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Showing 60 of 280 problems in Geometry and topology

ALG-RING-004

Kaplansky Zero-Divisor Conjecture

MajorConjecturec. 1940

K

is a field and

G

is a torsion-free group, then the group algebra

K [G]

has no nonzero zero divisors: for all

a, b \in K [G]

ab = 0

implies

a = 0

b = 0

ALG-RING-006

Kaplansky Direct-Finiteness Conjecture

MajorConjecturec. 1940

For every field

K

, every group

G

, and all

a, b \in K [G]

, if

ab = 1

, then

ba = 1

ALG-GROUP-014

Andrews–Curtis Conjecture

MajorConjecture1965

For every integer

n \geq 1

, let

F_{n} = ⟨ x_{1}, \dots, x_{n} ⟩

be the free group. If

r = (r_{1}, \dots, r_{n}) \in F_{n}^{n}

has normal closure

F_{n}

, then

r

can be transformed to

(x_{1}, \dots, x_{n})

by finitely many moves

r_{i} \mapsto r_{i}^{- 1}, r_{i} \mapsto r_{i} r_{j} (i \neq = j), r_{i} \mapsto w r_{i} w^{- 1} (w \in F_{n}),

their inverses, and permutations of the entries.

ALG-L2-017

Strong Atiyah Conjecture

MajorConjecture1976

Let

G

be a discrete group for which the orders of finite subgroups are bounded, and put

L = lcm {∣ H ∣ : H \leq G finite}

. For all

m, n \geq 1

and every matrix

A \in M_{m, n} (C G)

, let

r_{A} : ℓ^{2} (G)^{m} \to ℓ^{2} (G)^{n}

be the bounded

G

-equivariant operator given by right convolution by

A

. Then

L dim_{N (G)} ker r_{A} \in Z,

where

N (G)

is the group von Neumann algebra and

dim_{N (G)}

is its von Neumann dimension.

AG-CYCLE-001

Abundance Conjecture

LandmarkConjecturec. 1985

Let

(X, Δ)

be a projective Kawamata-log-terminal pair over

C

, with

K_{X} + Δ

Q

-Cartier

Q

-divisor. If

K_{X} + Δ

is nef, then it is semiample: some positive integral multiple is Cartier and base-point-free.

AG-MMP-012

Termination of Flips Conjecture

LandmarkConjecturec. 1988

Let

(X_{0}, Δ_{0})

be a projective

Q

-factorial Kawamata-log-terminal pair over

C

. Every sequence

(X_{0}, Δ_{0}) ⇢ (X_{1}, Δ_{1}) ⇢ (X_{2}, Δ_{2}) ⇢ \dots

in which each arrow is a

(K_{X_{i}} + Δ_{i})

-flip terminates after finitely many steps.

AG-CYCLE-002

Hartshorne Complete-Intersection Conjecture

MajorConjecture1974

X \subset P_{C}^{n}

is a smooth closed subvariety with

dim X > 2 n /3

, then

X

is a complete intersection in

P^{n}

: its homogeneous ideal is generated by

codim_{P^{n}} X

homogeneous polynomials.

AG-ADJ-015

Irrationality of a Very General Cubic Fourfold

MajorOpen problemc. 1969

A very general smooth cubic hypersurface

X \subset P_{C}^{5}

, meaning one outside a countable union of proper Zariski-closed subsets of the parameter space, is not rational over

C

AG-CONE-019

Morrison–Kawamata Cone Conjecture

MajorConjecture1993–1997

Let

(X, Δ)

be a projective

Q

-factorial Kawamata-log-terminal pair over

C

with

K_{X} + Δ \equiv 0

, and put

Nef^{e} (X) = Nef (X) \cap Eff (X)

inside

N^{1} (X)_{R}

, where

Eff (X)

is the cone generated by classes of effective Cartier divisors. There is a rational polyhedral cone

Π \subset Nef^{e} (X)

such that

Nef^{e} (X) = g \in Aut (X, Δ) ⋃ g^{*} Π,

and interiors of

g^{*} Π

and

h^{*} Π

are disjoint unless the cones coincide.

