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Showing 55 of 280 problems in Analysis and functional analysis

NT-AN-001

Generalized Riemann Hypothesis for Dirichlet $L$ -functions

LandmarkConjecturec. 1884

For every primitive Dirichlet character

χ

modulo an integer

q \geq 1

, every zero

ρ

of the analytically continued Dirichlet function

L (s, χ) = n = 1 \sum \infty \frac{χ ( n )}{n ^{s}} (ℜ s > 1)

satisfying

0 < ℜ ρ < 1

has

ℜ ρ = \frac{1}{2}

NT-PRIME-009

Montgomery’s Pair-Correlation Conjecture

LandmarkConjecture1973

Assume the Riemann hypothesis, write the nontrivial zeros of the Riemann zeta function as

\frac{1}{2} + iγ

, counted with multiplicity, and put

N (T) = # {γ : 0 < γ \leq T}

. For every fixed

0 < α < β

\frac{1}{N ( T )} # {(γ, γ^{'}) : 0 < γ, γ^{'} \leq T, α \leq \frac{( γ - γ ^{'} ) lo g T}{2 π} \leq β} ⟶ \int_{α}^{β} (1 - (\frac{sin π u}{π u})^{2}) d u

T \to \infty

NT-CORR-035

Chowla’s Correlation Conjecture

LandmarkConjecture1965

Let

λ (n) = (- 1)^{Ω (n)}

be the Liouville function, where

Ω (n)

is the number of prime factors of

n

, counted with multiplicity. For every integer

k \geq 2

and every choice of distinct nonnegative integers

h_{1}, \dots, h_{k}

n \leq x \sum j = 1 \prod k λ (n + h_{j}) = o (x) (x \to \infty) .

NT-AN-004

Landau’s $n^2+1$ Prime Conjecture

MajorConjecture1912

There are infinitely many positive integers

n

for which

n^{2} + 1

is prime.

NT-PRIME-008

Irrationality of the Euler–Mascheroni Constant

MajorOpen problemUnknown

The number

γ = n \to \infty lim (k = 1 \sum n \frac{1}{k} - lo g n)

is irrational.

NT-AN-028

Dirichlet Divisor-Problem Exponent Conjecture

MajorExact constant problemc. 1916

Let

d (n) = # {a \in Z_{> 0} : a ∣ n}

and

γ = lim_{N \to \infty} (\sum_{k = 1}^{N} 1/ k - lo g N)

, and put

Δ (x) = n \leq x \sum d (n) - x lo g x - (2 γ - 1) x .

For every

ε > 0

Δ (x) = O_{ε} (x^{1/4 + ε}) (x \to \infty) .

ALG-RING-005

Kaplansky Idempotent Conjecture

MajorConjecturec. 1950

G

is a torsion-free group and

e \in C [G]

satisfies

e^{2} = e

, then

e = 0

e = 1

AG-HYP-022

Green–Griffiths–Lang Conjecture

LandmarkConjecture1980

Let

X

be a smooth projective complex variety of general type. There exists a proper Zariski-closed subset

Y ⊊ X

such that every nonconstant holomorphic map

f : C \to X

satisfies

f (C) \subseteq Y

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-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-013

Invariant subspace problem for Hilbert space

LandmarkOpen problem1935–1950

For every separable infinite-dimensional complex Hilbert space

H

and every bounded linear operator

T : H \to H

, there is a closed linear subspace

M

such that

{0} ⊊ M ⊊ H, T (M) \subseteq M .

ANAL-FA-006

Kakeya set conjecture in dimensions $n\ge4$

LandmarkConjecturec. 1971

Let

n \geq 4

and let

E \subset R^{n}

be a Borel set such that for every

v \in S^{n - 1}

there is

x_{v} \in R^{n}

with

{x_{v} + t v : 0 \leq t \leq 1} \subset E

. Then

dim_{H} E = n

ANAL-FA-007

Fourier restriction conjecture for the sphere

LandmarkConjecturec. 1967

Let

n \geq 3

, let

d σ

be surface measure on

S^{n - 1}

, and for

1 \leq p, q < \infty

define

E g (x) = \int_{S^{n - 1}} e^{2 πi x \cdot ω} g (ω) d σ (ω) .

q > \frac{2 n}{n - 1} and \frac{n + 1}{q} \leq \frac{n - 1}{p ^{'}}, p^{'} = \frac{p}{p - 1}

