Unsolved Problems

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

Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=...

What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?...

Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?...

If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplici...

For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite?...

Is every finite group the Galois group of some Galois extension of $\mathbb{Q}$?...

Does generalized moonshine exist for all elements of the Monster group?...

Is every finitely presented periodic group finite?...

Is every discrete countable group sofic?...

What is the structure of the discrete spectrum of automorphic forms on reductive groups?...

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

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

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....

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

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

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

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?...

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

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?...