A word of explanation is in order for this page. There was once an eccentric Hungarian mathematician named Paul Erdös. Erdös used to advertize prized problems in his lectures around the world. Namely, he used to offer specific dollar inducements for certain problems he considered particularly important. While there is no proof that any problems were actually solved because of such inducement, he certainly succeeded in amusing the audience. There is perhaps nothing wrong with amusing the public. Furthermore, offering a prize draws attention to specific problems that could stimulate further research. A list of over half a dozen such problems may be found below.

Berger prize (systolic ratio of quaternionic projective plane)

Filling prize (filling area conjecture)

Loewner prize (Loewner's torus inequality in genus 3)

Lusternik-Schnirelmann prize (equality of LS and systolic category
in dimension 4)

Poincaré prize (Ricci flow and systoles)

Rodin prize (asymptotic behavior of systole of surfaces of large
genus)

More: systoles of 4-manifolds

Joyce prize: systoles of 8-manifolds and Spin(7) holomony