GEO-TOP-001

Smooth four-dimensional Poincaré conjecture

IconicConjecturec. 1960

Every closed, connected, smooth

4

-manifold

M

that is homotopy equivalent to the standard sphere

S^{4}

is diffeomorphic to

S^{4}

GEO-TOP-002

Smooth four-dimensional Schoenflies conjecture

LandmarkConjecturec. 1960

For every smooth embedding

e : S^{3} ↪ S^{4}

, each of the two closures of the components of

S^{4} ∖ e (S^{3})

is diffeomorphic to the standard closed

4

-ball

D^{4}

GEO-TOP-004

Borel topological rigidity conjecture

LandmarkConjecture1953

Let

M

and

N

be closed, connected, topological

n

-manifolds with

n \geq 5

and contractible universal covers. Every homotopy equivalence

f : N \to M

is homotopic to a homeomorphism.

GEO-TOP-007

Novikov higher-signature conjecture

LandmarkConjecture1970

Let

M

be a closed connected oriented smooth manifold, let

Γ = π_{1} (M)

, let

u : M \to B Γ

classify its universal cover, and let

α \in H^{*} (B Γ; Q)

. The higher signature

⟨ L (M) ⌣ u^{*} α, [M] ⟩ \in Q

is invariant under orientation-preserving homotopy equivalence. Here

L (M)

is the total Hirzebruch

L

-class of

TM

, and only the top-degree component is evaluated.

GEO-TOP-015

Weinstein conjecture

LandmarkConjecture1978

Let

M^{2 n - 1}

be a closed smooth manifold and let

α

be a contact form, so

α \land (d α)^{n - 1}

is nowhere zero. The Reeb vector field

R_{α}

, uniquely defined by

α (R_{α}) = 1

and

d α (R_{α}, \cdot) = 0

, has a nonconstant periodic orbit.

GEO-TOP-003

K-theoretic Farrell–Jones isomorphism conjecture

MajorConjecture1993

For every discrete group

G

, every associative unital ring

R

, and every integer

n

, the assembly map

H_{n}^{G} (E_{VCyc} G; K_{R}) ⟶ K_{n} (RG)

is an isomorphism. Here

E_{VCyc} G

is the terminal

G

-CW complex whose

H

-fixed points are contractible for virtually cyclic

H \leq G

and empty otherwise, and

H_{*}^{G} (-; K_{R})

is the equivariant homology theory whose value at

G / H

K_{*} (R H)

GEO-TOP-009

Bott rational-ellipticity conjecture

MajorConjecturec. 1982

M

is a closed simply connected smooth manifold that admits a Riemannian metric of nonnegative sectional curvature, then

M

is rationally elliptic, meaning

i \geq 2 \sum dim_{Q} (π_{i} (M) \otimes Q) < \infty.

GEO-TOP-011

Cartan–Hadamard isoperimetric conjecture

MajorConjecture1976

Let

(M^{n}, g)

be a complete simply connected Riemannian

n

-manifold with sectional curvature

K \leq 0

, and let

Ω \subset M

be a relatively compact domain with smooth boundary. Then

Area_{g} (\partial Ω) \geq n ω_{n}^{1/ n} Vol_{g} (Ω)^{(n - 1) / n},

where

ω_{n}

is the Euclidean volume of the unit ball in

R^{n}

GEO-TOP-012

Positive sectional curvature on $S^2\times S^2$

MajorOpen problemc. 1965

There exists a smooth Riemannian metric

g

S^{2} \times S^{2}

such that every

2

-plane

σ \subset T_{x} (S^{2} \times S^{2})

