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Leschke is a coauthor of the book Conformal Geometry of Surfaces in and Quaternions (Springer, 2002), developing the theory of quaternionic analysis. [8]
Katrin Leschke (born 1968) [1] is a German mathematician specialising in differential geometry and known for her work on quaternionic analysis and Willmore surfaces. She works in England as a reader in mathematics at the University of Leicester, [2] where she also heads the "Maths Meets Arts Tiger Team", an interdisciplinary group for the popularisation of mathematics, [3] and led the "m:iv" project of international collaboration on minimal surfaces. [4]
Leschke did her undergraduate studies at Technische Universität Berlin, and continued there for a PhD, [2] which she completed in 1997. Her dissertation, Homogeneity and Canonical Connections of Isoparametric Manifolds, was jointly supervised by Dirk Ferus and Ulrich Pinkall. [5]
She was a postdoctoral researcher at Technische Universität Berlin from 1997 to 2002, a visiting assistant professor at the University of Massachusetts Amherst from 2002 to 2005, and a researcher and temporary associate professor at the University of Augsburg from 2005 to 2007. [6] At Augsburg, she completed her habilitation, [2] working in the group of Katrin Wendland. [7] She joined the University of Leicester as New Blood Lecturer in 2007 and became reader there in 2016. [6]
Leschke is a coauthor of the book Conformal Geometry of Surfaces in and Quaternions (Springer, 2002), developing the theory of quaternionic analysis. [8]
In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere, the boundary of a ball in four dimensions is a 3-sphere. Topologically, a 3-sphere is an example of a 3-manifold, and it is also an n-sphere.
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by H, or in blackboard bold by Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form
In the mathematical field of differential topology, the Hopf fibration describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct great circle of the 3-sphere. Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-sphere.
In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.
Christian Hugo Eduard Study was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also Introduction to systolic geometry.
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some . Here Sp(n) is the sub-group of consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic matrix, while the group of unit-length quaternions instead acts on quaternionic -space by right scalar multiplication. The Lie group generated by combining these actions is then abstractly isomorphic to .
In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions define integrable almost complex structures.
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups.
In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-formφ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:
Robert "Bob" Osserman was an American mathematician who worked in geometry. He is specially remembered for his work on the theory of minimal surfaces.
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
In mathematics, hypercomplex analysis is the extension of complex analysis to the hypercomplex numbers. The first instance is functions of a quaternion variable, where the argument is a quaternion. A second instance involves functions of a motor variable where arguments are split-complex numbers.
Georg Scheffers was a German mathematician specializing in differential geometry.
Katrin Wendland is a German mathematical physicist who works as a professor at Trinity College Dublin.
Paul C. Yang is a Taiwanese-American mathematician specializing in differential geometry, partial differential equations and CR manifolds. He is best known for his work in Conformal geometry for his study of extremal metrics and his research on scalar curvature and Q-curvature. In CR Geometry he is known for his work on the CR embedding problem, the CR Paneitz operator and for introducing the Q' curvature in CR Geometry.
In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of G-structures on a manifold. Specifically, a quaternionic n-manifold can be defined as a smooth manifold of real dimension 4n equipped with a torsion-free -structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds.
William Hamilton Meeks III is an American mathematician, specializing in differential geometry and minimal surfaces.
Ulrich Pinkall is a German mathematician, specializing in differential geometry and computer graphics.
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