User anweshi - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:14:45Z http://mathoverflow.net/feeds/user/2938 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12657/proving-almost-all-matrices-over-c-are-diagonalizable Proving "almost all matrices over C are diagonalizable". Anweshi 2010-01-22T16:46:27Z 2013-01-31T20:00:45Z <p>This is an elementary question, but a little subtle so I hope it is suitable for MO.</p> <p>Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$. </p> <p>The characteristic polynomial $T - \lambda I$ splits into linear factors like $T - \lambda_iI$, and we have the Jordan canonical form:</p> <p>$$J = \begin{bmatrix} J_1 \\ &amp; J_2 \\ &amp; &amp; \ddots \\ &amp; &amp; &amp; J_n \end{bmatrix}$$</p> <p>where each block $J_i$ corresponds to the eigenvalue $\lambda_i$ and is of the form</p> <p>$$J_i = \begin{bmatrix} \lambda_i &amp; 1 \\ &amp; \lambda_i &amp; \ddots \\ &amp; &amp; \ddots &amp; 1 \\ &amp; &amp; &amp; \lambda_i \end{bmatrix}$$</p> <p>and each $J_i$ has the property that $J_i - \lambda_i I$ is nilpotent, and in fact has kernel strictly smaller than $(J_i - \lambda_i I)^2$, which shows that none of these Jordan blocks fix any proper subspace of the subspace which they fix. Thus, Jordan canonical form gives the closest possible to a diagonal matrix. The elements in the superdiagonals of the Jordan blocks are the obstruction to diagonalization.</p> <p>So far, so good. What I want to prove is the assertion that "Almost all square matrices over $\mathbb{C}$ is diagonalizable". The measure on the space of matrices is obvious, since it can be identified with $\mathbb{C}^{n^2}$. How to prove, perhaps using the above Jordan canonical form explanation, that almost all matrices are like this?</p> <p>I am able to reason out the algebra part as above, but is finding difficulty in the analytic part. All I am able to manage is the following. The characteristic equation is of the form</p> <p>$$(x - \lambda_1)(x - \lambda_2) \cdots (x - \lambda_n)$$</p> <p>and in the space generated by the $\lambda_i$'s, the measure of the set in which it can happen that $\lambda_i = \lambda_j$ when $i \neq j$, is $0$: this set is a union of hyperplanes, each of measure $0$.</p> <p>But here I have cheated, I used only the characteristic equation instead of using the full matrix. How do I prove it rigorously?</p> http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory Approaches to Riemann hypothesis using methods outside number theory Anweshi 2010-08-05T22:50:11Z 2012-12-14T15:12:46Z <p>Background: Once an analytic number theorist remarked to me that all attempts to prove the Riemann hypothesis using number theoretic methods have failed. Since then that remark stuck in my mind.</p> <p>The veracity of the above alluded number theorist's opinion does not really matter for the question to make sense; I just included it for background.</p> <p>Question:</p> <blockquote> <p>What are some promising methods from outside number theory to approach Riemann hypothesis?</p> </blockquote> <p>I know two:</p> <ol> <li><p>The geometric approach of Artin, Hasse, Weil and Deligne, the most important result being the proof of the Weil Conjectures.</p></li> <li><p>The Bost-Connes approach. This is outlined by <a href="http://www.neverendingbooks.org/index.php/the-bost-connes-coset-space.html" rel="nofollow">Lieven Le Bruyn</a> for instance and has a hint of thermodynamics . </p></li> </ol> <p>I imagine that both of the above are cited by some people as the basis for the hopes that the theory of the field with one element will prove the Riemann hypothesis. Again, this question formally has no need to be connected the theory of field with one element to make sense. Other than just mentioning the above, let us not get into that aspect.</p> <p>I am interested in other possible and promising methods. I am not interested in an equivalent formulation of Riemann hypothesis which is no better than the original. Both the above are very promising in terms of undiscovered things and might give a much better "big picture".</p> <p>An approach I am ambivalent about, is that of Baez-Duarte. Though it does provide some evidence. I do not know whether it is any easier to prove Riemann hypothesis that way, rather than the original statement.</p> <p>Please give me examples of any other methods; preferably very "promising" ones.</p> <p>Edit 1: The meaning of "methods outside number theory" is the following: Nothing in the book of Ivic or Titchmarsch and Heath-Brown. More precisely, methods outside the traditional sybjects of elementary number theory and analytic number theory. I have given two examples above. One with algebraic geometry and one with thermodynamics.</p> http://mathoverflow.net/questions/10514/teichmuller-theory-introduction Teichmuller Theory introduction Anweshi 2010-01-02T20:01:31Z 2012-09-18T12:36:53Z <p>What is a good introduction to Teichmuller theory, mapping class groups etc., and relation to moduli space of curves or Riemann surfaces?