In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces are formal moduli.
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.
In algebraic geometry, a hyperelliptic curve of genus g > 1 is an algebraic curve given by an equation of the form
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field. From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus are k-isomorphic for k any perfect field, so are the curves.
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
In theoretical physics, Seiberg–Witten theory is a theory that determines an exact low-energy effective action of a supersymmetric gauge theory—namely the metric of the moduli space of vacua.
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations.
In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.
This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.
Kefeng Liu, is a Chinese mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Calabi-Yau manifolds. He is a professor of mathematics at University of California, Los Angeles, as well as the Executive Director of the Center of Mathematical Sciences at Zhejiang University.
In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. It was proved for 2-dimensional tori by Appell (1891) and Humbert (1893), and in general by Lefschetz (1921)
In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation
Anton V. Zorich is a Russian mathematician at the Institut Mathématiques de Jussieu. He received his Ph.D. from Moscow State University under the supervision of Sergei Novikov.
In algebraic geometry, a level structure on a space X is an extra structure attached to X that shrinks or eliminates the automorphism group of X, by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of X.
Klaus Hulek is a German mathematician, known for his work in algebraic geometry and in particular, his work on moduli spaces.
