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Showing 31 of 280 problems in Mathematical physics

GEO-TOP-003

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.

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.

PDE-002

Global regularity for three-dimensional incompressible Euler

LandmarkOpen problemc. 1930

For every divergence-free Schwartz vector field

u_{0} \in S (R^{3}; R^{3})

, the maximal classical solution of

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

exists for all

t \geq 0

and belongs to

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

PDE-003

Global regularity for inviscid surface quasi-geostrophic flow

LandmarkOpen problem1994

For every

θ_{0} \in S (R^{2})

, the unique maximal classical solution of

\partial_{t} θ + u \cdot \nabla θ = 0, u = \nabla^{⊥} (- Δ)^{- 1/2} θ = (- R_{2} θ, R_{1} θ), θ (\cdot, 0) = θ_{0}

exists for every

t \geq 0

and is smooth, where

R_{j}

is the

j

-th Riesz transform.

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-009

De Giorgi conjecture in dimensions $4$ through $8$

LandmarkConjecture1978

Let

n \in {4, 5, 6, 7, 8}

. If

u \in C^{2} (R^{n})

is bounded, satisfies

Δ u = u^{3} - u and \partial_{x_{n}} u > 0 on R^{n},

then

u

is one-dimensional: there are

ν \in S^{n - 1}

and

h \in C^{2} (R)

such that

u (x) = h (ν \cdot x)

for every

x \in R^{n}

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.

PDE-004

Global regularity for three-dimensional incompressible MHD

MajorOpen problemc. 1950

For every pair of smooth divergence-free vector fields

u_{0}, b_{0} : T^{3} \to R^{3}

, the unique local smooth solution of

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

extends smoothly for all

t \geq 0

PDE-005

Global regularity for the two-dimensional inviscid Boussinesq system

MajorOpen problemc. 1980

For every

u_{0} \in S (R^{2}; R^{2})

and

θ_{0} \in S (R^{2})

with

\nabla \cdot u_{0} = 0

, the maximal classical solution of

\partial_{t} u + (u \cdot \nabla) u + \nabla p = θ e_{2}, \partial_{t} θ + u \cdot \nabla θ = 0, \nabla \cdot u = 0, (u, θ) ∣_{t = 0} = (u_{0}, θ_{0})

exists for all

t \geq 0

and is smooth.

PDE-006

Global classical solutions of the relativistic Vlasov–Maxwell system

MajorOpen problemc. 1980

Let

f_{0}^{\pm} \in C_{c}^{\infty} (R_{x}^{3} \times R_{p}^{3})

be nonnegative and let

E_{0}, B_{0} \in C_{c}^{\infty} (R^{3}; R^{3})

satisfy

\nabla \cdot E_{0} = \int (f_{0}^{+} - f_{0}^{-}) d p

and

\nabla \cdot B_{0} = 0

. With

v (p) = p / 1 + ∣ p ∣^{2}

, the system

\partial_{t} f^{\pm} + v \cdot \nabla_{x} f^{\pm} \pm (E + v \times B) \cdot \nabla_{p} f^{\pm} = 0, \partial_{t} E - \nabla \times B = - j, \partial_{t} B + \nabla \times E = 0, \nabla \cdot E = ρ, \nabla \cdot B = 0, ρ = \int_{R^{3}} (f^{+} - f^{-}) d p, j = \int_{R^{3}} v (p) (f^{+} - f^{-}) d p

has a unique classical solution for all

t \geq 0

DYN-008

Positive metric entropy for the standard map

LandmarkOpen problemc. 1980

There exists

K \in R ∖ {0}

such that the area-preserving standard map

S_{K} (x, y) = (x + y + K sin (2 π x), y + K sin (2 π x)) (mod Z^{2})

T^{2}

has positive Kolmogorov--Sinai entropy with respect to normalized Lebesgue measure.

