These are those rare occasions in life when it becomes possible to concentrate on things completely surreal without getting too much affected by the mortal issues. In general I do find it difficult to concentrate on doing my science after I have a had a walk along say the streets of Mumbai and seeing people living on them in abject poverty. And I have no reason to feel optimistic that any science I ever do will help them. This depresses me at some subconscious level. But then probably my love for ethereal surpasses my mortal pains. This ICM Satellite meet at IISc on "Geometric Topology and Riemannian Geometry" that I happened to be selected to attend was not free of such aspects. The key facet was the intensity of the mathematics and the opportunity to meet people with similar interests. (as I have never done! It is quite a rejuvenating experience to spend hours at a stretch talking to people who are all interested and intrigued by the same questions and issues.) The palpable thrill of being able to discuss geometry with professional geometers did help me take my mind off such nagging issues like the food which was unhealthy enough to make me and some others sick (looking at the correlation of events the suspect is the restaurant called Nesara inside IISc), that the bathrooms very often had no soap and that for some queer reason students from Princeton in similar stage of their PhD as me were put up in hotels. (whereas I got a student hostel to live in which was at least way better than the terrible accommodation that I had gotten when I last visited IISc as a KVPY fellow some 7 years ago) I have no clue why this discrimination was done but when I pointed out this seemingly racist arrangement to one of the organizers I got back some gibberish like "you are not equal to them" and "foreigners are not allowed to live inside the campus" etc. I was almost being tried to be given the impression that my fellow Princeton student was paying for his hotel (clearly this is false as I found by cross checking with them) whereas I on the other hand was being given some 800Rs as DA. (with no harm to me I could have easily returned back the 3500Rs odd of TA/DA but given the scenario such an act of mine might have been considered indecent!) As far as I can infer from experiences (may be I am wrong) even TIFR probably as some policy like this whereby every random grad student from outside India when lands up here will get a guest house room in the campus which is almost always out of bounds for similar standing Indian student visitors. Anyway when I first visited TIFR as a summer student I was given an accommodation so far away from the campus that I found it more beneficial for academics to spend my nights sleeping on the sofas and taking bath in the nightguard's bathroom.
The conference did start with some very dubious facets like an Internet arrangement that was yet to be setup when I arrived. (apparently I was "very early" by arriving one day before the start). The travel instructions somehow seemed to say that there will be someone waiting to receive the visitor at the airport if one arrived by plane. Of course people like me who can't afford to travel by planes had to struggle around to get to the conference location. (Interestingly 7years ago KVPY organizers had arranged for a pickup when I arrived on plane for the summer camp} {some fancy for the "skypeople"? (Avatar!)} Even when the Internet facility was setup it was under lock and key! So there was no way I could have accessed the Internet without first wasting quite a bit of time running around to figure out which volunteer had the key. Clearly at night it was impossible to do this. By the time the lectures ended and I got some breathing space to check my emails the mathematics office considered it to be the end of working hours and hence locked the computer room meant for the visitors! Thankfully some of the volunteers after a day or two realized that I am the only one around who is using the Internet facility and they let me keep the key. (Most others had laptops who connected using the wifi in the conference hall)
Some students in IISc apparently mistook me for a prof! (happy to know that somehow my face didn't look like some teenager's to them like many strangers think!) and some people observed to me,
"Why are you so happy the whole day? Why do you always have a smile?"
Never heard that from anyone! Anyway coming from a formal background in Physics it was an interesting experience to attend a mathematics conference for the first time. It was fun being the only Physics student in the midst.
It was quite an experience to attend a Hindustani Classical music (on Clarinets) concert by Narsimhalu Vadavati with a bunch of some of the best mathematicians. (weirdly enough the Japanese people seem to have been visibly enjoying what looked quite cacophonous to me!) The very low quality of the music left me depressed. Definitely India has far far better music to showcase to the world. Add to that the fact that the acoustics of the hall at National Institute of Advanced Studies (NIAS) were pretty bad with the ear splitting at the higher pitches. And then the power fails for a few minutes during the concert drowning us all in total darkness. Though not the first time. It happened the same way when Hutchings (from U.C.Berkeley) was explaining to a full packed hall the mysteries of the geometries of soap bubbles. (Interesting that we still don't have even a conjecture about what the shape should be for more than 4 bubbles together!) Such organizational fiascoes from the Indian side leaves me pretty much embarrassed while sitting in the crowd amidst an international crowd.
