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Posts Tagged ‘math’

Apologies to the reader for the small number of posts here this week. I was at the University of Oklahoma for a math visit from Tuesday to Friday.

It was a fruitful visit. I gave a couple of talks and discussed a lot of number theory. In many ways the whole thing, especially the math discussion, was reminiscent of my Olympiad days when we would think about math problems all the time, sometimes in groups, sometimes individually and explain our bright ideas to each other with a particular kind of passion that few people ever experience. The trouble with research is that everything is so specialized that it is hard to find people to talk to who really understand your stuff. So the last few days were almost from another planet — waking up at 8 AM, going over to Starbucks to meet the other two, and mathiness for the rest of the day. And did I mention that I was put up in a luxury suite and everything I ate or drank (including every cup of coffee) during my stay was paid for by them?

I am back now, so blogging should resume. Cannot promise a flood of posts though. Math has a way of reminding you of its superior status from time to time.

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Frenchmen and math

Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language, and forthwith it means something entirely different.

Goethe

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They keep cranking them out, don’t they?

Of course, there is the minutest possibility that this time its for real, but just going by history, I am prepared to bet all my savings against it.

Update: I apologize if my post seems to suggest that Xian-Jin Li, the author of the purported proof, is a crank. In fact, he is a competent mathematician who has done good work in the past. Nonetheless, I think that the chances of this proof being correct are extremely low; in fact Terry Tao claims to have already found a mistake.

Update 2: Xian-Jin Li has posted a new version (actually two three new versions!) of his preprint on the Arxiv. Most pertinently, the definition of the function h on page 20 has changed; so perhaps this addresses Tao’s objection above.
The reason I will be very surprised if this proof turns out to be correct is that it involves mostly functional analysis on the adeles. It has been generally believed that such techniques are not sufficient to prove Riemann. It would be a stunning achievement if Riemann is solved using only such elementary tools; will be following this news closely over the next few days.

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Most people, on coming to know that I do research in pure math, respond with a nod or a wide-eyed, “Ohh, that must be so hard!” Occasionally however, someone goes further and asks me what my research is really about. And then, I am usually in a fix.

How do I respond? There’s no way to explain that I study special values of L-functions for automorphic forms to a person who is not already familiar with all these words. So I usually take refuge in generalities like “Prime numbers”. Sometimes when I am in the mood, I explain to them what Fermat’s last theorem says (if I am lucky they already know this) and add that I work with methods ‘related to’ how Fermat was proven.

It was therefore a source of great joy to me to read Barry Mazur’s excellent article in the Bulletin on error-terms in number theory and the Sato-Tate conjecture. While the article isn’t quite about what I do research on, it comes fairly close. More importantly, it is engrossing, beautifully written, mathematically solid and accessible to anyone who knows some college-level mathematics and statistics. Perhaps, I should start carrying a copy of it in my pocket for exigencies like described above.

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The reason there were no posts over the weekend is that I went solo-camping in the San Gabriel mountains just north of Pasadena.

The idea of spending a weekend alone in the wilderness first occurred to me during a hike I did in the San Gabriels with a couple of friends last summer. I was awestruck by the utter beauty and solitude of a particular spot that we passed along the way. It was a place to remember, highlighted by a little brook that flowed arrestingly between the trees — bouncing over shiny round stones and creating little cascades. That’s when I got intrigued by the idea of spending a few days in these mountains alone — observing, musing, relaxing, and yes, doing math.  Last weekend I finally decided to do it.

After some research I settled on Hoegee’s campground. It seemed to fit the bill — beautiful, secluded, relatively close. I already had a tent and a sleeping bag. I rented other essential stuff —  cooking utensils, a stove, a sleeping pad and a bear-proof container (yes, there are bears in the San Gabriels). I also packed a lot of math books and print-outs of research articles.

Hoegee’s is three miles away from Chantry flats, the closest car-accessible place. I started hiking around noon on Saturday. Almost immediately I realised that these three miles would not be fun. The trail was uphill, and I was carrying too much load. I had on my back a large, rather poorly balanced backpack behind which was attached another backpack full of books and under which was tied my sleeping bag. I held the tent bag in one of my hands and the sleeping pad in the other. It felt like I was hauling up rocks. What made things worse was that the backpack wasn’t really the correct size for me — as a result it put all its weight on my shoulders. Everyone I passed remarked about the amount of stuff I was carrying (after I returned from the trip, I checked the weight of my load — it was a shade above seventy pounds).

The hike to Hoegee’s was nice otherwise. The trail wound its way along the forested slope high above a canyon before descending down. The campground was situated next to a shaded creek that flowed along the bottom. I reached to find several tents and many people already out there. I had hoped for a quiet, nearly empty place and this was rather unexpected and disappointing; however I also felt a mild sensation of relief — spending the night completely alone in the jungle amidst bears and other wild animals may have been scary.

I set up camp, had lunch, and then wandered around for a while. There were two small dams on the creek very near where I had set up tent. I settled myself on top of one of them and worked for a while. Later, I returned to the campground,  had dinner, and went inside my tent. I read a math paper in the lantern light for about an hour before going off to sleep.

         That's the dam I sat on

The next morning, I woke up very early. I ate some breakfast and then decided to hike up to the top of Mount Zion, a little over a mile away and about a thousand foot elevation gain. It was a foggy morning and as I hiked up, the clouds came rolling in, blocking everything from view. It looked almost surreal in the early morning light.

         On the way to Mt. Zion  

I didn’t meet anyone else during the journey, which lasted a little over a half-hour. The peak itself was a flat little area — there I spread out a sheet and sat down. For a while I read a book on automorphic forms. Then I listened to some music on my headphones. I realized again that Mozart sounds most beautiful when heard amidst nature. Finally I came down around noon. I would have stayed even longer had the weather not been so cold and wet up there.

         The peak

I could hardly recognize the campground when I returned. Everyone else had left (check out the contrast in the pictures below) and it looked a totally different place. There was no sound to be heard except the constant gurgle of the stream and the occasional chirp of a bird. Little sunlight reached the ground and the overall effect was one of total seclusion. I stood transfixed for a few minutes before proceeding to cook some lunch.

                                   The campground last evening    The campground that noon

After I finished eating, I packed up everything and headed back.

Pack-up time

The return was less torturous since much of the way was downhill — though my shoulders still suffered. My car was as I had left it and I drove off Chantry flats with memories of a beautiful weekend down at Hoegee’s, one I’d probably do again, though I’d carry less stuff next time and also stay for a bit longer.

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“The essence of mathematics lies in its freedom.”

George Cantor.

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I haven’t had much time to blog this weekend. Ideas for posts came and went. News broke, and got stale. I gave them all a haughty ignore and, with single minded devotion, concentrated on my L-functions.

One of the drawbacks of being a fourth year grad student is that you need to do a lot of research quickly enough to produce a decent body of work by next winter — for the perusal of the grim committee that will go through your job application. And I am a fourth year grad math student. My research consists of proving things — by the power of thought. Which means I work when I think and I … umm … think when I walk. So when do I blog?

Yet, being a student comes with its perks. One of them is that I get student-priced tickets for concerts. So I went to the Pasadena symphony yesterday to hear an evening of music. They were playing three piano concertos by Mozart, my favourite composer.

The pianist was superb. He played beautifully. The music was pure and simple and true. It was mostly joyous, sometimes moving and deep, but without an iota of negativity. It was a bit like the best kind of mathematics.

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