My favorite unsolved problems

Prize policies: when applicable, a prize for the problem will be payed with a check in US dollars to the first person who sends me a complete solution which I will verify and recognize as correct.

DIFFERENTIAL EQUATIONS AND ITERATION IN THE COMPLEX DOMAIN:

pdf Malmquist's general theorem of 1941.
pdf Fuchsian equations with PSU(2) monodromy (2019).
pdf Heun's equation with PSU(2) monodromy (2014). Updated in May 2019.
pdf A remarkable identity (March 2011). Progress of April 11, 2011. Now solved by Mukhin and Tarasov.
pdf Meromorphic functions satisfying certain differential equations (E. Hille) Updated on May 22, 2007.
pdf Wandering domains of entire functions.
pdf Escaping sets for entire functions. On 3.15.2022, Rempe, Waterman and Marti-Pete announced a counterexample.
pdf Two problems on completely invariant domains.

PARTIAL DIFFERENTIAL EQUATIONS:

pdf Backward uniqueness for the heat equation (2014).

GEOMETRY in 2 DIMENSIONS:

pdf Possible shapes of spherical quadrilaterals (2014). Recent progress: Classification of generic spherical quadrilaterals by A. Gabrielov.

ENTIRE AND MEROMORPHIC FUNCTIONS:

pdf Entire functions with zeros and ones on three rays and a functional equation (September 2015).
pdf A problem of B. Ya. Levin. A counterexample by Lev Buhovski.
pdf Exceptional set in Gross' Theorem.
pdf Gross' property of implicit functions.
pdf Three problems on level sets of subharmonic functions arising in the value distribution theory.
pdf A conjecture of Fuchs $200.
pdf Zeros of sums of simple fractions. (Solved by F. Nazarov in March 2015, unpublished.)
pdf Does every universe contain a place where you can stay at rest? (Lee Rubel) $200

GEOMETRIC FUNCTION THEORY AND POTENTIAL THEORY:

pdf A problem of potential theory arising in biology (2015) (Negative) solution of Paata Ivanishvili
pdf Goldberg's constant and its relatives (2014).
pdf A general conjecture about meromorphic functions in the plane (2013). (Counterexamples were recently constructed by David Sixsmith and Lasse Rempe).
pdf Some constants studied by Littlewood (Updated Oct 2002).
pdf "Hawaii Conjecture" (attributed to Gauss). Now solved by Michail Tyaglov.
pdf Equilibrium points of Newtonian potentials (Maxwell)
pdf Erdos' problem on the length of lemniscates (at least $200). Substantial recent progress is due to F. Nazarov and A. Fryntov.
pdf Find the maximum of one function of one variable.

SUBHARMONIC FUNCTIONS

An inequality for subharmonic functions

TRANSCENDENTAL HOLOMORPHIC CURVES:

pdf Deficiencies of targets of codimension greater than 1.
pdf Holomorphic curves with bounds on the spherical derivative.
pdf Modified Cartan's Conjecture. $500
pdf Holomorphic curves with few inflection points. $500
pdf Reversion of Cartan's SMT.

RATIONAL FUNCTIONS AND RATIONAL CURVES:

pdf Rational curves with real inflection points. (B. and M. Shapiro, for more info, see F. Sottile's page)
Now solved by Mukhin, Tarasov and Varchenko.

pdf Disconjugacy and Secant conjecture
pdf Three real polynomials (a short and simple restatement of the previous problem).
pdf Analytic invariant curves for rational functions. (Recent progress by Peter Mueller on Math Overflow).

POLYNOMIAL MATRICES (Control theory)

pdf Generic stabilizability by static output feedback.
pdf Simultaneous stabilizability of 3 systems.

NONE OF THE ABOVE

pdf Traces of elements of the modular group. Now solved by Bin Wang and Xunyun Zhu

Other interesting items in this site:

Progress report on some problems from Hayman's Collection

When function theory became an obsolete subject?
(Excerpt from a letter of Mittag-Leffler to Kowalevski.)

What is mathematics? Some expert's opinions.

jokes related to complex analysis

some problems, whose solutions I do know (level: undergraduate+)

stories about ODE, calculus and history of science

This page is partially supported by the NSF grants.