Hutchings amazed me by his conspicuous silence throughout the conference whether during lectures or during teabreaks. It is hard to find someone who so rarely ever talks! He never asked a single question nor did I ever see him interacting with many others. But it was endearing to see his childish enthusiasm and smile when on stage explaining even utterly complicated stuff like his recent discoveries of new obstructions to symplectic embeddings in 4dimensions.
I had a quite interesting conversations with Fernando, a Riemannian Geometry prof from Brazil. (After all one of the standard texts in the subject was written by doCarmo, a Brazilian). Fernando's talk was very neatly organized which made it very appealing to beginners like me.
I had some very fruitful conversations with this guy Aron Naber . He did his bachelors in aeronautical engineering from Penn State and then did a PhD in Riemannian Geometry at Princeton under Gang Tian! During his PhD he completely classified all Ricci Solitons on 4manifolds. He is just 1 year into his postdoc and he has been invited to speak at ICM satellite meet!
Interestingly he wanted me to explain to him what are sigma models! That is a very exotic QFT thing. Wonder why is he interested in it. He tells me that people are smelling that the good old idea of sigma models in QFT have some deep Riemannian Geometry hidden in them! I explained to him whatever I knew about sigma models.
Aaron's knowledge of geometry is awesome and I gained a lot from him.
He shared with me his various insights about Ricci flows and solitons.
I got pretty much charged up to follow up further on this topics.
Interestingly Shiraz (with whom I have very regular interactions in theoretical physics) had some insights about what are called rotationally invariant gradient solitons years back in this paper of his. It seems that there quite a nontrivial interest in being able to see if there is a causeeffect relationship between existence of a KahlerRicci soliton on a manifold and flat (in any sense) metrics on the cone over it. There are some technical conditions on this base manifold like it has to becompact complex manifolds whose first Chern number is positive (Fano Manifolds)(examples being the projective space probably) Apparently if the manifold is Einstein then the cone is probably CalabiYau and that is apparently of interest in physics.
Some other examples of interesting open questions which excited me were,
{not that I understand the exact issues but something tells me that these are interesting}
* If you start with a negative sectional curvature metric then is/when the constant scalar curvature metric in its conformal class also of negative sectional curvature? The problem seems to be that being of negative sectional curvature is an open condition and hence it is not clear as to whether it is preserved under Ricci Flow.
* Are there hyperbolic 3manifolds with a foliation whose leaves are minimal surfaces?
* Seems to me that there is quite a serious problem in finding an analogue of HeegardFloer homology for 4manifolds. (The motivation being that SeibergWitten invariant for 4manifolds is analogous to the Alexander polynomial on 3manifolds)
* People seem to want to know whether a hyperbolic 3manifold is always finitely covered by a fibered 3manifold.
* It seems that there are some topological restriction to putting hyperbolic metrics on spaces and it is not known whether one can put an hyperbolic metric on surface bundles over surfaces.
I hope that at some point in future I will get time and opportunity to spend time thinking about these.
Here I mention some basic issues that I discussed with Aaron and gained clarity about,
* Transitive action of the isometry group on the tangent spaces does not guarantee that it will take an orthonormal basis to another. In some vague sense the number of orthonormal basis in a vector space grows exponentially with the dimension and just transitivity can't match up. Being "maximally isotropic" or "maximally symmetric" (both meaning that they have n(n+1)/2 Killing fields or that is the dimension of the Isometry group) is necessary to make the isometry action be transitive on the set of all orthonormal basis of a tangent space (and make sectional curvature constant). Thats what happens for the euclidean planes, spheres and the hyperbolic planes and also for the FRW metric in usual cosmology.
Basically the idea is that homogeneity and isotropy is not enough to get constant sectional curvature. Homogeneity alone gets only scalar curvature constant and isotropy at a point gets only Ricci curvature constant. (it basically maps every 1dim subspace isometrically to another and Ricci basically depends only on a basis)
Homogeneity and Isotropy together hence give only a constant Ricci curvature manifold.
* If you can act on a constant curvature (probably in any sense) by a discrete subgroup of the isometry group then you end up as a quotient another constant curvature Riemannian manifold whose fundamental group is isomorphic to the discrete subgroup you started with. But whether or not the quotient is compact and whether what its genus will be (if definable) are much harder questions.
I had come across this concept while I was trying to understand some paper of Witten's where he seemed to be creating solutions of Einstein's equations by acting on the hyperbolic plane by discrete subgroup of isometries of that.
One more interesting piece of insight from Fernando's talk that I ruminated over during these is that from the GaussBonnet it is not so hard to see that on a torus if the sectional curvature is always nonnegative then it is surely 0.
Now it seems that this kind of "rigidity" theorems also hold in higher dimensions but they get enormously harder and only work case by case.