, at every

x

, has sectional curvature

K_{g} (σ) > 0

GEO-TOP-013

Purely cosmetic surgery conjecture

MajorConjecture1990

Let

K \subset S^{3}

be a nontrivial knot and, for each slope

t \in Q \cup {\infty}

in the meridian--longitude basis, let

S_{t}^{3} (K)

denote the result of

t

-Dehn filling its exterior. If

r, s \in Q \cup {\infty}

and

S_{r}^{3} (K)

and

S_{s}^{3} (K)

are orientation-preservingly homeomorphic, then

r = s

GEO-TOP-019

$L$ -space conjecture

MajorConjecture2013–2015

Let

Y

be a closed, connected, orientable, irreducible rational-homology

3

-sphere, and let

H F (Y; F_{2})

denote its hat Heegaard Floer homology. The following are equivalent: (i)

rank_{F_{2}} H F (Y; F_{2}) > ∣ H_{1} (Y; Z) ∣

, so

Y

is not an

L

-space; (ii)

π_{1} (Y)

admits a total order

<

satisfying

g < h \Rightarrow f g < f h

for all

f, g, h

; (iii)

Y

admits a coorientable taut codimension-one foliation.

GEO-TOP-020

Hyperbolic volume conjecture

MajorConjecture1997

Let

K \subset S^{3}

be a hyperbolic knot and let

J_{N} (K; q)

be its

N

-colored Jones polynomial normalized by

J_{N} (unknot; q) = 1

. Then

N \to \infty lim \frac{2 π}{N} lo g J_{N} (K; e^{2 πi / N}) = Vol (S^{3} ∖ K),

where the right-hand side is the complete finite-volume hyperbolic volume.

GEO-DIFF-021

Carath\u00e9odory Umbilic Conjecture

MajorConjecture1924

Every closed

C^{2}

, strictly convex surface embedded in

R^{3}

has at least two umbilic points, where the two principal curvatures are equal.

ANAL-FA-017

Baum–Connes conjecture without coefficients

LandmarkConjecture1982

For every countable discrete group

G

, the analytic assembly map

μ_{G} : K_{*}^{G} (\underline{E} G) ⟶ K_{*} (C_{r}^{*} (G))

is an isomorphism. Here

\underline{E} G = E_{Fin} G

is the terminal

G

-CW complex whose

H

-fixed-point space is contractible for finite subgroups

H \leq G

and empty for infinite

H

;

K_{*}^{G}

is equivariant topological

K

-homology with

G

-compact supports; and

C_{r}^{*} (G)

is the operator-norm closure of the left regular representation of

C G

ℓ^{2} (G)

ANAL-FA-009

Weak cosmic censorship (future-null-infinity form)

IconicConjecture1969

Let

D_{AF}

be the space of smooth, complete vacuum initial data

(h, K)

R^{3}

satisfying

R_{h} - ∣ K ∣_{h}^{2} + (tr_{h} K)^{2} = 0

and

div_{h} K - d (tr_{h} K) = 0

. In fixed asymptotic coordinates, put

r = ∣ x ∣

and require

\partial^{α} (h_{ij} - δ_{ij}) = O (r^{- 1 - ∣ α ∣})

and

\partial^{α} K_{ij} = O (r^{- 2 - ∣ α ∣})

for every multi-index

α

. Give

D_{AF}

the relative weighted

C^{\infty}

topology induced by the seminorms

q_{m} (γ, L) = ∣ α ∣ \leq m \sum x sup (1 + r)^{1 + ∣ α ∣} ∣ \partial^{α} γ ∣ + ∣ α ∣ \leq m \sum x sup (1 + r)^{2 + ∣ α ∣} ∣ \partial^{α} L ∣

on differences

(γ, L) = (h - h^{'}, K - K^{'})

. There is an open dense subset of

D_{AF}

whose maximal globally hyperbolic developments have a conformal completion with complete future null infinity

I^{+}

, meaning that every physical null geodesic ending at

I^{+}

has infinite affine length.

