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 June 19, 2023.

pdf
Wandering domains of entire functions.

pdf
Escaping sets for entire functions. On 3.15.2022, Rempe, Waterman and Marti-Pete announced a counterexample, their paper on arXiv

pdf Two problems on completely invariant domains.

pdf Analytic
invariant curves for rational functions.

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 Composite periodic entire functions
(October 2023).

pdf
Entire functions with zeros and ones on three rays and a functional equation
(September 2015).

pdf A problem of B. Ya. Levin.

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unpublished.)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 (Duval and da Costa).

pdf
Modified Cartan's Conjecture. $500. Recent progress by
Katsutoshi Yamanoi

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.
Steven Karp and Kevin Purbhoo announced
a proof of this conjecture.

pdf Three real polynomials (a short and
simple restatement
of the previous problem).

POLYNOMIAL MATRICES (Control theory)

pdf
Generic stabilizability by static output feedback (Byrns and Anderson).

pdf
Simultaneous stabilizability of 3 systems (Blondel).

NONE OF THE ABOVE

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

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 was partially supported by the NSF grants.