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Showing 49 of 280 problems in Algebra and representation theory

NT-AUTO-030

Generalized Ramanujan Conjecture for $\mathrm{GL}_n$

LandmarkConjecturec. 1967

For every

n \geq 2

, every irreducible unitary cuspidal automorphic representation

π = ⨂_{v}^{'} π_{v}

GL_{n} (A_{Q})

, where

A_{Q}

is the adèle ring of

Q

, and every place

v

Q

, the local representation

π_{v}

is tempered; equivalently, all its

K_{v}

-finite matrix coefficients belong to

L^{2 + ε} (GL_{n} (Q_{v}) / Z_{v})

for every

ε > 0

, where

K_{v}

is a maximal compact subgroup and

Z_{v}

is the center.

NT-LFUNC-033

Artin Holomorphy Conjecture

LandmarkConjecture1923

Let

L / K

be a finite Galois extension of number fields with group

G

, and let

ρ : G \to GL (V)

be a nontrivial irreducible finite-dimensional complex representation. For every nonzero prime ideal

p

K

, choose a prime of

L

above it, let

I_{p}

be its inertia group, and let

Frob_{p}

denote an arithmetic Frobenius element acting on

V^{I_{p}}

. The Artin

L

-function

L (s, ρ, L / K) = p \prod det (1 - ρ (Frob_{p}) N p^{- s} ∣ V^{I_{p}})^{- 1}, ℜ s > 1,

has an analytic continuation to an entire function on

C

NT-PRIME-010

Artin’s Primitive Root Conjecture

MajorConjecture1927

Let

g \in Z ∖ {- 1, 0, 1}

be not a perfect square. For

n \geq 1

, let

ζ_{n} = e^{2 πi / n}

, let

μ (n)

be the Möbius function (zero if a prime square divides

n

, and

(- 1)^{r}

n

is a product of

r

distinct primes), and put

δ (g) = n = 1 \sum \infty \frac{μ ( n )}{[ Q ( ζ _{n} , g ^{1/ n} ) : Q ]} .

Then

δ (g) > 0

and, as

x \to \infty

# {p \leq x : p prime, p ∤ g, g generates (Z / p Z)^{\times}} \sim δ (g) Li (x),

where

Li (x) = \int_{2}^{x} d t / lo g t

; the field in the denominator is independent of the choice of the

n

-th root of

g

NT-MOD-019

Lehmer’s Nonvanishing Conjecture for Ramanujan’s $\tau$ -function

MajorConjecture1947

Define integers

τ (n)

n = 1 \sum \infty τ (n) q^{n} = q m = 1 \prod \infty (1 - q^{m})^{24} (∣ q ∣ < 1) .

Then

τ (n) \neq = 0

for every integer

n \geq 1

NT-IWAS-023

Leopoldt’s Conjecture

MajorConjecture1962

Let

K

be a number field and

p

a rational prime. For each place

v ∣ p

, let

K_{v}

be the completion and let

lo g_{v} : O_{K_{v}}^{\times} \to K_{v}

be the local

p

-adic logarithm: if

m_{v}

is the maximal ideal of

O_{K_{v}}

, its restriction to

1 + m_{v}

lo g_{v} (1 + x) = m = 1 \sum \infty \frac{( - 1 ) ^{m + 1} x ^{m}}{m} (x \in m_{v}),

and it vanishes on torsion units. Then the map

O_{K}^{\times} \otimes_{Z} Q_{p} ⟶ v ∣ p \prod K_{v}, u ⟼ (lo g_{v} u)_{v ∣ p},

is injective.