PDE-008

$C^2$ strong cosmic censorship

LandmarkConjecturec. 1979

Fix a closed smooth

3

-manifold

Σ

and a smooth background Riemannian metric

b

. Let

D_{vac} (Σ)

be the space of smooth pairs

(h, K)

, with

h

Riemannian and

K

symmetric, satisfying

R_{h} - ∣ K ∣_{h}^{2} + (tr_{h} K)^{2} = 0

and

div_{h} K - d (tr_{h} K) = 0

. Give it the relative Fréchet topology induced by

q_{m} (γ, L) = ∥ γ ∥_{C^{m} (b)} + ∥ L ∥_{C^{m} (b)}

m = 0, 1, \dots

, on differences

(γ, L) = (h - h^{'}, K - K^{'})

. In every nonempty connected component of

D_{vac} (Σ)

, the data whose maximal globally hyperbolic development admits no proper isometric embedding into a connected Lorentzian manifold with

C^{2}

metric form a residual set, meaning a countable intersection of open dense sets.

PDE-010

Nonlinear Stability of Subextremal Kerr Black Holes

LandmarkConjecturec. 1965

For every fixed Kerr exterior with mass

m > 0

and angular momentum parameter

∣ a ∣ < m

, all sufficiently small asymptotically flat vacuum perturbations of compatible initial data have a complete exterior development that, after gauge adjustment, remains close to and asymptotically approaches a nearby Kerr exterior.

DYN-005

Birkhoff billiard conjecture

LandmarkConjecture1927

Let

Ω \subset R^{2}

be bounded with

C^{\infty}

, strictly convex boundary. Its billiard phase space is the open annulus

A = {(x, v) : x \in \partial Ω, ∣ v ∣ = 1, v points inward and is not tangent to \partial Ω}

, with the billiard map sending one reflected state to the next. If

A

is foliated by invariant circles homotopic to its boundary, then

\partial Ω

is an ellipse.

DYN-006

Ivrii periodic-billiard conjecture

MajorConjecture1980

Let

Ω \subset R^{n}

be a bounded domain with

C^{\infty}

boundary, and let

B^{*} \partial Ω = {(x, ξ) \in T^{*} \partial Ω : ∣ ξ ∣ < 1}

, equipped with its canonical Liouville measure. The set of points of

B^{*} \partial Ω

that are periodic under the billiard map has measure zero.

DYN-007

Anosov-manifold conjecture

MajorConjecturec. 1970

Every closed connected smooth manifold that admits an Anosov diffeomorphism is homeomorphic to an infranilmanifold, namely a quotient

Γ\ N

, where

N

is a simply connected nilpotent Lie group and

Γ

is a torsion-free discrete subgroup of

N ⋊ C

for some compact subgroup

C \leq Aut (N)

, acting freely and cocompactly on

N

GRAPH-012

Conway's Thrackle Conjecture

MajorConjecture1969

Let a finite simple graph

G

be drawn in the plane with vertices as distinct points and edges as simple arcs, with no edge through a nonincident vertex and no three edges meeting at an interior point. Suppose every pair of distinct edges meets exactly once, either at their common endpoint or in one proper crossing. Then

∣ E (G) ∣ \leq ∣ V (G) ∣.

GRAPH-013

Harary–Hill Conjecture for the Crossing Number of $K_n$

MajorExact constant problem1960–1963

Let

cr (G)

be the minimum number of pairwise edge crossings over plane drawings of a graph

G

in general position. For every integer

n \geq 1

cr (K_{n}) = \frac{1}{4} ⌊ \frac{n}{2} ⌋ ⌊ \frac{n - 1}{2} ⌋ ⌊ \frac{n - 2}{2} ⌋ ⌊ \frac{n - 3}{2} ⌋ .

GRAPH-014

Albertson's Crossing-Number Conjecture

MajorConjecture2007

Every finite simple graph

G

satisfies

cr (G) \geq cr (K_{χ (G)}) .

Here

cr (G)

is the minimum number of edge crossings in a plane drawing of

G

χ (G)

is its chromatic number, and

K_{q}

is the complete graph on

q

vertices.