DYN-013

Improbability of noncollision singularities

MajorConjecturec. 1984

Fix

N \geq 5

and masses

m_{1}, \dots, m_{N} > 0

. For

q_{i} (t) \in R^{3}

, consider

\overset{q}{¨}_{i} = j \neq = i \sum m_{j} \frac{q _{j} - q _{i}}{∣ q _{j} - q _{i} ∣ ^{3}}, i = 1, \dots, N,

and the collision-free phase space

P = {(q_{1}, \dots, q_{N}, \overset{q}{˙}_{1}, \dots, \overset{q}{˙}_{N}) : q_{i} \neq = q_{j} for i \neq = j} \subset R^{6 N}

. With respect to Lebesgue measure on

P

, the set of initial conditions whose maximal solution has a finite endpoint

T < \infty

while

i \neq = j min ∣ q_{i} (t) - q_{j} (t) ∣ \neq ⟶ 0 as t ↑ T

has measure zero.

PROB-003

No Infinite Cluster at Critical Percolation on $\mathbb Z^d$

LandmarkConjecturec. 1960

For Bernoulli bond percolation on

Z^{d}

, independently declare each nearest-neighbor edge open with probability

p

, and let

p_{c} (d)

be the threshold for an infinite open cluster. Then for every

d \geq 2

P_{p_{c} (d)} (0 \leftrightarrow \infty) = 0.

PROB-001

Two-dimensional self-avoiding-walk critical exponents

LandmarkConjecture1972–1982

For

n \geq 0

, let

S_{n} = {ω = (ω_{0}, \dots, ω_{n}) \in (Z^{2})^{n + 1} : ω_{0} = 0, ∥ ω_{j + 1} - ω_{j} ∥_{1} = 1 (0 \leq j < n), ω_{i} \neq = ω_{j} (0 \leq i < j \leq n)}

and put

c_{n} = ∣ S_{n} ∣

. Let

μ = lim_{n \to \infty} c_{n}^{1/ n}

, whose existence is known, and let

P_{n}

and

E_{n}

denote the uniform probability law on

S_{n}

and expectation with respect to that law. There exist constants

A, D \in (0, \infty)

such that, as

n \to \infty

c_{n} \sim A μ^{n} n^{11/32} and E_{n} [∥ ω_{n} ∥_{2}^{2}] \sim D n^{3/2} .

Thus the counting and metric critical exponents are respectively

γ = 43/32

and

ν = 3/4

PROB-002

KPZ fluctuation exponent for two-dimensional first-passage percolation

LandmarkConjecturec. 1986

Give each unoriented nearest-neighbor edge

e

Z^{2}

an independent exponential random variable

τ_{e}

of mean

1

. For

x, y \in Z^{2}

, define

T (x, y) = π : x \to y min e \in π \sum τ_{e},

where the minimum is over finite nearest-neighbor paths from

x

y

. If

e_{1} = (1, 0)

, then

n \to \infty lim \frac{lo g Var T ( 0 , n e _{1} )}{lo g n} = \frac{2}{3} .

TCS-014

Quantum PCP Conjecture

LandmarkConjecturec. 2006

There exist constants

q \in N

and

0 \leq a < b \leq 1

such that the following promise problem is

QM A

-hard: given an

n

-qubit Hamiltonian

H = \sum_{i = 1}^{m} H_{i}

, where

m = poly (n)

, each

H_{i}

acts on at most

q

qubits and

0 ⪯ H_{i} ⪯ I

, distinguish

λ_{m i n} (H) \leq am from λ_{m i n} (H) \geq bm,

where

λ_{m i n} (H)

is the least eigenvalue of

H

. Here

QM A

is bounded-error quantum polynomial-time verification with a polynomial-size quantum witness.

TCS-023

$BQP$ versus $BPP$

LandmarkOpen problemc. 1994

BPP \neq = BQP ?

Here

BPP

is classical probabilistic polynomial time with error at most

1/3

, and

BQP

is the analogous class decided by polynomial-time-uniform, polynomial-size quantum circuits over a fixed universal gate set, measured at the end, with error at most

1/3

TCS-029

Quantum Parallel Repetition Conjecture

MajorConjecture2004

For every finite two-player one-round game

G

whose entangled value satisfies

ω^{*} (G) < 1

, there is a constant

c_{G} > 0

such that the entangled value of the

n

-fold parallel repetition obeys

ω^{*} (G^{\otimes n}) \leq e^{- c_{G} n} (n \geq 1) .