</p> http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra Ways to prove the fundamental theorem of algebra Anweshi 2010-01-02T22:11:26Z 2012-08-08T18:09:01Z <p>This seems to be a favorite question everywhere, including Princeton quals. How many ways are there?</p> <p>Please give a new way in each answer, and if possible give reference. I start by giving two:</p> <ol> <li><p>Ahlfors, Complex Analysis, using Liouville's theorem.</p></li> <li><p>Courant and Robbins, What is Mathematics?, using elementary topological considerations.</p></li> </ol> <p>I won't be choosing a best answer, because that is not the point.</p> http://mathoverflow.net/questions/10516/uniformization-theorem-for-riemann-surfaces Uniformization theorem for Riemann surfaces Anweshi 2010-01-02T20:13:23Z 2012-08-05T05:36:22Z <p>How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to each other?</p> <p>I would like to know about different ways of proving it, and appropriate references. This is not to know the best way; but to know about various possible approaches. Therefore I wouldn't be choosing a best answer.</p> http://mathoverflow.net/questions/33281/effective-proofs-of-siegels-theorem-using-arithmetic-geometry Effective proofs of Siegel's theorem using arithmetic geometry Anweshi 2010-07-25T15:01:24Z 2012-01-20T14:17:07Z <p>This is a speculation and perhaps naive. The theorem of Siegel that</p> <blockquote> <p>There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a finite set of places in a number field $K$</p> </blockquote> <p>has proofs from arithmetic geometry, for instance by <a href="http://www.ams.org/mathscinet-getitem?mr=257086" rel="nofollow">Parshin</a> and <a href="http://www.ams.org/mathscinet-getitem?mr=718935" rel="nofollow">Faltings</a>, or by <a href="http://www.ucl.ac.uk/~ucahmki/siegelinv.pdf" rel="nofollow">Kim</a>. These proofs are interesting in that they use the modern developments in algebraic geometry and arithmetic to prove a theorem in number theory. But it is dissatisfying that as far as I am aware these proofs do not give an effective bound as opposed to the proofs using Baker's theorem. For instance, Silverman's book feels that the Parshin-Faltings proof is indirect. </p> <p>But I do not know about the more recent developments and philosophy. I am wondering whether there is any hope that the more sophisticated proofs can be made effective, as it would give the best of both worlds. I hope the experts here can answer this. </p> http://mathoverflow.net/questions/10419/depressed-graduate-student Depressed graduate student. Anweshi 2010-01-02T00:07:33Z 2012-01-02T00:08:43Z <p>How does a depressed graduate student go about recovering his enthusiasm for the subject and the question at hand?</p> <p>Edit: I am not that grad student; it is a very talented friend of mine.</p> <p><strong>Moderator's update:</strong> The discussion about this question should happen <a href="http://meta.mathoverflow.net/discussion/125/dispensing-psychological-advice/" rel="nofollow">at this link on meta</a>. This page is reserved for answers to the question as stated.</p> http://mathoverflow.net/questions/14257/injective-modules-and-pontrjagin-duals Injective modules and Pontrjagin duals Anweshi 2010-02-05T11:54:16Z 2011-02-23T13:25:10Z <p>Forgive me for this naive question. </p> <p>We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784.</p> <blockquote> <p>Every module is a submodule of an injective module.</p> </blockquote> <p>For a module $M$, Lang defines its dual to be $M$^ = $Hom(M, \mathbb{Q}/\mathbb{Z})$.</p> <p>This "dual", if anything, is an algebraic version of the Pontrjagin dual of a topological group $G$, which is $Hom(G, \bf{S}^1)$, where we consider the continuous homomorphisms. $\mathbb{Q}/\mathbb{Z}$ is the torsion part of $\bf{S}^1$, and it seems this restriction is done for preventing too big a dual.</p> <p>The proof goes on and shows that any module is a submodule of an injective module.</p> <p>However, is there any conceptual explanation of why this works, and why this mysterious appearance of a dual which looks so much like the Pontrjagin dual?</p> http://mathoverflow.net/questions/33665/are-there-motivic-proofs-of-weil-conjectures-in-special-cases Are there "motivic" proofs of Weil conjectures in special cases? Anweshi 2010-07-28T15:04:07Z 2010-12-06T17:12:31Z <p>This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the <a href="http://en.wikipedia.org/wiki/Standard_conjectures_on_algebraic_cycles" rel="nofollow">standard conjectures</a> on vanishing of cycles would imply the Weil conjectures. So, are there proofs of Weil conjectures in special cases using partial results on the standard conjectures? If so, which cases, and what are the references?</p> <p>Background: Borcherds mentions <a href="http://mathoverflow.net/questions/31963/weil-conjecture-for-algebraic-surfaces/32005#32005" rel="nofollow">here</a> that Manin proved a few special cases in higher dimensions using motives.