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Symmetric spaces are a class of homogeneous spaces characterized by their symmetrical properties, where for each point there exists an isometry reversing geodesics through that point. They play a crucial role in differential geometry and representation theory, as they generalize the notion of constant curvature spaces and provide a rich structure for studying Lie groups and algebraic groups.
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An isometry is a transformation in geometry that preserves distances between points, meaning the original shape and the transformed shape are congruent. Isometries include translations, rotations, reflections, and glide reflections, and are fundamental in understanding symmetry and rigid motions in Euclidean spaces.
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Differential geometry is the field of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry, particularly those involving curves and surfaces. It plays a crucial role in understanding the geometry of differentiable manifolds and has applications in physics, particularly in the theory of general relativity and modern theoretical physics.
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Lie groups are mathematical structures that combine algebraic and geometric properties, allowing for the study of continuous symmetries. They are fundamental in many areas of mathematics and physics, providing a framework for analyzing objects that are invariant under transformations like rotations and translations.
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A Riemannian manifold is a smooth manifold equipped with an inner product on the tangent space at each point, allowing for the generalization of geometric concepts such as angles, distances, and curvature. This structure enables the application of calculus and analysis techniques to study the manifold's geometric properties and its intrinsic shape.
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Algebraic groups are mathematical structures that combine the properties of both algebraic varieties and group theory, allowing for the study of symmetries in algebraic geometry through group actions. They are fundamental in understanding the solutions to polynomial equations and have applications in number theory, representation theory, and beyond.
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Representation Theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. This approach allows complex algebraic problems to be translated into more manageable linear algebra problems, providing deep insights into the symmetry and structure of mathematical objects.
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A double coset is a generalization of a coset in group theory, where instead of considering the left or right multiplication by a single subgroup, we consider both left and right multiplication by two subgroups. This structure is essential in understanding the decomposition of groups and plays a significant role in representation theory and the study of symmetric spaces.
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An Exceptional Jordan algebra is a non-associative algebra over the real numbers, known as the Albert algebra, which is a 27-dimensional algebra of Hermitian 3x3 matrices over the octonions. It is significant in the study of certain mathematical structures, including the classification of simple Jordan algebras and connections to exceptional Lie groups and quantum mechanics.
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Automorphic forms are complex analytic functions that are invariant under the action of a discrete group, typically arising in the context of number theory and representation theory. They play a crucial role in modern mathematics, particularly in the Langlands program, which seeks to connect Galois groups in algebraic number theory with automorphic representations of adelic groups.
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The modular group, denoted as PSL(2, Z), is a fundamental object in the study of complex analysis and number theory, consisting of 2x2 matrices with integer entries and determinant one, modulo its center. It acts on the upper half-plane via fractional linear transformations and is central to the theory of modular forms, which are crucial in understanding various aspects of arithmetic geometry and string theory.
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