"In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi"
(Georg Cantor, a thesis in his PhD defense, Dissertation Berlin 1867, Theses. III. In: Gesammelte Abhandlungen, Hrsg. Ernst Zermelo, Verlag von Julius Springer, Berlin 1932, S. 31)
I used to have monetary prizes for some problems. In summer 2026 I decied to withdraw them, in view of the AI revolution which is going on.
I try to do my best to keep this list up-to-date. In the last week (September 20-25, 2026) this required daily updates, sometimes more than that.
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). Solved completely
by several people using AI in August-September 2026. Will be posted on arXiv.
pdf
Wandering domains of entire functions. Solved completely by Lasse Rempe, assisted by AI. See also
Drach,Pardo-Simon,Usacar and
Zihao Ye.
pdf
Escaping sets for entire functions. Solved.
Rempe, Waterman and Marti-Pete constructed a
counterexample,
J. Amer. Math. Soc. 38 (2025),
877-918
pdfTwo problems on completely invariant domains.
pdfAnalytic
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 A problem proposed by AI.
pdf Linear combinations with prescribed
indicators (July 2026)
pdf Composite periodic entire functions
(October 2023).
pdf
Entire functions with zeros and ones on three rays and a functional equation
(September 2015).
pdf Exceptional
set in Gross' Theorem.
A preprint by Sina Nadi
is posted with an example of the Gross exceptional set of Hausdorff
dimension one. So Gross's theorem is best possible!
pdf Gross'
property of implicit functions. This conjecture is completely disproved
now in the
preprint by Sina Nadi
pdf
Three problems on level sets of subharmonic
functions arising in the
value distribution theory.
Problem 1 solved
using AI in September 2026.
pdf A conjecture of Fuchs.
Solved
using AI.
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)
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). solved by Michail Tyaglov.
pdf
Equilibrium points of Newtonian potentials (Maxwell)
A counterexample
to Maxwell's conjecture is found, though it holds for 3 points.
All AI assisted.
Positive solution of Question 1 is
announced.
pdf Erdos'
problem on the length of lemniscates.
Substantial
recent progress is due to F. Nazarov and A. Fryntov. Further progress by T. Tao
pdf
Find the maximum of one function of one variable. Solved
in September 2026 by Teng Zhang, using AI, with computer-assisted proof
(a rigorous numerical computation).
SUBHARMONIC FUNCTIONS
pdf 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. Recent progress by
Katsutoshi Yamanoi
Now Solved copmpletely, AI-assisted.
pdf Holomorphic curves with few
inflection points.
Now solved using AI, September 2026.
pdf Reversion of Cartan's SMT. Now solved
by Y. Chen, An asymptotic equality of Cartan's Second Main Theorem
and some generalizations, Analysis Mathematica 51 (2025) 129-163.
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 solved it completely,
accepted in JAMS.
pdf Three real polynomials (a short and
simple restatement
of the previous problem).
Solved.
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 Estimates of the number of solutions of
some non-holomorphic equations. Solved using AI, by reducing to R. Thom's
Genus Inequality proved by P. Kronheimer and T. Mrowka, Math. Res. Letters,
1, 1994, 797-808.
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.