(with

p^{'} = \infty

for

p = 1

), then there is

C = C (n, p, q)

such that

∥ E g ∥_{L^{q} (R^{n})} \leq C ∥ g ∥_{L^{p} (S^{n - 1}, d σ)}

for every

g \in L^{p} (S^{n - 1}, d σ)

ANAL-FA-008

Kannan–Lovász–Simonovits (KLS) conjecture

LandmarkConjecture1995

There is a universal constant

C < \infty

such that for every

n

, every log-concave probability measure

μ

R^{n}

satisfying

\int x d μ (x) = 0

and

\int x_{i} x_{j} d μ (x) = δ_{ij}

, and every locally Lipschitz

f : R^{n} \to R

Var_{μ} (f) \leq C \int_{R^{n}} ∥\nabla f (x) ∥_{2}^{2} d μ (x) .

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-018

Free-group-factor isomorphism problem

LandmarkOpen problemc. 1943

For each integer

n \geq 2

, let

F_{n}

be the free group on

n

generators, let

λ_{n}

be its left regular representation on

ℓ^{2} (F_{n})

, and define the free group factor

L (F_{n}) = {λ_{n} (g) : g \in F_{n}}^{''}

. Determine which of the following alternatives holds:

L (F_{m}) ≅ L (F_{n})

for every

m, n \geq 2

, or

L (F_{m}) \neq ≅ L (F_{n})

whenever

m \neq = n

, where

≅

denotes a unital normal

*

-isomorphism of von Neumann algebras.

ANAL-FA-002

Separable quotient problem

MajorOpen problem1932

Every infinite-dimensional Banach space

X

has a closed linear subspace

Y \subset X

such that the quotient Banach space

X / Y

is both separable and infinite-dimensional.

ANAL-FA-004

Mazur rotation problem

MajorOpen problem1932

Let

X

be a separable real Banach space such that for every

x, y \in X

with

∥ x ∥ = ∥ y ∥ = 1

, there is a surjective linear isometry

U : X \to X

with

Ux = y

. Then

X

is linearly isometric to a real Hilbert space.

ANAL-FA-005

Beurling–Ahlfors operator norm conjecture

MajorExact constant problem1982

For

1 < p < \infty

, let

d A

be Lebesgue area measure and define the Beurling--Ahlfors transform on Schwartz functions by

B f (z) = - \frac{1}{π} p.v. \int_{C} \frac{f ( w )}{( z - w ) ^{2}} d A (w), p^{*} = max {p, \frac{p}{p - 1}} .

Then its

L^{p} (C)

-operator norm is

∥ B ∥_{L^{p} \to L^{p}} = p^{*} - 1

ANAL-FA-011

Corona problem for the unit ball

MajorOpen problemc. 1962

Let

n \geq 2

and

m \geq 1

be integers, let

B^{n} = {z \in C^{n} : ∥ z ∥_{2} < 1}

, and let

H^{\infty} (B^{n})

be the algebra of bounded holomorphic functions on

B^{n}

. If

f_{1}, \dots, f_{m} \in H^{\infty} (B^{n})

satisfy

z \in B^{n} in f j = 1 \sum m ∣ f_{j} (z) ∣ > 0,

then there exist

g_{1}, \dots, g_{m} \in H^{\infty} (B^{n})

such that

\sum_{j = 1}^{m} f_{j} (z) g_{j} (z) = 1

for every

z \in B^{n}

ANAL-FA-013

Two-dimensional Fuglede conjecture

MajorCanonical finite case1974

Let

Ω \subset R^{2}

be a bounded Lebesgue measurable set of positive measure. The following are equivalent: (i) there is a countable

Λ \subset R^{2}

such that

{∣Ω ∣^{- 1/2} e^{2 πiλ \cdot x} : λ \in Λ}

is an orthonormal basis of

L^{2} (Ω)

; (ii) there is a countable

T \subset R^{2}

such that

\sum_{t \in T} 1_{Ω} (x - t) = 1

for almost every

x \in R^{2}

ANAL-FA-016

Diagonal Fourier extension conjecture for compact paraboloids

MajorConjecturec. 1967

Let

d \geq 3

, let

P^{d - 1} = {(ξ, ∣ ξ ∣^{2}) : ξ \in R^{d - 1}} \subset R^{d}

, and let

d σ = η d S

, where

d S

is surface measure and

0 \leq η \in C_{c}^{\infty} (P^{d - 1})

is nonzero. Define

E_{σ} f (x) = \int_{P^{d - 1}} e^{2 πi x \cdot ω} f (ω) d σ (ω) .

