This is the 4764th most frequent Japanese word.
極める
To master or reach the extreme.
In this sentence, '極める' is used to mean 'to master' or 'to achieve a high level of skill in' mathematics.
彼は数学を極めるために毎日勉強しています。
He studies every day to master mathematics.
Here, '極められた' as the past passive form of '極める' conveys that the techniques are highly refined, indicating excellence achieved in the skills.
その画家は極められた技法で美しい作品を作り上げている。
That artist creates beautiful works with refined techniques.
'極める旅' uses '極める' in the sense of 'to explore deeply' or 'to investigate thoroughly,' describing the depth of engagement in the journey.