DG-006

Hadwiger–Levi Illumination Conjecture

LandmarkConjecture1957–1960

Let

d \geq 2

, let

S^{d - 1}

be the unit sphere, and let

K \subset R^{d}

be a convex body. Say that a direction

u \in S^{d - 1}

illuminates

x \in \partial K

x + t u \in int K

for some

t > 0

. There exist at most

2^{d}

directions such that every point of

\partial K

is illuminated by at least one of them.

DG-007

Mahler Volume-Product Conjecture

LandmarkConjecture1939

For a convex body

K \subset R^{d}

containing

0

in its interior, put

K^{\circ} = {y : ⟨ x, y ⟩ \leq 1 \forall x \in K}

. If

K = - K

, then

vol (K) vol (K^{\circ}) \geq \frac{4 ^{d}}{d !} .

For arbitrary

K

, let

s (K)

be its Santaló point, the minimizer of

vol ((K - x)^{\circ})

over

x \in int K

. Then

vol (K) vol ((K - s (K))^{\circ}) \geq \frac{( d + 1 ) ^{d + 1}}{( d ! ) ^{2}} .

DG-016

Toeplitz Square-Peg Conjecture

LandmarkConjecture1911

For every continuous injective map

γ : S^{1} \to R^{2}

, the Jordan curve

γ (S^{1})

contains four distinct points that are the vertices of a nondegenerate Euclidean square.

DG-017

Kneser–Poulsen Conjecture

LandmarkConjecture1954–1955

Let

d, N \geq 1

r > 0

, and

p_{1}, \dots, p_{N}, q_{1}, \dots, q_{N} \in R^{d}

satisfy

∥ q_{i} - q_{j} ∥_{2} \leq ∥ p_{i} - p_{j} ∥_{2} for all i, j .

Writing

B (x, r)

for the closed Euclidean ball and

vol_{d}

for

d

-dimensional Lebesgue measure, one has

vol_{d} (i = 1 ⋃ N B (q_{i}, r)) \leq vol_{d} (i = 1 ⋃ N B (p_{i}, r))

and

vol_{d} (i = 1 ⋂ N B (q_{i}, r)) \geq vol_{d} (i = 1 ⋂ N B (p_{i}, r)) .

DG-010

Reinhardt's Smoothed-Octagon Conjecture

MajorConjecture1934

For a centrally symmetric convex disk

K \subset R^{2}

, define

δ_{L} (K) = Λ sup \frac{area ( K )}{det Λ},

where the supremum is over full-rank lattices

Λ \subset R^{2}

for which the interiors of the translates

K + λ

λ \in Λ

, are pairwise disjoint. Let

O

be Reinhardt's smoothed octagon, obtained from a regular octagon by replacing each vertex by the hyperbola arc tangent to its two incident sides and asymptotic to the two adjacent nonincident sides. Then

δ_{L} (K) \geq δ_{L} (O) = \frac{8 - 4 2 - lo g 2}{2 2 - 1} .

DG-011

Kalai's $3^d$ Conjecture

MajorConjecture1989

Every centrally symmetric convex

d

-polytope has at least

3^{d}

nonempty faces in total, counting faces of every dimension

0, \dots, d

and counting the polytope itself.

DG-013

Optimal Sphere Packing in Dimension Five

MajorConjecturec. 1900

For a packing

P

of congruent closed balls in

R^{5}

with disjoint interiors, define its upper asymptotic density by

\overline{ρ} (P) = R \to \infty lim sup x \in R^{5} sup \frac{vol _{5} ( B ( x , R ) \cap ⋃ _{B \in P} B )}{vol _{5} ( B ( x , R ))},

where

B (x, R)

is the radius-

R

ball centered at

x

. Let

Δ_{5}

be the supremum of

\overline{ρ} (P)

over all such packings. Then

Δ_{5} = \frac{π ^{2}}{15 2},

the density attained by the

D_{5}

root-lattice packing.