</p> http://mathoverflow.net/questions/14404/serres-fac-in-english Serre's FAC in English Anweshi 2010-02-06T17:34:44Z 2010-10-25T13:51:20Z <p>Has somebody translated J.-P. Serre's "Faisceaux algébriques cohérents" into English? At least part of it?</p> <p>In a fit of enthusiasm, I started translating it and started TeXing. But after section 8, I got tired and stopped.</p> <p>However if somebody else already took the trouble, I would be most grateful. I do not know a word of French(except maybe faisceau), and forgot whatever I learned in the process of translation very quickly.</p> <p>This is made community wiki, as I do not want to get into rep issues. Please feel free to close this if you think this qn is inappropriate for MO(I have added my own vote for closing, in case this helps). I would be happy to receive answers in comments.</p> http://mathoverflow.net/questions/30156/demystifying-complex-numbers/34696#34696 Answer by Anweshi for Demystifying complex numbers Anweshi 2010-08-05T22:16:31Z 2010-08-05T22:16:31Z <p>From the perspective of complex analysis, the theory of Fourier series has a very natural explanation. I take it that the students had seen Fourier series first, of course. I had mentioned this <a href="http://mathoverflow.net/questions/11761/why-sin-and-cos-in-the-fourier-series/11763#11763" rel="nofollow">elsewhere</a> too. I hope the students also know about Taylor theorem and Taylor series. Then one could talk also of the Laurent series in concrete terms, and argue that the Fourier series is studied most naturally in this setting.</p> <p>First, instead of cos and sin, define the Fourier series using complex exponential. Then, let $f(z)$ be a complex analytic function in the complex plane, with period $1$. </p> <p>Then write the substitution $q = e^{2\pi i z}$. This way the analytic function $f$ actually becomes a meromorphic function of $q$ around zero, and $z = i \infty$ corresponds to $q = 0$. The Fourier expansion of $f(z)$ is then nothing but the Laurent expansion of $f(q)$ at $q = 0$.</p> <p>Thus we have made use of a very natural function in complex analysis, the exponential function, to see the periodic function in another domain. And in that domain, the Fourier expansion is nothing but the Laurent expansion, which is a most natural thing to consider in complex analysis.</p> <p>I <em>am</em> am electrical engineer; I have an idea what they all study; so I can safely override any objections that this won't be accessible to electrical engineers. Moreover, the above will reduce their surprise later in their studies when they study signal processing and wavelet analysis.</p> http://mathoverflow.net/questions/30220/abstract-thought-vs-calculation/34693#34693 Answer by Anweshi for Abstract Thought vs Calculation Anweshi 2010-08-05T21:51:53Z 2010-08-05T21:51:53Z <p>I mentioned this <a href="http://mathoverflow.net/questions/32968/slick-ways-to-make-annoying-verifications/34690#34690" rel="nofollow">somewhere else</a> too.</p> <p>Many general statements in algebraic geometry can be proved via direct tedious verification or by abstract thought. In fact, the notions of abstract algebraic variety and scheme is created precisely for this purpose. I will illustrate this with an example of showing that the elliptic curve is a group.</p> <p>Method 1: Define an elliptic curve over a field as a curve in the Weierstrass form with nonzero determinant. Upon this define the addition and inverse laws using the chord-and-tangent process, obtaining algebraic expressions. To show that the elliptic curve is a group, you have to show the addition is associative. Then do a very tedious verification of the identities. </p> <p>Method 2: Another way is to use elliptic functions to prove the identity in the complex case. Since the algebraic group law holds true over the complex numbers, it is satisfied by an infinite number of algebraically independent solutions, and therefore the group law must be true in universality, over any field whatsoever. Of course this needs to be made precise with <a href="http://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry%23The_Lefschetz_principle" rel="nofollow">Lefschetz principle</a> .</p> <p>Method 3:(My favorite) Later algebraic geometry developed and it was possible to prove statements without relying on the Lefschetz principle. For instance, the group law on elliptic curve is always a consequence of the Riemann-Roch, which was proved in its full power by Weil, Hirzebruch and Grothendieck. But this might be seen as a sledgehammer by some; in any case it is a remarkable sledgehammer. </p> http://mathoverflow.net/questions/32968/slick-ways-to-make-annoying-verifications/34690#34690 Answer by Anweshi for Slick ways to make annoying verifications Anweshi 2010-08-05T21:32:16Z 2010-08-05T21:32:16Z <p>Before algebraic geometry was developed sufficiently and the Riemann-Roch was proved with its present power, the <a href="http://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry%23The_Lefschetz_principle" rel="nofollow">Lefschetz principle</a> was used to dispose of many statements in algebraic geometry.