Several prizes are waiting to be awarded for outstanding mathematical achievement. The most prestigious mathematical award is the Fields medal. Other awards are the Wolf Prize from the Wolf Foundation of Israel, the Leroy P. Steele Prize of the American Mathematical Society, the Bôcher Prize, Cole Prizes in algebra and number theory, and the Delbert Ray Fulkerson Prize, all presented by the American Mathematical Society.
In 2000, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) named seven “Millennium Prize Problems,” selected by focusing on important classic questions in mathematics that have resisted solution over the years. A $7 million prize fund was established for the solution to these problems, with $1 million allocated to each. The problems are:
1. Riemann hypothesis
2. Poincaré conjecture (the correct solution was confirmed in 2006, the prize was turned down)
3. Hodge conjecture
4. Swinnerton-Dyer conjecture
5. solution of the Navier-Stokes equations
6. formulation of Yang-Mills theory
7. determination of whether NP-problems are actually P-problems.
There are too many unsolved mathematical problems to include them all in this single article. Listed below are some of the unsolved mathematical problems by category:
Homological conjectures in commutative algebra
Hilbert’s fifteenth and sixteenth problem
Serre’s conjecture II
Serre’s multiplicity conjectures
Grothendieck–Katz p-curvature conjecture
Resolution of singularities in characteristic p
Standard conjectures on algebraic cycles
Zariski multiplicity conjecture
Algebraic number theory
Are there infinitely many real quadratic number fields with unique factorization?
Characterize all algebraic number fields that have some power basis.
Stark conjectures (including Brumer–Stark conjecture)
The Jacobian conjecture
Schanuel’s conjecture and four exponentials conjecture
Khabibullin’s conjecture on integral inequalities
Hilbert’s thirteenth problem
Solving the happy ending problem for arbitrary n
Finding matching upper and lower bounds for k-sets and halving lines
The Hadwiger conjecture
The Kobon triangle problem
The McMullen problem
Ulam’s packing conjecture
Filling area conjecture
The Einstein problem
Inscribed square problem
Moser’s worm problem
The moving sofa problem
The Thomson problem
Circle packing in an equilateral triangle
Circle packing in an isosceles right triangle
Is every reversible cellular automaton in three or more dimensions locally reversible?
The Erdős–Gyárfás conjecture
The Erdős–Hajnal conjecture
The Hadwiger conjecture
The Erdős–Faber–Lovász conjecture
The total coloring conjecture
Hadwiger conjecture (graph theory)
The list coloring conjecture
The Ringel–Kotzig conjecture
The Hadwiger–Nelson problem
Petersen coloring conjecture
The reconstruction conjecture
The cycle double cover conjecture
Does a Moore graph with girth 5 and degree 57 exist?
Conway’s thrackle conjecture
The Blankenship–Oporowski conjecture
Is every finitely presented periodic group finite?
The inverse Galois problem
Is every group surjunctive?
Does generalized moonshine exist?
The Cherlin–Zilber conjecture
Determine the structure of Keisler’s order
The stable field conjecture
The Stable Forking Conjecture for simple theories
For which number fields does Hilbert’s tenth problem hold?
Shelah’s eventual Categority conjecture
Lachlan’s decision problem
Do the Henson graphs have the finite model property?
The universality problem for C-free graphs
The universality spectrum problem
Partial differential equations
Regularity of solutions of Vlasov–Maxwell equations
Regularity of solutions of Euler equations
The values of the Ramsey numbers
The values of the Van der Waerden numbers