The statement for ndimensional torus was due to none other than Gromov and Lawson.
A particularly interesting generalization of this is by Schoen and Tau who showed that "mass" as defined in General Relativity always has to be positive. (intuitively this should always be true but this was far from obvious from just the Einstein's equations). Apparently there is a more sophisticated proof of it by Witten using Penrose's twistors. (I guess this is the same Schoen who proved the famous Yamabe conjecture that on compact manifolds every conformal class of metrics has a constant scalar curvature metric)
A general idea behind these rigidity theorems is that given a manifold with a metric a compactly supported metric deformation cannot change the scalar curvature. This is something that I would like to understand further.
I also met this very eloquent PhD student at Princeton called Nick who is looking into Conformal Geometry (under Wang). I got along very well with him. He happens to be a batchmate of Arul (he was one year senior to be at CMI). Among the many interesting quips he made about Maths and doing a PhD, one was, "If you really like someone then you should get married otherwise one marries either before or after a PhD". {Of course getting a PhD is simpler than that!} This came up in the context of Aron having already married while only 1 year out of his PhD. (of course there were rumours about who will get the Fields but now talking about them is pointless, but definitely it is completely exciting that half the Fields medals have been given for proving conjectures in condensed matter physics and one can read a short very insightful review of them here by none other than Terence Tao, another Fields medalist himself) Though Michael Usher said, "Fields medal? Who cares?" (Usher gave a very nice talk with lots of motivations!)
Interestingly Nick tells me that given the hugely depleting number of geometry students in his place he is very lonely in his working. (On my side geometry definitely looks almost extinct!). And he tells me that many students in Princeton work for long hours alone and rarely ever talk to others. (interestingly he says he is one of them though I found him quite loquacious!)
Good to know that I am not the only lonely one around. (as one often feels in TIFR where the rest are always huddled into groups doing some calculation which I don't find interesting)
Nick and I chatted about everything under the sun from windmills to conformal metrics and a lot about studentadvisor relationships and the hiccups about beginning research. We talked quite a bit about the issue of building prerequisites for research. (so many things in Riemannian geometry end up being questions in Cartan's classification of semisimple Lie Groups. How many geometry PhD students know the proof of Cartan's classification? Nick tells me none!) It was nice to find someone to talk to about some of these issues which have been also puzzling me offlate.
Nick did his undergrad from WashingtonSeattle.
Nick tells me that at Princeton many people work completely alone and since people don't have to attend classes together there is very little bonding between the people. He says that people at U.Chicago form a very wellknit group since they go together to classes. We talked about which is better, to have regular classes and courses or to be left alone to read as in Princeton.
Aaron's wife works as a counselor in some "Residential Home" for "crazy teenage girls". They are basically taking care of lunatics and helping them rehabilitate them. Aaron says that such social work is simply horrible life since there is lot of work and she gets paid zilch for it. He tells me that he would for nothing on earth take up social work as a career. Apparently it has been listed at the top of some recently brought out "never take up" jobs list. Apparently Mathematics is at the top of best jobs in the US! He didn't lose chance poke at me by pointing out that Physics features somewhere in the 6th or 7th position.
I met my CMI prof KV at the joint dinner between the geometry and the representation theory conference. He asked about my whereabouts and seemed pretty friendly to me. He has also been following on the Vinay Deolalikar stuff and he says that people are apparently convinced that the proof doesn't work but something interesting might be going on.
I also got saw the legendary Kashiwara around. Though I didn't talk to him.
On the mathematical front I was quite enthralled by the talks given by Ron Stern and Ron Fintushel. Awesome people. It was quite exciting to talk to them too. Stern tells me, "One learns SeibergWitten Theory if one really wants to learn it". And anyway one upshot of many of the talks was that I should get excited everytime I hear of a lagrangian submanifold of a symplectic manifold. Apart from just a rewriting of good old classical mechanics I had not taken symplectic forms that seriously. Seems I should. And also I had never taken a keen interest in spaces where the dimension is not constant even on connected components. These are of course not manifolds. But then they are very exciting in Riemannian Geometry. For one thing these don't have a welldefined notion of "tangent spaces" but have "tangent cones". And more curiously these might not even be unique. From whatever Aaron says it seems to me that these are known in Physics as "orbifolds". I wonder if they are the same things. Aaron has proved some rate of growth theorems about the dimensions of these stuff for lower bounded Ricci curvature spaces and the formulas looks like Holder Inequalities. (strange!) Andras Stipsicz made this very thought provoking statement while analyzing the pathbreaking work of Sucharit,
{Sucharit and 3manifolds have become almost synonymous now!}
"3manifold are like prime numbers"
Sucharit's talk was one blitzkrieg of intelligence overflow.
