --- Milnor-Schwarz lemma. --- Möbius transformation. --- Nielsen-Schreier Subgroup theorem. --- Perron-Frobenius theorem. --- Riemannian manifold. --- Schottky lemma. --- Thompson's group. --- asymptotic dimension. --- automorphism group. --- automorphism. --- bi-Lipschitz equivalence. --- braid group. --- braids. --- coarse isometry. --- combinatorics. --- compact orientable surface. --- cone type. --- configuration space. --- context-free grammar. --- curvature. --- dead end. --- distortion. --- endomorphism. --- finite group. --- folding. --- formal language. --- free abelian group. --- free action. --- free expansion. --- free group. --- free nonabelian group. --- free reduction. --- generators. --- geometric group theory. --- geometric object. --- geometric space. --- graph. --- group action. --- group element. --- group ends. --- group growth. --- group presentation. --- group theory. --- group. --- homeomorphism. --- homomorphism. --- hyperbolic geometry. --- hyperbolic group. --- hyperbolic space. --- hyperbolicity. --- hyperplane arrangements. --- index. --- infinite graph. --- infinite group. --- integers. --- isoperimetric problem. --- isoperimetry. --- jigsaw puzzle. --- knot theory. --- lamplighter group. --- manifold. --- mapping class group. --- mathematics. --- membership problem. --- metric space. --- non-free action. --- normal subgroup. --- path metric. --- ping-pong lemma. --- ping-pong. --- polynomial growth theorem. --- product. --- punctured disks. --- quasi-isometric equivalence. --- quasi-isometric rigidity. --- quasi-isometry group. --- quasi-isometry invariant. --- quasi-isometry. --- reflection group. --- reflection. --- relators. --- residual finiteness. --- right-angled Artin group. --- robotics. --- semidirect product. --- space. --- surface group. --- surface. --- symmetric group. --- symmetry. --- topological model. --- topology. --- train track. --- tree. --- word length. --- word metric. --- word problem.