</p> <p>For instance: Define an elliptic curve over a field as a curve in the Weierstrass form with nonzero determinant. Upon this define the addition and inverse laws using the chord-and-tangent process, obtaining algebraic expressions. To show that the elliptic curve is a group, you have to show the addition is associative. One way is a very tedious verification of the identities. </p> <p>Another way is to use elliptic functions to prove the identity in the complex case. Since the algebraic group law holds true over the complex numbers, it is satisfied by an infinite number of algebraically independent solutions, and therefore the group law must be true in universality, over any field whatsoever. Of course this needs to be made precise with Lefschetz principle.</p> <p>But later algebraic geometry developed and it was possible to prove statements without relying on the Lefschetz principle. For instance, the group law on elliptic curve is always a consequence of the Riemann-Roch.</p> http://mathoverflow.net/questions/31358/can-a-mathematical-definition-be-wrong/34687#34687 Answer by Anweshi for Can a mathematical definition be wrong? Anweshi 2010-08-05T21:16:38Z 2010-08-05T21:16:38Z <p>The <a href="http://en.wikipedia.org/wiki/Euler_characteristic" rel="nofollow">Euler Characteristic</a> in the statement of <a href="http://en.wikipedia.org/wiki/Riemann-Roch" rel="nofollow">Riemann-Roch</a> was redefined by Grothendieck to be in the <a href="http://en.wikipedia.org/wiki/K-group" rel="nofollow">K-group</a>, which he presumably(ie, I imagine so) constructed for this purpose.</p> http://mathoverflow.net/questions/33865/indeterminate-x-in-abstract-algebra-ring-theory/34438#34438 Answer by Anweshi for Indeterminate "x" in Abstract Algebra/Ring Theory Anweshi 2010-08-03T20:36:59Z 2010-08-03T20:40:16Z <p>I feel duty-bound to add an answer since I had desired that this question be re-opened. There is nothing much I am able to contribute in addition to the well-known things. </p> <p>The functorial definition of the polynomial algebra $R[x]$ over a ring $R$ is the following. It is a ring $R[x]$ together with a homomorphism $R \to R[x]$ such that given any ring homomorphism $\phi \colon R \to S$ and any fixed element $a \in S$, there is a unique homomorphism $\phi^\prime \colon R[x] \to S$ such that $\phi^\prime (x) =s$.</p> <p>This is a proffered way of saying that the indeterminate "x" should be free to vary without any restriction whatsoever except that it is an element of a ring. A concrete manifestation is given by the free $R$-module on the set of symbols $x^k$ where $k \geq 0$ together with a certain multiplication operation. Since we do not a priori know what is "x", we make this precise with a machinery of sequences; and when we are done, we call a particular element of the resulting ring to be "x".</p> <p>I contend that the ring $\mathbb Z$ and the polynomial rings over it are very important objects in the category of commutative rings with identity. For one, any commutative ring with identity $R$ admits a unique homomorphism $\mathbb Z \rightarrow R$. Moreover, if we let $r$ run through the elements of $R$, then there is an obvious surjective map from $\mathbb Z[(X_r)_{r\in R}]$ to $R$ and this realization of every comm. unital ring as a quotient of some polynomial ring over $\mathbb Z$ can be used to construct the co-product in this category.</p> <p>I should mention that fixing "x" in the polynomial algebra is in a sense fixing some "co-ordinate". The <a href="http://en.wikipedia.org/wiki/Symmetric_algebra" rel="nofollow">Symmetric Algebra</a> over a vector space is interesting in this sense.</p> http://mathoverflow.net/questions/32397/vector-spaces-without-natural-bases/34431#34431 Answer by Anweshi for Vector spaces without natural bases Anweshi 2010-08-03T20:06:39Z 2010-08-03T20:13:08Z <p>The vector space $\mathbb C / \mathbb R$ does not have a preferred basis. Among the two bases ${1, i}$ and ${1, -i}$, there is no reason to prefer one over the other. The choice of one of these amounts to a choice of an orientation for the plane.</p> http://mathoverflow.net/questions/34390/on-proving-that-a-certain-set-is-not-empty-by-proving-that-it-is-actually-large/34392#34392 Answer by Anweshi for On proving that a certain set is not empty by proving that it is actually large Anweshi 2010-08-03T14:41:33Z 2010-08-03T15:25:15Z <p>There are non-Borel sets that are Lebesgue measurable. This is proved in the following way. First show that the Borel sigma algebra for the real line is uncountable with cardinality of the real line. On the other hand, you have the Cantor set which is uncountable(cardinality = $\mathbb R$) and is of Lebesgue measure zero. Since Lebesgue measure is complete, every subset of the Cantor set belongs to the Lebesgue sigma algebra and therefore the Lebesgue sigma algebra has cardinality of the power set of the reals. </p> <p>Construction of explicit examples <a href="http://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice" rel="nofollow">would require</a> axiom of choice.