Fdtxjr

The best known lower bound for the minimal length of superpermutations was originally posted anonymously to 4chan.

The story is told at Mystery Math Whiz and Novelist Advance Permutation Problem, and a publication with a cleaned-up version of the proof is at A lower bound on the length of the shortest superpattern, with "Anonymous 4chan Poster" as the first author. The original 4chan source is archived here.

I'll just let below (famous) 1966 article from the Bulletin of the American Mathematical Society speak for itself...

COUNTEREXAMPLE TO EULER'S CONJECTURE ON SUMS OF LIKE POWERS

BY L. J. LANDER AND T. R. PARKIN

Communicated by J. D. Swift, June 27, 1966

A direct search on the CDC 6600 yielded

$$ 27^5 + 84^5 + 110^5 + 133^5 = 144^5 $$

as the smallest instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler [1] that at least $ n $ $ n $th powers are required to sum to an $ n $th power, $ n > 2 $.

REFERENCE

1. L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1952, p. 648.

Perelman solving the Poincare "conjecture," posting it only on the arXiv, leaving math, and refusing the Clay prize could be interpreted as a kind of "mic drop."

Not math but in physics the statistical interpretation of the wave-function was announced by Max Born in a footnote.

From his paper Zur Quantenmechanik der Stoßvorgänge,

(1) Anmerkung bei der Korrektur: Genauere Überlegung zeigt, daß die
Wahrscheinlichkeit dem Quadrat der Größe $Phi_n_tau m$ proportional ist.

This can be translated as

(1) Addition in proof: More careful consideration shows that the probability is proportional to the square
of the quantity $Phi_n_tau m.$

Because of its implications this is probably the most important footnote in the history of physics. Max Born was awarded the Nobel prize "for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction".

Applications of algebra to a problem in topology (YouTube) at Atiyah80 was a talk by Mike Hopkins. In it he announced the solution to the Kervaire invariant one problem in all but one dimension (arXiv, Annals).

I consider this manner as a mark of a professional mathematician: let others convey the excitement of a discovery. A good recent example was the submission of a paper on bounded gaps between primes. Much of the public excitement was generated by people other than the author, Yitang Zhang.

Gerhard "Can Be Excited In Private" Paseman, 2019.03.10.

From the Wikipedia article on Frank Nelson Cole:

On October 31, 1903, Cole famously made a presentation to a meeting of
the American Mathematical Society where he identified the factors of
the Mersenne number $2^67$ − 1, or M67.[5] Édouard Lucas had demonstrated
in 1876 that M67 must have factors (i.e., is not prime), but he was
unable to determine what those factors were. During Cole's so-called
"lecture", he approached the chalkboard and in complete silence
proceeded to calculate the value of M67, with the result being
147,573,952,589,676,412,927. Cole then moved to the other side of the
board and wrote 193,707,721 × 761,838,257,287, and worked through the
tedious calculations by hand. Upon completing the multiplication and
demonstrating that the result equaled M67, Cole returned to his seat,
not having uttered a word during the hour-long presentation. His
audience greeted the presentation with a standing ovation.

Kurt Gödel, only a few days before Hilbert gives his famous "We must know – We will know!" quote, had just proven that we cannot know.

Namely, any reasonably strong foundation of mathematics, if it has a finitary proof verification process, cannot decide all the true statements. Mathematics, in its essence, is incomplete.

Philosophically speaking, perhaps one of the biggest mic drop moments. Metaphorically, this virtual coinciding with Hilbert's lecture just makes the room even more silent afterwards.

My favorite is non-mathematician Marjorie Rice challenging the proof of "No other pentagon tilings exist" with multiple new pentagon tilings. Schattschneider's article was the primary announcement of the results.

I think the following anecdote fits well in this category. Note however, that other participants may have experienced these things differently, since they will have had a better background knowledge of the topic.

At a conference in Uppsala in September 2012, Geordie Williamson was scheduled to give a talk. I can unfortunately not recall the precise topic, as I can no loner find the program for the conference.

He starts his talk by apologizing that he is in fact going to talk about a completely different topic, since he had very recently finished some work on this with his collaborator Ben Elias.

He then goes on to describe Soergel's conjecture and some of the ideas that he and Ben have been working on, hoping to make progress on the conjecture.

The talk is quite technical, involving a lot of quite deep ideas and descriptions of how certain geometrical ideas, such as Hodge theory, can be given more algebraic analogues and how these may be put together to make progress on the conjecture.

As is typical of any technical talk, it is very hard to keep track of all the details and how they fit together along the way, so he provides a nice summary in the end:

"In conclusion, Soergel's conjecture is true".

Onsager announced in 1948 that he and Kaufman had found a proof for the fact that the spontaneous magnetization of the Ising model on the square lattice with couplings $J_1$ and $J_2$ is given by

$M = left(1 - left[sinh (2beta J_1) sinh (2beta J_2)right]^-2right)^frac18$

But he kept the proof a secret as a challenge to the physics community. The proof was obtained by Yang in 1951