**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:**

**Algebra**

Homological conjectures in commutative algebra

Hilbert’s fifteenth and sixteenth problem

Hadamard conjecture

Jacobson’s conjecture

Zauner’s conjecture

Wild Problem

Köthe conjecture

Birch–Tate conjecture

Serre’s conjecture II

Bombieri–Lang conjecture

Farrell–Jones conjecture

Bost conjecture

Uniformity conjecture

Kaplansky’s conjecture

Kummer–Vandiver conjecture

Serre’s multiplicity conjectures

Pierce–Birkhoff conjecture

Eilenberg–Ganea conjecture

Green’s conjecture

Grothendieck–Katz p-curvature conjecture

Sendov’s conjecture

**Algebraic geometry**

André–Oort conjecture

Bass conjecture

Deligne conjecture

Fröberg conjecture

Fujita conjecture

Hartshorne conjectures

Manin conjecture

Nakai conjecture

Resolution of singularities in characteristic p

Standard conjectures on algebraic cycles

Section conjecture

Tate conjecture

Virasoro conjecture

Whitehead conjecture

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)

**Analysis**

The Jacobian conjecture

Schanuel’s conjecture and four exponentials conjecture

Lehmer’s conjecture

Pompeiu problem

Khabibullin’s conjecture on integral inequalities

Hilbert’s thirteenth problem

Vitushkin’s conjecture

**Discrete geometry**

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

Hopf conjecture

Kakeya conjecture

**Euclidean geometry**

The Einstein problem

Inscribed square problem

Moser’s worm problem

The moving sofa problem

Dürer’s conjecture

The Thomson problem

Pentagonal tiling

Falconer’s conjecture

g-conjecture

Circle packing in an equilateral triangle

Circle packing in an isosceles right triangle

**Dynamical systems**

Furstenberg conjecture

Margulis conjecture

MLC conjecture

Weinstein conjecture

Is every reversible cellular automaton in three or more dimensions locally reversible?

**Graph theory**

Barnette’s conjecture

The Erdős–Gyárfás conjecture

The Erdős–Hajnal conjecture

The Hadwiger conjecture

The Erdős–Faber–Lovász conjecture

Harborth’s conjecture

The total coloring conjecture

Hadwiger conjecture (graph theory)

The list coloring conjecture

The Ringel–Kotzig conjecture

The Hadwiger–Nelson problem

Lovász conjecture

Tutte’s conjectures

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

Negami’s conjecture

The Blankenship–Oporowski conjecture

Hedetniemi’s conjecture

Vizing’s conjecture

**Group theory**

Is every finitely presented periodic group finite?

The inverse Galois problem

Is every group surjunctive?

Andrews–Curtis conjecture

Herzog–Schönheim conjecture

Does generalized moonshine exist?

**Model theory**

Vaught’s conjecture

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

Kueker’s 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

**Ramsey theory**

The values of the Ramsey numbers

The values of the Van der Waerden numbers

Erdős–Burr conjecture