</p> http://mathoverflow.net/questions/34384/omplete-homogeneous-space-which-is-not-locally-compact/34386#34386 Answer by Anweshi for Сomplete homogeneous space which is not locally compact Anweshi 2010-08-03T13:56:36Z 2010-08-03T14:21:57Z <p>A Banach space is homogeneous since the metric is arising from a norm. An infinite dimensional Banach space has the property that its unit ball is not compact; therefore the space is not locally compact.</p> <p>For a concrete example, take the space of continuous real functions on an interval with the supremum norm.</p> http://mathoverflow.net/questions/34110/algebraic-geometry-examples/34127#34127 Answer by Anweshi for Algebraic geometry examples Anweshi 2010-08-01T17:26:56Z 2010-08-01T17:26:56Z <p>For the more arithmetically minded people, an illustration of the Weil conjectures in the simple cases of the <a href="http://en.wikipedia.org/wiki/Weil_conjectures#Projective_space" rel="nofollow">projective space</a> by direct checking and also proving the Hasse-Weil theorem for elliptic curves could be very instructive.</p> <p>Again, for the more arithmetically minded people who are also open to some speculation, one could use counting of the number of points on the projective space over $\mathbb F_q$ and use the observation of Tits to introduce the <a href="http://en.wikipedia.org/wiki/Field_with_one_element" rel="nofollow">Field with one element</a>.</p> <p>The Mordell-Weil theorem and Faltings theorem could be mentioned(without proof, of course) and compared to the case of genus 0, ie lines, to show that the geometry of a curve affects its arithmetical behavior significantly.</p> http://mathoverflow.net/questions/34120/sylows-theorem-3rd-proof-page-96-i-n-herstein/34122#34122 Answer by Anweshi for Sylow's theorem 3rd Proof Page 96 I.N.Herstein Anweshi 2010-08-01T16:47:40Z 2010-08-01T17:01:57Z <p>As Robin Chapman has given an elegant and compact proof, I content myself with answering your query about applications. Sylow's three theorems are a very interesting tool to classify groups of low cardinality. Some exercises are on this given in Herstein's book itself. Upto groups of order 60, you can use just the three theorems of Sylow and classify them as direct or semi-direct products. Here all three theorems are needed; only the third proof of Herstein proves all three.</p> <p>The case of groups of order 60 is a bit intricate; the appropriate reference is M. Artin's Algebra book.</p> http://mathoverflow.net/questions/29323/math-puzzles-for-dinner/34000#34000 Answer by Anweshi for Math puzzles for dinner Anweshi 2010-07-31T10:33:44Z 2010-07-31T10:33:44Z <p>A puzzle(rather, a tale to lure the reader into the domain of complex numbers) lifted from George Gamow's "<a href="http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642" rel="nofollow">One, Two, Three, Infinity</a>":</p> <p>There was a young and adventurous man who found among his great-grandfather’s papers a piece of torn parchment that revealed the precise location of a hidden treasure. The instruction reads:</p> <blockquote> <p>Sail to North latitude __ and West longitude __ where thou wilt find a deserted island. There lieth a large meadow, not pent, on the north shore of the island where standeth a lonely oak and a lonely pine tree. There thou wilt see also an old gallows on which we once were wont to hang traitors. Start thou from the gallows and walk to the oak counting thy steps. At the oak thou must turn right by a right angle and take the same number of steps. Put here a spike in the ground. Now must thou return to the gallows and walk to the pine counting thy steps. At the pine thou must turn left by a right angle and see that thou takest the same number of steps, and put another spike into the ground. Dig halfway between the spikes; the treasure is there.</p> </blockquote> <p>The instructions being quite clear and explicit, our young man chartered a ship and sailed to the South Seas. He found the island, the field, the oak and the pine, but to his great sorrow, the gallows was gone. Too long a time had passed: rain and sun and wind had disintegrated the wood and returned it to the soil, leaving no trace of the place where once it had stood. Our adventurous man fell into despair. Digging all over the field at random, he found nothing and sailed back empty-handed.</p> <p>A sad story for sure, but sadder to think that he might have easily located the treasure had he known a little about the arithmetic of complex numbers!!