Marcos Dajczer (born 19 November 1948, in Buenos Aires) is an Argentine-born Brazilian mathematician whose research concerns geometry and topology. [1]
Dajczer obtained his Ph.D. from the Instituto Nacional de Matemática Pura e Aplicada in 1980 under the supervision of Manfredo do Carmo. [2]
In 2006, he received Brazil's National Order of Scientific Merit honour for his work in mathematics. [3] He was a Guggenheim Fellow in 1985. [4]
Do Carmo–Dajczer theorem is named after his teacher and him. [5] [6] [7]
In the mathematical field of geometric topology, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries.
In the mathematical field of Riemannian geometry, the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
In differential geometry, the Gaussian curvature or Gauss curvatureΚ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2, at the given point: For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow, sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
In mathematics, the Chern theorem states that the Euler–Poincaré characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature form.
Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:
In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.
Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984 and his works on harmonic maps.
In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex.
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924. Carathéodory never committed the conjecture into writing, but did publish a paper on a related subject. In John Edensor Littlewood mentions the conjecture and Hamburger's contribution as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in the formal analogy of the conjecture with the Four Vertex Theorem for plane curves. Modern references to the conjecture are the problem list of Shing-Tung Yau, the books of Marcel Berger, as well as the books.
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949. Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The systolic area of a metric is defined to be the ratio area/sys2. The systolic ratio SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry.
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case.
Fernando Codá dos Santos Cavalcanti Marques is a Brazilian mathematician working mainly in geometry, topology, partial differential equations and Morse theory. He is a professor at Princeton University. In 2012, together with André Neves, he proved the Willmore conjecture. Since then, among proving other important conjectures, Marques and Neves greatly extended Almgren–Pitts min-max theory to prove theorems about minimal surfaces.
Manfredo Perdigão do Carmo was a Brazilian mathematician. He spent most of his career at IMPA and is seen as the doyen of differential geometry in Brazil.
In mathematics, the Almgren–Pitts min-max theory is an analogue of Morse theory for hypersurfaces.
William Hamilton Meeks, III is an American mathematician, specializing in differential geometry and minimal surfaces.