NT-CYC-024

Vandiver’s Conjecture

MajorConjecturec. 1929

For every prime

p

, let

ζ_{p} = e^{2 πi / p}

and let

h_{p}^{+}

be the ideal-class number of the maximal real subfield

Q (ζ_{p} + ζ_{p}^{- 1})

of the

p

-th cyclotomic field. Then

p ∤ h_{p}^{+}

NT-SPECTRAL-029

Selberg Eigenvalue Conjecture

MajorConjecture1965

Let

Γ \leq SL_{2} (Z)

be a congruence subgroup, meaning that

Γ

contains

{g \in SL_{2} (Z) : g \equiv I (mod N)}

for some

N \geq 1

. If

λ > 0

is a discrete

L^{2}

-eigenvalue of the positive hyperbolic Laplacian

- y^{2} (\partial_{x}^{2} + \partial_{y}^{2})

Γ\ H

, where

H = {x + i y \in C : y > 0}

, then

λ \geq \frac{1}{4}

ALG-POLY-001

Plane Jacobian Conjecture

LandmarkCanonical finite case1939

F = (F_{1}, F_{2}) : C^{2} \to C^{2}

is a polynomial map and

det (\frac{\partial F _{i}}{\partial x _{j}})_{1 \leq i, j \leq 2} \in C^{\times},

then there is a polynomial map

G : C^{2} \to C^{2}

satisfying

G \circ F = F \circ G = id_{C^{2}}

ALG-GALOIS-002

Inverse Galois Conjecture over $\mathbb Q$

LandmarkOpen problemc. 1892

For every finite group

G

, there exists a finite Galois extension

L / Q

such that

Gal (L / Q) ≅ G

ALG-GALCOH-018

Serre’s Conjecture II

LandmarkConjecture1962

Let

k

be a perfect field of Galois cohomological dimension at most

2

, and let

G

be a semisimple simply connected linear algebraic group over

k

. Then

H^{1} (k, G) = {1}

; equivalently, every

G

-torsor over

Spec k

has a

k

-rational point.

ALG-GROUP-021

Soficity Conjecture

LandmarkConjecture2000

Every countable group is sofic. Explicitly, for every countable group

G

, finite

F \subseteq G

, and

ε > 0

, there are an integer

n

and a map

σ : G \to Sym (n)

such that

σ (1) = 1

d_{H} (σ (g h), σ (g) σ (h)) < ε (g, h \in F),

and

d_{H} (σ (g), 1) > 1 - ε

for every

g \in F ∖ {1}

, where

d_{H}

is normalized Hamming distance.

ALG-NC-003

Dixmier Conjecture for the First Weyl Algebra

MajorCanonical finite case1968

Let

A_{1} (C) = C ⟨ x, y ⟩ / (y x - x y - 1)

be the first Weyl algebra. Every unital

C

-algebra endomorphism

A_{1} (C) \to A_{1} (C)

is an automorphism.

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-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

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-RING-007

Köthe Conjecture

MajorConjecture1930

For every associative ring

R

(not necessarily unital), if

I, J \subseteq R

are left ideals such that every element of

I

and every element of

J

is nilpotent, then every element of

I + J

is nilpotent.

ALG-REP-008

Finitistic Dimension Conjecture

MajorConjecture1960

Let

A

be a finite-dimensional algebra over a field, and let

pd_{A} M

denote the projective dimension of a finitely generated left

A

-module

M

. Then

findim (A) = sup {pd_{A} M : M \in A - mod, pd_{A} M < \infty} < \infty.

ALG-REP-009

Auslander–Reiten Conjecture

MajorConjecture1975

Let

A

be an Artin algebra and

M

a finitely generated left

A

-module. If

Ext_{A}^{i} (M, M \oplus A) = 0 for every i > 0,

then

M

is projective.

ALG-REP-010

Nakayama Conjecture

MajorConjecture1958

Let

A

be a finite-dimensional algebra over a field and

0 ⟶_{A} A ⟶ I^{0} ⟶ I^{1} ⟶ I^{2} ⟶ \dots

its minimal injective resolution as a left module. If every

I^{i}

is projective, then

A

is self-injective.

ALG-MODREP-011

Broué’s Abelian Defect Group Conjecture

MajorConjecture1990

Let

p

be a prime, let

(K, O, k)

be a splitting

p

-modular system for a finite group

G

O

is a complete discrete valuation ring with characteristic-

0

fraction field

K

and algebraically closed residue field

k

of characteristic

p

, and both fields split every subgroup of

G

. Let

B

be a block algebra of

O G

with abelian defect group

D

. If

b

is the Brauer-correspondent block of

O N_{G} (D)

, then

D^{b} (B - mod_{fg}) ≃ D^{b} (b - mod_{fg})

as triangulated categories, where these are bounded derived categories of finitely generated left modules.