</p> <p>Question: How??? </p> <p>Answer: Read on from <a href="http://books.google.com/books?id=EZbcwk6SkhcC&amp;pg=PA35&amp;lpg=PA35&amp;dq=gamow+one+two+three+infinity+island&amp;source=bl&amp;ots=tLZaT7YazU&amp;hl=en&amp;sa=X#v=onepage&amp;q&amp;f=false" rel="nofollow">Here</a>.</p> http://mathoverflow.net/questions/33911/why-linear-algebra-is-funor/33999#33999 Answer by Anweshi for Why linear algebra is fun!(or ?) Anweshi 2010-07-31T10:17:48Z 2010-07-31T10:17:48Z <p>Since you are going to address undergraduates, there's a book <a href="http://www.maa.org/reviews/LAgems.html" rel="nofollow">Linear algebra gems</a> that might give you lots of simple, cool stuff to present. It is also available at <a href="http://www.amazon.com/Linear-Algebra-Gems-Undergraduate-Mathematics/dp/0883851709/" rel="nofollow">amazon</a>.</p> http://mathoverflow.net/questions/28803/what-is-the-proper-initiation-to-the-theory-of-motives-for-a-new-student-of-algeb/33645#33645 Answer by Anweshi for What is the proper initiation to the theory of motives for a new student of algebraic geometry? Anweshi 2010-07-28T11:59:42Z 2010-07-28T12:23:15Z <p><a href="http://www.math.tifr.res.in/~sujatha/ihes.pdf" rel="nofollow">Here</a> is a very understandable introductory article by R. Sujatha. For a beginning student this is good.</p> <p>In my case, after that article, my next encounter with motives was with the more precise definition of a motive from the initial parts of Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points”. It even sort of defines a mixed motive; in fact it is the only definition of mixed motive that I know.</p> <p>Read the <a href="http://www.ams.org/mathscinet-getitem?mr=1012168" rel="nofollow">Mathscinet review</a> and also, <a href="http://quomodocumque.wordpress.com/2009/07/24/le-groupe-fondamental-de-la-droite-projective-moins-trois-points-is-now-online/" rel="nofollow">Jordan Ellenberg's opinion</a> on this remarkable paper of Deligne. I myself was astonished when I first looked into it and saw how much stuff was contained in it.</p> <p>Deligne's paper "<a href="http://www.numdam.org/numdam-bin/fitem?id=SB_1968-1969__11__139_0" rel="nofollow">Formes modulaires et représentations $l$-adiques</a>" proving that the Weil conjectures imply the Ramanujan conjecture, is almost close to the theory of motives even though it does not explicitly mention motives. Here the representations of the absolute Galois group on the étale, or rather on the $\ell$-adic, cohomology is considered. This might give some starting insight into the Galois representations approach to motives.</p> http://mathoverflow.net/questions/33555/striking-applications-of-bakers-theorem Striking applications of Baker's theorem Anweshi 2010-07-27T18:58:58Z 2010-07-28T02:21:16Z <p>I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with the tag diophantine-approximation, while there are almost 900 questions on number theory overall. It is my intention to promote the important subject a little bit by asking one more question.</p> <p>Question: </p> <blockquote> <p>What are some striking applications of Baker's theorem on lower bounds for linear forms on logarithms of algebraic numbers?</p> </blockquote> <p>If, for example, I were in a discussion with a person who has no experience with diophantic approximation, to impress upon the person the importance of Baker's theorem I would cite the following two examples:</p> <ol> <li><p>Giving effective bounds for solutions of (most of the time exponential) diophantine equations under favorable condition. For example, Tijdeman's work on the Catalan conjecture, or giving effective bounds for Siegel's theorem, Fermat's last theorem, Falting's theorem, etc., in certain cases.</p></li> <li><p>Transcendence results which are significant improvements over Gelfond-Schneider. In particular, the theorem that if $\alpha_1, \ldots, \alpha_n$ are $\mathbb{Q}$-linearly independent, then their exponentials are algebraically independent over $\mathbb Q$. I would cite the <a href="http://www.math.jussieu.fr/~miw/articles/pdf/GDL/SemBxMSGB.pdf" rel="nofollow">expose of Waldschmidt</a> for more details. </p></li> </ol> <p>These are, to me, quite compelling reasons to study Baker's theorem. But as I do not know much more on the subject, I would run out of arguments after these two. I would appreciate any more striking examples of the power of Bakers' theorem. This is 1. for my own enlightenment, 2., for future use if such an argument as I hypothesized above actually happens, 3. To promote the subject of diophantine approximation in this forum, especially in the form of Baker's theorem.</p> http://mathoverflow.net/questions/31395/functions-of-several-complex-variables-book-recommendations/33566#33566 Answer by Anweshi for Functions of several complex variables: book recommendations? Anweshi 2010-07-27T20:10:47Z 2010-07-27T20:10:47Z <p>For a great introduction, try Raghavan Narasimhan, "Several complex variables", Chicago lectures in mathematics.