{"id":1646,"date":"2015-12-21T18:48:16","date_gmt":"2015-12-22T00:48:16","guid":{"rendered":"http:\/\/blogs.ams.org\/blogonmathblogs\/?p=1646"},"modified":"2015-12-21T18:48:16","modified_gmt":"2015-12-22T00:48:16","slug":"mind-blowing-math-reminiscence","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/blogonmathblogs\/2015\/12\/21\/mind-blowing-math-reminiscence\/","title":{"rendered":"Diagonalization and Other Mathematical Wonders"},"content":{"rendered":"

It\u2019s only a slight exaggeration to say I\u2019m a mathematician because of Cantor\u2019s diagonalization arguments (both the proof that the rationals are countable and the proof that the reals aren\u2019t). I was already enjoying my intro to proofs class when we got to it, but it was the first theorem in the class (or in math generally) that truly astonished me.\u00a0<\/span><\/p>\n

\"In<\/a>

In a highly scientific survey of all the mathematicians who live in my house about mind-blowing\u00a0mathematics, diagonalization made a strong showing. Image: Jochen Burghardt, via Wikimedia Commons.<\/p><\/div>\n

Mike Lawler<\/a><\/span> recently wrote a post about math that made him go whoa!<\/span><\/a> that links to a recent Reddit thread on the same subject<\/span><\/a>. Lawler\u2019s list includes the residue theorem and Galois theory along with Polya\u2019s theory of counting, a topic I was unfamiliar with. Unsurprisingly, I am not alone<\/span><\/a> in being astonished by the diagonalization argument, but people love a lot of other mathematics as well.<\/span><\/p>\n

If you\u2019re feeling a little blah after a long semester and months of dwindling\u00a0daylight (Southern Hemisphere-dwellers, just imagine you\u2019re reading this in six months), a trip through that Reddit thread might cheer you up. It\u2019s got some of the greatest hits of undergraduate math and a good dose of heavier research math to boot. (Incidentally, maybe I need to bite the bullet and read the proof of the Riemann-Roch theorem in Hartshorne<\/span><\/a>, which comes highly recommended by at least one Redditor.)<\/span><\/p>\n

Reading through other people\u2019s top mathematical moments naturally led me to reflect on my favorite bits of mathematics. I was a late mathematical bloomer. Not a lot of math really blew my mind in college because my attitude at the time tended towards the\u00a0utilitarian. Diagonalization notwithstanding, I didn\u2019t often appreciate the beauty of what I was learning or even know that I should be surprised by it. As time passes, I gain more and more respect for many ideas in math, even ones I\u2019ve been familiar with for years. <\/span><\/p>\n

For me, teaching complex analysis this semester was refreshing. Revisiting the basic material and really feeling like I understood how all the pieces fit together was gratifying. I have never appreciated the Cauchy integral formula<\/a> more. Another recent time I really felt like I had a new appreciation for a theorem came last semester, when I was teaching undergraduate geometry and topology. The Gauss-Bonnet<\/a> theorem seemed to tumble from the classification of surfaces in a way that felt so much more natural to me than it had when I first learned it.<\/span><\/p>\n

Finally, I can\u2019t reminisce about mind-blowing math without pointing to my favorite piece of mathematics communication of the year, and SMBC comic about Kempner series. (Kevin Knudson\u2019s Forbes post<\/span><\/a>\u00a0on the topic is fun, too, although it doesn\u2019t use the word \u201cballs\u201d as many times.)<\/span><\/p>\n

\"Image:<\/a>

Image: Zach Weinersmith, SMBC Comics.<\/p><\/div>\n

When was the last time math made you say \u201cWhat the balls?!\u201d<\/span><\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

It\u2019s only a slight exaggeration to say I\u2019m a mathematician because of Cantor\u2019s diagonalization arguments (both the proof that the rationals are countable and the proof that the reals aren\u2019t). I was already enjoying my intro to proofs class when … Continue reading →<\/span><\/a><\/p>\n

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