Unsolved Problems

For a finite group $G$ and prime $p$, is the number of irreducible characters of degree not divisible by $p$ equal to the corresponding number for the...

Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...

Can all regular languages be expressed with generalized regular expressions of bounded star height?...

Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function?...

Is $\|f(A)\| \leq 2\sup_{z \in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?...

Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal?...

Do symmetric informationally complete POVMs exist in all dimensions?...

Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?...

Can a finite system of left cosets forming a partition of a group have distinct indices?...

Is the number of countable models of a complete first-order theory finite, $\aleph_0$, or $2^{\aleph_0}$?...

Is every simple group with $\aleph_0$-stable theory an algebraic group over an algebraically closed field?...

Relate the order of the center of the Steinberg group of the ring of integers to the Dedekind zeta function....

If a polynomial of degree d over a field of characteristic 0 shares a factor with each of its first d-1 derivatives, must it be $(x-a)^d$?...

Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?...

Is $\|f(A)\| \leq 2 \sup_{z \in W(A)} |f(z)|$ for any matrix A and analytic function f on the numerical range W(A)?...

Characterize the determinant of the sum of two normal matrices....

Does every group with cohomological dimension 2 have a 2-dimensional Eilenberg-MacLane space K(G,1)?...

Are the assembly maps in algebraic K-theory and L-theory isomorphisms?...

Is every finite lattice isomorphic to the congruence lattice of some finite algebra?...

Does a Hadamard matrix of order 4k exist for every positive integer k?...

If a ring has no nil two-sided ideal besides {0}, does it also have no nil one-sided ideal besides {0}?...

Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?...

Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?...

Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...

Can all regular languages be expressed with generalized regular expressions having bounded star height?...

For which number fields is there an algorithm to determine if a Diophantine equation has solutions?...

Does every complete first-order theory in a countable language have countably many, $\aleph_0$, or $2^{\aleph_0}$ countable models?...

Is the theory of the real numbers with addition, multiplication, and exponentiation decidable?...

Is every infinite field with a stable first-order theory separably closed?...

Do Henson graphs have the finite model property?...

Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function?...

Is every infinite minimal field of characteristic zero algebraically closed?...

Determine the structure of Keisler's order on first-order theories....

For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$?...

If R is a regular local ring and P,Q are prime ideals with $\dim(R/P) + \dim(R/Q) = \dim(R)$, is $\chi(R/P, R/Q) > 0$?...

Is there a bound N(g,d) such that all curves of genus g≥2 over degree d number fields have at most N(g,d) rational points?...