DG-014

Four-Dimensional Borsuk Problem

MajorCanonical finite case1933

Let

b (4)

be the least

m

such that every bounded set

S \subset R^{4}

of positive diameter can be partitioned into

m

subsets, each having diameter strictly smaller than

diam (S)

. Is

b (4) = 5 ?

DG-015

Lebesgue's Universal Covering Problem

MajorExact constant problem1914

Let

U = in f {area (K) : K \subset R^{2} is convex and every planar set S with diam (S) \leq 1 has an isometric copy contained in K} .

Determine the exact value of

U

, and whether the infimum is attained.

DG-019

Ehrhart Volume Conjecture

MajorConjecture1964

Let

K \subset R^{n}

be a convex body whose barycenter is the origin and whose only interior lattice point is

0

. Then

vol (K) \leq \frac{( n + 1 ) ^{n}}{n !} .

Moreover, equality holds only for unimodular images of the centered simplex

(n + 1) conv (0, e_{1}, \dots, e_{n}) - (1, \dots, 1) .

PROB-004

Strict Convexity of the First-Passage Percolation Limit Shape

MajorConjecturec. 1965

Assign independent mean-one exponential passage times to the nearest-neighbor edges of

Z^{d}

d \geq 2

. Let

B \subset R^{d}

be the deterministic compact convex limit shape from the first-passage percolation shape theorem. Then

B

is strictly convex: its boundary contains no nontrivial line segment.

MATH-PHYS-001

Yang–Mills existence and mass gap

IconicOpen problem2000

For every compact simple gauge group

G

, construct gauge-invariant Euclidean Yang--Mills Schwinger functions on

R^{4}

satisfying the Osterwalder--Schrader axioms OS0--OS4: regularity/tempered growth, Euclidean covariance, reflection positivity, permutation symmetry, and clustering. Their Osterwalder--Schrader reconstruction must be a nontrivial relativistic quantum field theory whose joint energy--momentum spectrum consists of the vacuum

0

and a subset of

{p : p_{0} \geq 0, p_{0}^{2} - ∥ p ∥^{2} \geq Δ^{2}}

for some

Δ > 0

MATH-PHYS-008

Quantum unique ergodicity

LandmarkConjecture1994

Let

(M, g)

be a closed connected Riemannian manifold with strictly negative sectional curvature, and let

- Δ_{g} φ_{j} = λ_{j}^{2} φ_{j}

∥ φ_{j} ∥_{L^{2}} = 1

, with

λ_{j} \to \infty

. For every classical order-zero pseudodifferential operator

A \in Ψ^{0} (M)

⟨ A φ_{j}, φ_{j} ⟩ ⟶ \int_{S^{*} M} σ_{0} (A) d L .

Here

S^{*} M = {(x, ξ) \in T^{*} M : ∣ ξ ∣_{g} = 1}

σ_{0} (A)

is the degree-zero principal symbol restricted to

S^{*} M

, and

L

is normalized Liouville probability measure.

MATH-PHYS-009

General spacetime Penrose inequality

LandmarkConjecture1973

Let

(M^{3}, g, k)

be a smooth, connected, orientable, complete asymptotically flat initial data set. Define its energy and momentum densities by

16 π μ = R_{g} + (tr_{g} k)^{2} - ∣ k ∣_{g}^{2}, 8 π J = div_{g} (k - (tr_{g} k) g),

assume the dominant energy condition

μ \geq ∣ J ∣_{g}

, and fix an asymptotically flat end. Let

Σ

be an outermost apparent horizon relative to that end, allowing a disjoint union of marginally outer trapped components

H + tr_{Σ} k = 0

and marginally inner trapped components

H - tr_{Σ} k = 0

. Define

A

to be the infimum of the total

g

-areas of smooth closed surfaces enclosing

Σ

relative to the chosen end. If the ADM energy-momentum of the end is

(E, P)

and

m_{ADM} = E^{2} - ∣ P ∣^{2}

, then

m_{ADM} \geq \frac{A}{16 π} .

Equality should occur only when the exterior data arise from a spacelike slice of the Schwarzschild spacetime.