ALG-MODREP-012

Alperin Weight Conjecture

MajorConjecture1986

Let

G

be a finite group,

p

a prime, and

k

an algebraically closed field of characteristic

p

. A

p

-weight is a pair

(Q, ϕ)

, where

Q \leq G

is a

p

-subgroup and

ϕ

is an irreducible complex character of

N_{G} (Q) / Q

p

-defect zero. Here defect zero means that the

p

-part of

ϕ (1)

equals the

p

-part of

∣ N_{G} (Q) / Q ∣

. The number of isomorphism classes of simple

k G

-modules equals the number of

G

-conjugacy classes of

p

-weights.

ALG-MODREP-013

Donovan’s Conjecture

MajorConjecturec. 1976

Fix a prime

p

, an algebraically closed field

k

of characteristic

p

, and a finite

p

-group

D

. Among all finite groups

G

, there are only finitely many Morita-equivalence classes of block algebras

B

k G

whose defect groups are isomorphic to

D

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-GROUP-015

Boone–Higman Conjecture

MajorConjecture1974

Every finitely generated group with solvable word problem embeds into a finitely presented simple group.

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.

ALG-MODREP-019

Brauer’s $k(B)$ -Conjecture

MajorConjecture1946

Let

G

be a finite group, let

p

be a prime, and fix a splitting

p

-modular system for

G

. Let

B

be a

p

-block of

G

, meaning a two-sided summand of the modular group algebra cut out by a primitive central idempotent; let

Irr (B)

be the set of irreducible ordinary complex characters belonging to

B

, and let

D \leq G

be a defect group of

B

. Then

k (B) := # Irr (B) \leq ∣ D ∣.

ALG-ARR-020

Terao’s Freeness Conjecture

MajorConjecture1983

Let

A

be a finite central hyperplane arrangement in a finite-dimensional complex vector space

V

. For each

H \in A

, choose

α_{H} \in V^{*}

with

H = ker α_{H}

, and define the module of logarithmic derivations

D (A) = {θ \in Der_{C} C [V] : θ (α_{H}) \in α_{H} C [V] for every H \in A} .

Call

A

free when

D (A)

is a free

C [V]

-module, and let

L (A) = {H \in S ⋂ H : S \subseteq A}

be its intersection lattice, ordered by reverse inclusion. If two such arrangements

A

and

A^{'}

have isomorphic intersection lattices, then

A

is free if and only if

A^{'}

is free.

AG-SING-013

Resolution of Singularities in Positive Characteristic

LandmarkOpen problem1964

Let

k

be a perfect field of positive characteristic and

X

an integral separated scheme of finite type over

k

. There exist a smooth integral

k

-scheme

Y

and a proper birational morphism

f : Y \to X

that is an isomorphism over the regular locus

X_{reg}

AG-CYCLE-004

Lefschetz Standard Conjecture $B$

MajorConjecture1968

Let

X

be a smooth projective variety of dimension

d

over an algebraically closed field, let

H^{*}

be a Weil cohomology theory with characteristic-

0

coefficient field, and let

L (α) = h ⌣ α

for the class

h

of an ample divisor. For every

i < d

, the inverse of the hard-Lefschetz isomorphism

L^{d - i} : H^{i} (X) \sim H^{2 d - i} (X)

is induced by an algebraic correspondence in

CH^{i} (X \times X) \otimes Q

, where

CH^{i}

denotes codimension-

i

algebraic cycles modulo rational equivalence.

AG-SURF-009

Nagata Conjecture

MajorConjecture1959

Let

r \geq 10

, let

p_{1}, \dots, p_{r}

be very general points of

P_{C}^{2}

, meaning outside a countable union of proper Zariski-closed subsets of the configuration space, and let

m_{1}, \dots, m_{r} \geq 0

. If a plane curve of degree

d

has multiplicity at least

m_{i}

at every

p_{i}

, then

d r > i = 1 \sum r m_{i} .

