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This is a question that I thought about recently, and I thought would be interesting to the MO community.

What are some famous conjectures, more specifically those that attracted a lot of attention and had a non-trivial impact on the direction of at least one mathematical subject area for some time, that were named after particular mathematicians which were resolved within their lifetimes?

Two prominent examples in number theory include the following:

1) The Taniyama-Shimura conjecture, asserting that every elliptic curve over $\mathbb{Q}$ is modular. This conjecture was shown to imply Fermat's Last Theorem in the 1980s, and whose semi-stable case was famously resolved by Taylor and Wiles in 1995. Breuill, Conrad, Diamond, and Taylor then finished the remaining cases in 2000. While Taniyama tragically died not long after making the conjecture, Goro Shimura only passed away recently and certainly saw the resolution of this conjecture.

2) The Weil-conjectures, which can be viewed as a form of Riemann hypothesis for algebraic varieties over finite fields, was famously resolved by Deligne in the 1970s. Weil, who lived until 1998, certainly saw the resolution of these conjectures.

A non-example, off by about a decade, is Mordell's conjecture, which is the assertion that any algebraic curve defined over a number field having genus $g \geq 2$ has at most finitely many $K$-rational points for any number field $K$. Mordell died in 1972, 11 years before Faltings proved his conjecture in 1983.

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    $\begingroup$ There are too many. And unclear what does it exactly mean "famous". I vote to close. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 31, 2019 at 16:55
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    $\begingroup$ How many people should know or understand a conjecture to make it "famous"? $\endgroup$
    – Alexandre Eremenko
    Commented Aug 31, 2019 at 16:58
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    $\begingroup$ I would say “if you have to ask, then it’s not famous” would be a good metric. The examples I gave are certainly unambiguously famous $\endgroup$
    – Stanley Yao Xiao
    Commented Aug 31, 2019 at 17:01
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    $\begingroup$ @StanleyYaoXiao I disagree. Although I have heard of those theorems, people I have met in the more applied end of (real or functional) analysis or probability theory would not, AFAICT. If you just mean "unambiguously famous within a particular area" then I fear the question will just be too broad and attract many niche answers $\endgroup$
    – Yemon Choi
    Commented Aug 31, 2019 at 21:52
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    $\begingroup$ Some of Hilbert's 23 problems were solved during his lifetime. Do they count? $\endgroup$
    – Gerry Myerson
    Commented Aug 31, 2019 at 23:24

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The Pólya conjecture was stated in 1919 and disproved by C. B. Haselgrove in 1958. G. Pólya died in 1985.

For an example in the positive direction, we can consider the Adams conjecture, stated in 1963 and proved by D. Quillen in 1971. J. F. Adams died in 1989.

Another important example is the Milnor conjecture in algebraic K-theory, stated in 1970 and proved by V. Voevodsky in 1996. J. Milnor is still alive.

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Van der Waerden's conjecture was posed in 1926 and proved in 1980, within the lefttime of Van der Waerden (1903-1996).

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Thurston was still alive, when Perelman solved the Thurston conjecture.

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Serre's conjecture about projective modules over polynomial ring was proved by Quillen and Suslin independently. Though Serre is alive today (Thank God), sadly, Quillen passed away shortly after he proved the conjecture and Suslin passed away recently.

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    $\begingroup$ Quillen died in 2011... $\endgroup$
    – Dan Petersen
    Commented Aug 31, 2019 at 19:05
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The Bloch–Kato conjecture (the one on Milnor K-theory, not the one on special values of L-functions) was conjectured in 1986 and proven by Voevodsky in 2008 (published in 2011). Both Bloch and Kato are still alive.

The same goes for Milnor's conjecture (1970), proven by Voevodsky in 1996 (published in 2003), which is a special case of the Bloch–Kato conjecture, for which Voevodsky was awarded the Fields medal in 2002.

Voevodsky himself sadly passed away in 2017.

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  • $\begingroup$ Just a minor remark: I would say "prematurely" rather that "tragically", since Voevodsky's death was due to natural causes. $\endgroup$
    – Francesco Polizzi
    Commented Sep 1, 2019 at 9:49
  • $\begingroup$ @FrancescoPolizzi: you're probably right; I didn't think of all connotations of that word. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Sep 1, 2019 at 21:15

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