He made his intentions very clear by his opening lines,
"All this is very interesting to me but will be either known to some of you or will be simply too overcomplicated. Hence I will go fast anyway"
He set up an algorithm by which any link could be written as a grid diagram (like TicTacToe) and a set of moves on it which basically mimicked the Reidmaster moves and showed that all his moves give cobordant links. Then he constructed a hugely complicated derivative operator which sums over some very queerly chosen blocks inside that grid and miraculously its composition with itself is 0! And hence you have invented a homology theory.
Now the biggest miracle is that this is precisely the HeegardFloer homology.
Sucharit says,
"The definition is too complicated but it comes out naturally"
Except probably Andras and Siddharth nobody asked a question and the entire talk ended with the same eerie silence as it went with.
Anyway on a larger scale it does seems to be quite a big thing to do to be able to see in absolutely unexpected places interesting dimension lowering maps whose selfcomposition is 0. Thats the name of the game. You have discovered a homology theory! Things are miraculous if that gives a finite process of counting in the space of maps in some other well known homology theory.
Pursuit of geometry does bring one to very endearing events in life.This guy Nick from Princeton was very nice to me.
He said to me "I think you should definitely meet Gabai" and he introduced me to David Gabai! It was quite an opportunity to get to talk to this legend.
I have been startled by the exuberance and energy of Gabai. I don't know of of many students in their 20s who are so excited about mathematics as this some 60 year old guy! He always had a sparkling smile on his face and was always scribbling on with something or the other in mathematics. I was just stunned by this guy's never ending energy to talk about mathematics and always so with a huge smile on his face!
I also saw "Mahan Maharaj" around apparently one of the very famous "swamis" at Ramkrishna Mission/Ramkrishna Mission Vivekananda university. He allegedly secured a very high rank in the IIT entrance exam in 1987, the same year Rajesh Gopakumar topped it. Rajesh did his PhD from Princeton and Mahan did his from Berkeley. As far as I hear after some more years around as postdoc he gave up normal life and took "diksha" and became a "swami" living that stern life.
Mahan is pretty famous for his work on hyperbolic manifolds and what are called "ending lamination spaces". I have heard a lot about him from 2 of my acquaintances who have studied under him during their undergrad. He has apparently been working very hard with the organization at his place to make subjects are differential geometry compulsory for Physics undergrads and subjects like Conformal Field Theory and Classical Mechanics compulsory for maths undergrads. He has had partial success.
Finally I saw him around here. David Gabai seemed to be very fond of him. Mahan has an awesome physical built!
He gave his seminar wearing that typical saffron coloured kurta and beneath a saffron coloured wrap around dhoti. His talk was about some partial solution of the HilbertSmith conjecture. His was clearly the best Indian talk. (compared to unspeakably bad talk by Ramesh Sharma and add to that his distractingly irrelevant mention of some calculation tricks in "ancient Indian mathematics" at the beginning of the talk) Mahan had an awesome level of confidence and a charismatic speaker. His intelligence was almost palpable. Fintushel one day came up with a question as to whether every involution on the product of two Riemann surfaces of different genus splits as a product of involution on each. The very next day Mahan went up on the board and gave a proof for it. In fact he proved a stronger statement that it splits for any discrete group action. (or something like that). Interestingly it was Fintushel's birthday on which Gabai said ,"This is Mahan's birthday gift for him" Gabai said that "I won't reveal Fintushel's age but he now qualifies for medical aid"
Gabai led the congregation to sing a "Happy Birthday to You" song for Fintushel to which he replied, "I thank you all from the bottom of my heart" The camaraderie between these stalwarts in geometry was very inspiring.
I am not just irreligious but antireligious but it doesn't stop me from appreciating Mahan. This guy can go around as a brilliant mathematician and except for his dress there will be no reason to see or know of his such queer affiliations. Thankfully he doesn't wear his religion on his sleeve.
I planned out a dinner with Nick and I was wondering if he would make it to the appointed spot treading through this so complicated campus with innumerable lanes and forests. And I was overwhelmed when around the meeting time I saw Nick standing by the side of the nearby jungle poring over a map of the campus!
I went up to him and saved him further trouble to what already he had taken up to find his way through this labyrinth to come and meet me.
I was simply overwhelmed by this gesture. I don't remember the last time I was at the receiving end of such efforts!
Over dinner we managed to talk about a myriad of topics. Nick asked me about my family and he wondered "So does your entire family do mathematics?" !
