Multivalued functionIn mathematics, a multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function. Multivalued functions arise commonly in applications of the implicit function theorem, since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the inverse function of a differentiable function is a multivalued function, and is single-valued only when the original function is monotonic.
Morphism of algebraic varietiesIn algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties.
Goro Shimurawas a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem. Gorō Shimura was born in Hamamatsu, Japan, on 23 February 1930. Shimura graduated with a B.A. in mathematics and a D.
Poincaré half-plane modelIn non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry.
Standard conjectures on algebraic cyclesIn mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an that is . Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see .
J-invariantIn mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that Rational functions of j are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).
Congruence relationIn abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. The prototypical example of a congruence relation is congruence modulo on the set of integers.
Algebraic surfaceIn mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two).
Adequate equivalence relationIn algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the of pure motives with respect to that relation.
Modular lambda functionIn mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve , where the map is defined as the quotient by the [−1] involution.