AG-RAT-018

Kuznetsov’s Rationality Conjecture for Cubic Fourfolds

MajorConjecture2010

Let

X \subset P_{C}^{5}

be a smooth cubic fourfold. In the bounded derived category

D^{b} (Coh X)

of coherent sheaves on

X

, put

A_{X} = ⟨ O_{X}, O_{X} (1), O_{X} (2) ⟩^{⊥},

where

^{⊥}

denotes the full right-orthogonal subcategory. Then

X

is rational if and only if there is a complex K3 surface

S

with an exact

C

-linear triangulated equivalence

A_{X} ≃ D^{b} (Coh S)

AG-ZERO-020

Bloch Conjecture for Surfaces with $p_g=0$

MajorConjecture1975

Let

X

be a smooth projective complex surface with

p_{g} (X) = h^{0} (X, K_{X}) = 0

. The Albanese map

alb_{X} : CH_{0} (X)^{0} ⟶ Alb (X) (C)

from degree-zero zero-cycles modulo rational equivalence is an isomorphism.

GEO-TOP-004

Hilbert–Smith conjecture

LandmarkConjecturec. 1940

Let

G

be a locally compact Hausdorff topological group and

M

a connected finite-dimensional topological manifold. If

G \times M \to M

is a continuous faithful action, then

G

is a Lie group.

GEO-TOP-005

Cannon conjecture

LandmarkConjecture1994

G

is a Gromov-hyperbolic group whose Gromov boundary

\partial G

is homeomorphic to

S^{2}

, then

G

admits a properly discontinuous, cocompact action by isometries on hyperbolic

3

-space

H^{3}

GEO-TOP-006

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-008

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-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.

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-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.

DYN-015

Gottschalk Surjunctivity Conjecture

MajorConjecture1973

For every group

G

and every finite alphabet

A

, every continuous

G

-equivariant injective map

τ : A^{G} \to A^{G}

, with

A^{G}

carrying the product topology and the shift action, is surjective.

COMB-011

Alon–Tarsi Latin Square Conjecture

MajorConjecture1990

Let

n

be even. Regard every row and every column of a Latin square

L

of order

n

as a permutation of its symbol set, and define

sgn (L) = i = 1 \prod n sgn (row_{i}) j = 1 \prod n sgn (column_{j}) .

The number of Latin squares of order

n

with sign

+ 1

is different from the number with sign

- 1

TCS-007

Valiant's $VP$ versus $VNP$ Conjecture

LandmarkConjecture1979

Over

Q

V P \neq = V NP .

The class

V P

consists of polynomial families

(f_{n})

, with

f_{n} \in Q [x_{1}, \dots, x_{p (n)}]

for some polynomial

p

, whose degrees and arithmetic-circuit sizes are bounded by polynomials in

n

. The class

V NP

consists of polynomial families

(g_{n})

for which there are a polynomial

q

and a family

(h_{n}) \in V P

such that

g_{n} (x) = y \in {0, 1}^{q (n)} \sum h_{n} (x, y) .

TCS-018

Matrix Multiplication Exponent $\omega=2$

LandmarkConjecturec. 1969

Over the concrete field

Q

, let

ω = in f {τ : two n \times n matrices can be multiplied using O (n^{τ + ε}) field operations for every ε > 0} .

Then

ω = 2.

TCS-024

Deterministic Polynomial Identity Testing

LandmarkOpen problemc. 1980

Is there a deterministic polynomial-time algorithm which, given a division-free arithmetic circuit over

Q

with binary-encoded rational constants, decides whether the formal polynomial computed by the circuit is identically zero, in time polynomial in the circuit's binary encoding length?

LOG-MODEL-002

Cherlin–Zilber Algebraicity Conjecture

MajorConjecturec. 1979

Every infinite simple group

G

of finite Morley rank is, as an abstract group, isomorphic to the group of

K

-rational points of a simple algebraic group over an algebraically closed field

K

LOG-MODEL-003

Stable Fields Conjecture

MajorConjecturec. 1971

K

is an infinite field whose complete first-order theory in the language of rings is stable, then

K

is separably closed.

LOG-MODEL-004

NIP Fields Conjecture

MajorConjecturec. 2015

Let

K

be an infinite field whose complete first-order theory in the language of rings has no independence property (is NIP). Then at least one of the following holds:

K

is separably closed;

K

is real closed; or

K

admits a nontrivial henselian valuation.