Then he asked me whether I have brothers or sisters and whether they too do mathematics. He told me that he has a brother who is also a maths undergrad at Washington university. His father is a chemist who "works with fishes for the government" and his mother apparently used to check car license certificates but gave up the job after he was born. And his mother was 42 when he was born and he also a brother! (My mother is currently 42!) He gave me a long lecture on what he sees computers to be able to do by 2050. In this context he explained to me how using ZFC one can code every mathematics proof as a sequence of symbols drawn from a finite set and how using some simple rules a computer can be made to check in finite time whether this proof is correct and how this implies that in principle a computer can also come up with a proof given a statement. And how this entire thing relies on Godel's incompleteness and how this is impractical because of timecomplexity issues. Apparently Russel was the first person who showed that this can be done and using this coding technique he took 77 pages to prove that 1+1=2! (But this is important since that is apparently the only way known by which a computer can be made to understand the proof of 1+1=2. It is also apparently the first example of a computer comprehensible mathematical proof and very few such examples are known) Then he went on to explain to me how using the axiom of choice coupled to this technique one can make the computer understand statements like "take any 2 points in the topological space". And how this connects to "Model Theory"
Interestingly he has picked up all this while he did philosophy courses as a maths undergrad. This is very interesting that philosophy courses end up teaching so much of deep mathematics. (Philosophy courses as I have seen some Indian students do in Indian colleges is just pathetic nonsense)
He imagines that the timecomplexity issues of coding proofs will be the next big thing computer technology is going to achieve. All this while he was so excitedly and animatedly explaining it all this, he seemed to clearly forget that there are other people around in the restaurant! We went on to talk about Avatar, Harry Potter, Lord of the Rings, Celine Dione and Crisswell and Golem! (comparing the similarity of the technology between how Golem and Avatars were created) Then the discussion shifted to cooking and had a laugh comparing each other's astounding cooking skills. (He has gotten addicted to ChickenTandoori and GarlicNaan and thats what we ordered. I introduced him to Lassi which he found awesome) He seemed to have had a very interesting undergrad where he took 5 years to get an MSc from WashingtonSeattle. In those 5 years he did some 10 undergrad courses and some 20 graduate courses. And he had the opportunity to replace the undergrad algebra courses where Artin is covered by 3 quarters of Algebra courses where Dummite and Foote is covered. And he took one full graduate course on curves and surfaces! (Its a terrible state in India that people can get BSc. in Maths never having done any theory of curves and surfaces like the ones in the book by Pressley or Singer and Thorpe). While in Washington he did 3 courses in differential geometry where basically the 2 books by Lee were done. (I have read most of Lee's second book and it is beautiful) His complex analysis course looks very inspiring to me where they took 2 quarters to do Ahlfors' book thoroughly! They spent about a month learning how to do real integrals using contour integration and doing those curious summations using complex analysis. I was mentally comparing to that the terribly unspeakable Complex Analysis course which I did in CMI where that horrible compressed book by Remmert was declared to be over in 4 months and never ever was using contour integration to do real integrals discussed and that was called a misuse of complex analysis course! Thankfully I have picked up a lot of that given the efforts of Amol to make us go through that in details, Probably the most (of the rare few) important things I learnt in any TIFR classes. It was simply impossible for me to understand from CauchyRiemann Equations to CasorattiWeirstarss theorem in 4 months. (as was attempted in one of those rare bad maths courses in CMI) A first course can at most go up to Riemann Mapping Theorem or at best Picard Theorem but even that was not done (judiciously so) in this Nick's college even in 6 months.
Nick told me of his experience of meeting Terence Tao when he visited UCLA before applying there. He was completely awed by that guy (and why shouldn't one be!). Apparently Terence took him out to the botanical garden and showed him around all the curious plants they have. Expectedly Nick says that Terence was probably one of the most friendly and kind people he has ever met. They did discuss some mathematics about what Terence was doing then. Never did it become an issue that he was talking to probably one of the greatest thinkers ever.
I had decided to pay for the bill but Nick put up an enormous protest and he wanted to pay. He was under the impression that I was an undergrad and when I told him that I get a stipend he agreed to reduce his stake to 70% of the bill. He said that he will pay the full bill if I ever come around to his place.
Post dinner we had a walk where he came up to my accommodation to see where I live.
Probably one of the best of the rare few dinner outings I have ever had.
I ended this trip by meeting an acquaintance of mine in Bangalore whom I technically know for the last 22 years but haven't met for many years in between. Interestingly while out with him I witnessed the upcoming Bangla band "Backbenchers" perform life at PlanetM. It was quite impressive.
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