Unsolved Problems in Mathematics


Erdős conjecture on arithmetic progressions

Image via Wikipedia

Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six yet to be solved are:

Only the Poincaré conjecture has been solved. The smooth four dimensional Poincaré conjecture is still unsolved. That is, can a four dimensional topological sphere have two or more inequivalent smooth structures?

Other still-unsolved problems

Additive number theory

Number theory: prime numbers

General number theory

Algebraic number theory

Discrete geometry

Ramsey theory

General algebra

Combinatorics

  • Number of Magic squares (sequence A006052 in OEIS)
  • Finding a formula for the probability that two elements chosen at random generate the symmetric group Sn
  • Frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
  • The Lonely runner conjecture: if k + 1 runners with pairwise distinct speeds run round a track of unit length, will every runner be “lonely” (that is, be more than a distance 1 / (k + 1)from each other runner) at some time?
  • Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?

Graph theory

Analysis

Partial differential equations

  • Regularity of solutions of Vlasov-Maxwell equations
  • Regularity of solutions of Euler equations

Group theory

Set theory

Other

Advertisements

One thought on “Unsolved Problems in Mathematics

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s