</p> <p>Well-written, concise and accessible.</p> http://mathoverflow.net/questions/32418/grothendiecks-galois-theory-today/33532#33532 Answer by Anweshi for Grothendieck's Galois Theory today Anweshi 2010-07-27T15:28:37Z 2010-07-27T15:28:37Z <p>I suggest that you read Deligne's wonderful paper “Le Groupe Fondamental de la Droite Projective Moins Trois Points”. I read a little bit of it and was astonished. Please do take it up and read it without further loss of time, notwithstanding the French.</p> <p>Jordan Ellenberg's and Matthew Emerton's opinion is available <a href="http://quomodocumque.wordpress.com/2009/07/24/le-groupe-fondamental-de-la-droite-projective-moins-trois-points-is-now-online/" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/33510/christening-fermats-little-theorem/33515#33515 Answer by Anweshi for Christening Fermat's Little Theorem Anweshi 2010-07-27T14:20:02Z 2010-07-27T14:32:32Z <p>Compared to Fermat's two squares theorem, or Fermat's four squares theorem, Fermat's Little theorem is indeed Little.</p> <p>Not to mention the hard-to-prove Fermat Last Theorem, which goes under FLT; so that acronym, or a contraction Flt isn't suitable as it will cause confusion.</p> <p>Therefore one might as well stick with "Fermat's Little Theorem" itself. I have given above a reasoning that it is comparatively little.</p> http://mathoverflow.net/questions/33456/geometric-flavored-textbook-on-algebra/33457#33457 Answer by Anweshi for Geometric flavored textbook on algebra Anweshi 2010-07-27T00:23:19Z 2010-07-27T13:43:13Z <p>Try Artin's Geometric Algebra.</p> <p>You might also find interesting Shafarevich's book “Basic Notions of Algebra”, for understanding the philosophy behind algebra. I strongly recommend that you spend some time with that book.</p> <p>Also try Coxeter's various books such as Projective Geometry, etc..</p> <p>But of course if you really want to understand the "geometry behind the algebra", then you should first have a good footing in algebra, and then you should study algebraic geometry. Even for Artin's Geometric Algebra, you better first have an understanding of algebra before reading it. It appears to me that at the moment the best bet for you will be Shafarevich's "Basic Notions of Algebra".</p> http://mathoverflow.net/questions/29271/algebraic-geometry-used-externally-in-problems-without-obvious-algebraic-struc/33440#33440 Answer by Anweshi for Algebraic geometry used "externally" (in problems without obvious algebraic structure). Anweshi 2010-07-26T20:40:10Z 2010-07-26T20:46:24Z <p>Define the Ramanujan $\tau$-function from $\mathbb N \rightarrow \mathbb Z$ as the Fourier coefficients of the $\Delta$ function; i.e.,</p> <p>$$\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} .$$</p> <p>This is a pure number-theoretic function. Now the Ramanujan conjecture says that </p> <p>$$|\tau(p)| \leq 2p^{11/2}$$</p> <p>for every prime $p$, which is also a purely number theoretic statement.</p> <p>Pierre Deligne proved it as a consequence of the Weil conjectures. </p> http://mathoverflow.net/questions/33250/references-general-idea-of-kervaire-invariant-problem/33419#33419 Answer by Anweshi for references / general idea of kervaire invariant problem Anweshi 2010-07-26T17:26:43Z 2010-07-26T17:26:43Z <p>Have a look at the <a href="http://www-math.mit.edu/~hrm/ksem.html" rel="nofollow">MIT-Harvard Seminar</a> and the lecture notes given at this link.</p> <p>For some background, see <a href="http://www.math.rochester.edu/u/faculty/doug/akhmetiev.html" rel="nofollow">Petr. M. Akhmetev's work</a>.</p> <p>I see that you are starting with the wikipedia article. For very very introductory stuff, such as statement itself, there's the <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=4038" rel="nofollow">article of Teichner</a>, <a href="http://www.scientificamerican.com/article.cfm?id=hypersphere-exotica" rel="nofollow">article in Scientific American</a>, <a href="https://simonsfoundation.org/news/-/asset_publisher/bo1E/content/mathematicians-solve-45-year-old-kervaire-invariant-puzzle" rel="nofollow">a Simons foundation article with some animations</a> and <a href="http://www.math.rochester.edu/u/faculty/doug/news.2009.427.html" rel="nofollow">an article in Nature</a>.</p> <p>Most of the above are lifted shamelessly from Douglas Ravenel's homepage.