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-002

Haldane gap for the spin- $1$ Heisenberg chain

LandmarkConjecture1983

For every integer

L \geq 2

, on

(C^{3})^{\otimes L}

with periodic boundary conditions let

H_{L} = j = 1 \sum L S_{j} \cdot S_{j + 1}, S_{L + 1} = S_{1},

where

S_{j}

are the spin-

1

S U (2)

generators at site

j

. If

E_{0} (L) < E_{1} (L)

are its two lowest distinct eigenvalues, then

L \to \infty lim inf (E_{1} (L) - E_{0} (L)) > 0.

MATH-PHYS-003

Anderson delocalization in dimension $d\ge3$

LandmarkConjecture1958

Let

d \geq 3

and

(H_{λ, ω} ψ) (x) = - ∣ y - x ∣ = 1 \sum ψ (y) + λ ω_{x} ψ (x) on ℓ^{2} (Z^{d}),

where the

ω_{x}

are iid with a bounded compactly supported density that is positive near

0

. For all sufficiently small

λ > 0

, there is a nonempty open interval

I

in the interior of

[- 2 d, 2 d]

on which

H_{λ, ω}

has almost surely nonempty purely absolutely continuous spectrum.

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.

MATH-PHYS-011

Critical three-dimensional Ising spin-field scaling limit

LandmarkConjecturec. 1970

Let

⟨ \cdot ⟩_{β_{c}}

be the infinite-volume zero-field Gibbs state at the critical inverse temperature

β_{c}

of the nearest-neighbor ferromagnetic Ising model on

Z^{3}

, where

β_{c}

is the infimum of the inverse temperatures for which the infinite-volume plus state has positive magnetization. The spins are

σ_{x} \in {- 1, 1}

, and finite-volume weights are proportional to

exp (β \sum_{⟨ x, y ⟩} σ_{x} σ_{y})

, with the sum over nearest-neighbor edges. For

a > 0

, set

e_{1} = (1, 0, 0)

n_{a} = ⌊ a^{- 1} ⌋

c_{a} = ⟨ σ_{0} σ_{n_{a} e_{1}} ⟩_{β_{c}}^{- 1/2}

, and define a random tempered distribution by

Φ_{a} (f) = a^{3} c_{a} x \in Z^{3} \sum σ_{x} f (a x), f \in S (R^{3}) .

a ↓ 0

Φ_{a}

converges in law in

S^{'} (R^{3})

to a non-Gaussian random distribution

Φ

. For every integer

k \geq 1

, its

k

-point correlation distribution has a smooth restriction

S_{k}

to pairwise distinct points, and there is

Δ_{σ} > 0

such that every conformal diffeomorphism

φ : U \to V

between open subsets of

R^{3}

satisfies

S_{k} (φ (x_{1}), \dots, φ (x_{k})) = i = 1 \prod k λ_{φ} (x_{i})^{- Δ_{σ}} S_{k} (x_{1}, \dots, x_{k}), λ_{φ} (x) = ∣ det D φ (x) ∣^{1/3},

for all pairwise distinct

x_{1}, \dots, x_{k} \in U

MATH-PHYS-004

Maximal ionization conjecture

MajorConjecture1984

For integers

Z, N \geq 1

, on the fermionic space

⋀^{N} L^{2} (R^{3}; C^{2})

let

H_{Z, N} = i = 1 \sum N (- \frac{1}{2} Δ_{i} - \frac{Z}{∣ x _{i} ∣}) + 1 \leq i < j \leq N \sum \frac{1}{∣ x _{i} - x _{j} ∣}, E_{Z} (N) = in f σ (H_{Z, N}),

and set

E_{Z} (0) = 0

. There is a universal constant

C < \infty

such that

E_{Z} (N) < E_{Z} (N - 1)

implies

N \leq Z + C

for every

Z, N \geq 1

MATH-PHYS-005

Three-dimensional Coulomb crystallization

MajorConjecturec. 1934

Let

Λ_{L} = (R / L Z)^{3}

, let

N = L^{3} \in N

, and let

G_{L}

be the zero-mean periodic Coulomb Green function satisfying

- Δ G_{L} = 4 π (δ_{0} - L^{- 3})

