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- Unsolved problems in geometry

Fujita conjecture

In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved as of 2017. It is named after Takao Fujita, who formulated it in 1985.

Hopf conjecture

In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.

Lebesgue's universal covering problem

Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by defin

Moser's worm problem

Moser's worm problem (also known as mother worm's blanket problem) is an unsolved problem in geometry formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The problem asks for the regi

Weinstein conjecture

In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the conjecture claims that on a compact conta

Section conjecture

In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism , where is a complete smooth curve of genus at

Bellman's lost in a forest problem

Bellman's lost-in-a-forest problem is an unsolved minimization problem in geometry, originating in 1955 by the American applied mathematician Richard E. Bellman. The problem is often stated as follows

Bombieri–Lang conjecture

In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variet

Nakai conjecture

In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.It states that if V is a complex algeb

Falconer's conjecture

In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact -dimensional spaces. Intui

Yau's conjecture

In differential geometry, Yau's conjecture from 1982, is a mathematical conjecture which states that a closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces. I

Chern's conjecture for hypersurfaces in spheres

Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consi

Atiyah conjecture on configurations

In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by Atiyah stating that a certain n by n matrix depending on n points in R3 is always non-singular.

Fröberg conjecture

In algebraic geometry, the Fröberg conjecture is a conjecture about the possible Hilbert functions of a set of forms. It is named after Ralf Fröberg, who introduced it in Fröberg . The Fröberg–Iarrobi

Hadwiger conjecture (combinatorial geometry)

In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that

Hilbert–Smith conjecture

In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithful

Hilbert's sixteenth problem

Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original probl

Covering problem of Rado

The covering problem of Rado is an unsolved problem in geometry concerning covering planar sets by squares. It was formulated in 1928 by Tibor Radó and has been generalized to more general shapes and

Pompeiu problem

In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, as follows. Suppose f is a nonzero continuous function defined on a

Kalai's 3^d conjecture

In geometry, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric p

Abundance conjecture

In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program,stating that for every projective variety with Kawamata log terminal

Tripod packing

In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axi

Sato–Tate conjecture

In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all pr

Zeeman conjecture

In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex , the space is collapsible. The conjecture, due to Christo

Virasoro conjecture

In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro

Standard conjectures on algebraic cycles

In 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 o

Volume conjecture

In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements. Let O deno

Dissection into orthoschemes

In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex.

Resolution of singularities

In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V. For varieties over

Clifford's theorem on special divisors

In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford on algebraic curves, showing the constraints on special linear systems on a curve C.

Pierce–Birkhoff conjecture

In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first sta

Hilbert's fifteenth problem

Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. The problem is to put Schubert's enumerative calculus on a rigorous founda

Chern's conjecture (affine geometry)

Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem. Chern's conjecture st

Carathéodory conjecture

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 192

Novikov conjecture

The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965. The Novikov conjecture concerns the homot

Spherical Bernstein's problem

The spherical Bernstein's problem is a possible generalization of the original Bernstein's problem in the field of global differential geometry, first proposed by Shiing-Shen Chern in 1969, and then l

Unknotting problem

In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram. There are several types of unknotting algori

Bass conjecture

In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass.

Square packing in a square

Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side a. If a is an integer, the answer is

Moving sofa problem

In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area A that can

Kobon triangle problem

The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number N(k) of nonoverlapping triangles whose s

Borel conjecture

In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. I

Rota's basis conjecture

In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if X is either a vector space o

Ulam's packing conjecture

Ulam's packing conjecture, named for Stanislaw Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space. The conjecture says tha

Filling area conjecture

In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introduci

McMullen problem

The McMullen problem is an open problem in discrete geometry named after Peter McMullen.

Einstein problem

In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles, that is, a shape that can tessellate space, but only in a no

Jacobian conjecture

In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has Jaco

Inscribed square problem

The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of s

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