</p> http://mathoverflow.net/questions/14404/serres-fac-in-english/14417#14417 Comment by Anweshi Anweshi 2010-10-26T18:06:46Z 2010-10-26T18:06:46Z Very impressive. This is a great contribution from you to all students of algebraic geometry. Thank you very much. Bless your kind soul! Wish you all the best. http://mathoverflow.net/questions/34803/equivalence-of-borel-algebra-and-power-set-of-rn Comment by Anweshi Anweshi 2010-08-06T23:15:06Z 2010-08-06T23:15:06Z (Also this is given in the first chapter of Rudin's &quot;Real and complex analysis&quot;). http://mathoverflow.net/questions/34803/equivalence-of-borel-algebra-and-power-set-of-rn Comment by Anweshi Anweshi 2010-08-06T23:14:18Z 2010-08-06T23:14:18Z @unknown: It is allright. The policy here is that questions whose answers are in wikipedia are not encouraged. So I wouldn't mind if you delete the question. http://mathoverflow.net/questions/34803/equivalence-of-borel-algebra-and-power-set-of-rn Comment by Anweshi Anweshi 2010-08-06T23:06:10Z 2010-08-06T23:06:10Z Ah! I had just written this down in an answer a few days ago. There is a very simple way to see that the Borel sigma algebra is much sparser compared to the power set, by a cardinality argument. See here: <a href="http://mathoverflow.net/questions/34390/on-proving-that-a-certain-set-is-not-empty-by-proving-that-it-is-actually-large/34392#34392" rel="nofollow" title="on proving that a certain set is not empty by proving that it is actually large">mathoverflow.net/questions/34390/&hellip;</a> http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/34718#34718 Comment by Anweshi Anweshi 2010-08-06T18:00:15Z 2010-08-06T18:00:15Z (More senior and knowledgeable people were sparring on infinitude of primes and I thought I will at least direct the sparring over to on-topic stuff such as the zeta function). http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/34718#34718 Comment by Anweshi Anweshi 2010-08-06T17:58:31Z 2010-08-06T17:58:31Z @Bcnrd: I do not know about the merit of Fursternberg's proof. But even a small lemma, small proof by a different means can be a great thing. A simple step for a single man or a single lemma might turn out to be a giant leap for a whole subject. Here I mean Euler's proof of the infinitude of primes. His method of proving it introduced the zeta function and all the rest of the story(which is after all our topic of discussion). http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/34720#34720 Comment by Anweshi Anweshi 2010-08-06T16:40:13Z 2010-08-06T16:40:13Z I wanted to know about approaches that were not already known to me. Besides some overly strict people are raising objection to the question itself as being subjective and argumentative, in the comments over there. I was in fear of the question getting closed and so steered away from anything with any hint of speculation. Note that by such fears I already CW-ed the question. http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory Comment by Anweshi Anweshi 2010-08-06T16:34:19Z 2010-08-06T16:34:19Z Nothing in the book of Ivic or Titchmarsch and Heath-Brown. More precisely, methods outside the traditional sybjects of elementary number theory and analytic number theory. I have given two examples above. One with algebraic geometry and one with thermodynamics. http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/34718#34718 Comment by Anweshi Anweshi 2010-08-06T13:51:12Z 2010-08-06T13:51:12Z Around the time I joined here I briefly thought of asking a big-list question on proofs of infinitude of primes. Then I felt that it is pointless as it is given already in Ribenboim, and moreover I had asked other big list of proofs questions and I better stop. http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/34744#34744 Comment by Anweshi Anweshi 2010-08-06T13:40:14Z 2010-08-06T13:40:14Z Thanks. I didn't know about it. http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/34720#34720 Comment by Anweshi Anweshi 2010-08-06T13:37:46Z 2010-08-06T13:37:46Z I already mentioned the field with one element in thecquestion itself and it was with the hopes that the answers will not spend time on that topic. http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory Comment by Anweshi Anweshi 2010-08-06T13:35:37Z 2010-08-06T13:35:37Z I just asked for an approach that a person answering might think to be promising. We do not want to hear just every failes attempt, do we? I do not want to give the open problem tag because I am uncomfortable with the connotation that I am aking to prove the Riemann hypothesis. http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory Comment by Anweshi Anweshi 2010-08-06T00:29:10Z 2010-08-06T00:29:10Z @Tom Smith: Yes. Though I didn't mention it explicitly, what the professor told me was that they failed completely hopelessly. Whereas the non-number theoretic approaches require some theory-building and there is hope yet and for the meantime we try to do the groundwork. http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory Comment by Anweshi Anweshi 2010-08-05T23:18:35Z 2010-08-05T23:18:35Z @Gjergji(or others): Feel free to make whatever change you like(except forcing it community wiki, which I am not in favor of, and which would merit a discussion). http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory Comment by Anweshi Anweshi 2010-08-05T23:06:41Z 2010-08-05T23:06:41Z @David Hansen: I mean that even if they aren't successful they might unearth a lot of interesting math. Indeed, the proof of Weil conjectures is already a lot of interesting math.