For every

q > 2 d / (d - 1)

there is a constant

C = C (d, q, σ)

such that

∥ E_{σ} f ∥_{L^{q} (R^{d})} \leq C ∥ f ∥_{L^{q} (d σ)}

for every

f \in L^{q} (d σ)

ANAL-FA-019

Connes Rigidity Conjecture for Property- $(T)$ Groups

MajorConjecturec. 1982

G

and

H

are countable ICC groups with Kazhdan's property

(T)

, then every isomorphism of their group von Neumann algebras,

L (G) ≅ L (H),

implies that

G ≅ H

. Here ICC means that every nonidentity conjugacy class is infinite.

ANAL-FA-020

Hyperlinear Group Conjecture

MajorConjecturec. 2000

Every countable group

G

is hyperlinear: there are integers

n_{k}

and an injective homomorphism

G ⟶ k \prod U (n_{k}) / k ⨁ U (n_{k}),

where the metric ultraproduct uses normalized Hilbert--Schmidt distance and a nonprincipal ultrafilter.

ANAL-FA-021

Heil–Ramanathan–Topiwala Conjecture

MajorConjecture1996

For every nonzero

g \in L^{2} (R)

and every finite set of distinct points

(a_{j}, b_{j}) \in R^{2}

, the time--frequency translates

e^{2 πi b_{j} x} g (x - a_{j})

are linearly independent in

L^{2} (R)

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}

DYN-003

Density of hyperbolicity for rational maps

LandmarkConjecturec. 1920

For every

d \geq 2

, hyperbolic maps are dense in the space

Rat_{d}

of degree-

d

rational maps

f : C \to C

, with its coefficient topology. Here

f

is hyperbolic when the closure of the forward orbits of all its critical points is disjoint from its Julia set

J (f)

DYN-004

Mandelbrot local-connectivity conjecture

LandmarkConjecturec. 1982

The Mandelbrot set

M = {c \in C : {f_{c}^{n} (0) : n \geq 0} is bounded}, f_{c} (z) = z^{2} + c,

is locally connected: for every

c \in M

and every neighborhood

U

c

, there is a connected neighborhood

V

c

in the subspace

M

with

V \subset U

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-011

Banach simple-Lebesgue-spectrum problem

MajorOpen problem1932

There exist a standard nonatomic probability space

(X, B, μ)

and an invertible ergodic measure-preserving transformation

T : X \to X

such that the Koopman operator

U_{T} f = f \circ T

, restricted to

L_{0}^{2} (X, μ) = {f \in L^{2} (X, μ) : \int_{X} f d μ = 0},

is unitarily equivalent to the bilateral shift

S

ℓ^{2} (Z)

, defined by

S (e_{n}) = e_{n + 1}

DG-004

Falconer's Distance Set Conjecture

LandmarkConjecture1985

Let

d \geq 2

and let

E \subset R^{d}

be compact. Write

dim_{H}

for Hausdorff dimension. If

dim_{H} (E) > \frac{d}{2},

then its distance set

Δ (E) = {∥ x - y ∥_{2} : x, y \in E}

has positive one-dimensional Lebesgue measure.

DG-018

High-Dimensional Sphere-Packing Exponent

LandmarkOpen problemc. 1900

Let

Δ_{d}

be the supremal density of a packing of congruent balls in

R^{d}

. Determine the exponential asymptotics of

Δ_{d}

, equivalently determine

d \to \infty lim inf \frac{1}{d} lo g_{2} Δ_{d} and d \to \infty lim sup \frac{1}{d} lo g_{2} Δ_{d},

and decide whether these two quantities are equal.

STAT-002

Gaussian Product Inequality Conjecture

MajorConjecture2012

For every integer

n \geq 1

, every centered jointly Gaussian random vector

X = (X_{1}, \dots, X_{n})

with

Var (X_{i}) > 0

for all

i

, and every choice of real exponents

α_{1}, \dots, α_{n} > 0

E [i = 1 \prod n ∣ X_{i} ∣^{α_{i}}] \geq i = 1 \prod n E [∣ X_{i} ∣^{α_{i}}] .

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-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-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.

CTRL-001

Generalized Belgian Chocolate Problem

MajorExact constant problem1994

Call a nonzero real polynomial Hurwitz stable if all of its complex zeros have strictly negative real part. Define

δ_{*} = sup {δ \in (0, 1) : there exist nonzero Hurwitz-stable x, y \in R [s] such that (s^{2} - 2 δs + 1) x (s) + (s^{2} - 1) y (s) is Hurwitz stable} .

Determine

δ_{*}

exactly.