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Showing 12 of 280 problems

NT-AN-001

Birch–Swinnerton-Dyer Conjecture

IconicConjecture1965

Let

E / Q

be an elliptic curve. Write

E (Q) ≅ E (Q)_{tors} \oplus Z^{r}

, and let

L (E, s)

be the Hasse–Weil

L

-function of

E

, continued to an entire function by modularity. Then

ord_{s = 1} L (E, s) = r .

NT-ADD-005

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}

PDE-001

Three-dimensional Navier–Stokes existence and smoothness

IconicOpen problem1934

Let

u_{0} : R^{3} \to R^{3}

C^{\infty}

, divergence-free, and satisfy

sup_{x} (1 + ∣ x ∣)^{N} ∣ \partial^{α} u_{0} (x) ∣ < \infty

for every

N

and multi-index

α

. There exist

u \in C^{\infty} (R^{3} \times [0, \infty); R^{3})

and

p \in C^{\infty} (R^{3} \times [0, \infty))

satisfying

\partial_{t} u + (u \cdot \nabla) u = - \nabla p + Δ u, \nabla \cdot u = 0, u (\cdot, 0) = u_{0}

for all

t \geq 0

, with

sup_{0 \leq t \leq T} \int_{R^{3}} ∣ u (x, t) ∣^{2} d x < \infty

for every

T < \infty

PDE-007

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.

DYN-001

Uniform finiteness in Hilbert's sixteenth problem

IconicOpen problem1900

For every integer

d \geq 2

, there is an integer

H (d) < \infty

such that every planar polynomial vector field

\overset{x}{˙} = P (x, y), \overset{y}{˙} = Q (x, y), max {de g P, de g Q} \leq d,

having only finitely many limit cycles has at most

H (d)

limit cycles. A limit cycle is an isolated periodic orbit.

TCS-001

$P$ versus $NP$

IconicOpen problem1971

P \neq = NP ?

Here

P

is the class of decision languages recognized by deterministic Turing machines in polynomial time, and

NP

is the class recognized by nondeterministic Turing machines in polynomial time (equivalently, languages with polynomial-size certificates verifiable in deterministic polynomial time).

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