. For pairwise distinct

x_{1}, \dots, x_{N} \in Λ_{L}

, define the neutral-jellium energy, with the point self-energies omitted, by

E_{L} (x_{1}, \dots, x_{N}) = 1 \leq i < j \leq N \sum G_{L} (x_{i} - x_{j}) - i = 1 \sum N \int_{Λ_{L}} G_{L} (x_{i} - y) d y + \frac{1}{2} \iint_{Λ_{L}^{2}} G_{L} (x - y) d x d y .

Then

L \to \infty L^{3} \in N lim L^{- 3} x_{1}, \dots, x_{L^{3}} \in Λ_{L} x_{i} \neq = x_{j} (i \neq = j) in f E_{L} (x_{1}, \dots, x_{L^{3}}) = e_{BCC},

where

e_{BCC}

is the thermodynamic energy per unit volume of the unit-density body-centered-cubic lattice under the same normalization.

MATH-PHYS-006

Sharp three-dimensional Lieb–Thirring constant

MajorExact constant problem1976

For every real

V \in L^{5/2} (R^{3})

, if

{λ_{j}}

are the negative eigenvalues of

- Δ + V

, counted with multiplicity, then

j \sum ∣ λ_{j} ∣ \leq \frac{1}{15 π ^{2}} \int_{R^{3}} V_{-} (x)^{5/2} d x, V_{-} = max {- V, 0},

and

1/ (15 π^{2})

is the optimal universal constant.

MATH-PHYS-007

Optimal Lieb–Oxford constant

MajorExact constant problemc. 1981

For each

N \geq 1

, let

P_{N}

be the symmetric probability measures

P

(R^{3})^{N}

with finite Coulomb energy and one-particle density

ρ_{P}

, normalized by

\int_{R^{3}} ρ_{P} = N

. Define

E_{ind} (P) = \int_{(R^{3})^{N}} i < j \sum \frac{1}{∣ x _{i} - x _{j} ∣} d P (x_{1}, \dots, x_{N}) - \frac{1}{2} \iint_{R^{3} \times R^{3}} \frac{ρ _{P} ( x ) ρ _{P} ( y )}{∣ x - y ∣} d x d y

and

C_{LO} = N \geq 1 sup P \in P_{N} sup \frac{- E _{ind} ( P )}{\int _{R^{3}} ρ _{P} ( x ) ^{4/3} d x} .

Q_{L} = [- L /2, L /2]^{3}

, define

e_{UEG} = L \to \infty L^{3} \in N lim L^{- 3} in f {E_{ind} (P) : P \in P_{L^{3}}, ρ_{P} = 1_{Q_{L}}} .

Then

C_{LO} = - e_{UEG}

MATH-PHYS-010

Homogeneous interacting Bose-gas condensation

MajorConjecture1925

Let

v \geq 0

be a nonzero radial finite-range potential on

R^{3}

with scattering length

a

, let

Λ_{L} = (R / L Z)^{3}

, and let

v_{L}

be the

L

-periodic extension of

v

. For a normalized bosonic ground state of

H_{N, L} = - i = 1 \sum N Δ_{i} + i < j \sum v_{L} (x_{i} - x_{j}) on L_{sym}^{2} (Λ_{L}^{N}),

let

γ_{N, L}^{(1)}

be its one-particle density matrix, normalized by

Tr γ_{N, L}^{(1)} = N

. Then

ρ a^{3} ↓ 0 lim N, L \to \infty N / L^{3} = ρ lim \frac{λ _{m a x} ( γ _{N, L}^{(1)} )}{N} = 1.