12,594 reputation
22661
bio website math.stackexchange.com
location Princeton, NJ
age 19
visits member for 3 years, 4 months
seen 3 hours ago

Contact: amiteshdatta@gmail.com

I am a graduate student in pure mathematics at Princeton University.

I am interested in the following branches of pure mathematics and will generally answer questions in these areas:

General Topology, Finite Group Theory, Noncommutative Algebra, Commutative Algebra, Galois theory, Measure Theory, Functional Analysis, Complex Analysis, Fourier Analysis, Algebraic Geometry, Algebraic Topology, Topological K-theory, Differential Geometry, Algebraic Number Theory, Riemannian Geometry, Lie Groups and Lie Algebras, Spectral Sequences, Morse Theory and Morse Homology, Symplectic Geometry, Algebraic K-theory, Contact Geometry, Topological Quantum Field Theory

I have read, in part or completely, the following books in mathematics and recommend them all (the list is not exhaustive and is in no particular order):

Principles of Mathematical Analysis by Walter Rudin

Topology: A First Course by James Munkres

Algebra: A Graduate Course by Martin Isaacs

Real and Complex Analysis by Walter Rudin

Linear Algebra Done Right by Sheldon Axler

An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby

Elements of Algebraic Topology by James Munkres

Finite Group Theory by Martin Isaacs

Topics in Algebra by I.N. Herstein

Algebraic Number Fields by Gerald Janusz

Introduction to Commutative Algebra by Michael Atiyah and Ian Macdonald

Algebraic Geometry: A First Course by Joe Harris

Algebraic Geometry by Robin Hartshorne

Algebraic Geometry and Arithmetic Curves by Qing Liu

Classical Fourier Analysis by Loukas Grafakos

Finite Dimensional Vector Spaces by Paul Halmos

Problems in Algebraic Number Theory by Jody Esmonde and M. Ram Murty

Analysis on Manifolds by James Munkres

Noncommutative Rings by I.N. Herstein

K-theory by M. F. Atiyah

Lie Groups by Daniel Bump

Cohomology Operations by Norman Steenrod and David Epstein

Riemannian Geometry by M. P. do Carmo

Morse Theory by J. Milnor

Lectures on the h-corbodism theorem by J. Milnor

The local structure of algebraic K-theory by Bjorn Ian Dundas, Thomas Goodwillie, and Randy McCarthy

Symplectic Geometry by Ana Cannas da Silva


Sep
5
comment Is Serre's $S_1$ condition equivalent to having no embedded primes?
@Pedro I guess it could be because one copies the code from someone else who has used it incorrectly - I know sometimes I've made mistakes because I've copied the code from someone who used it incorrectly. (Of course, in this case, Ben may have other reasons.) Ben, how are things? I noticed you gave a talk at the Honours Conference last week. How is your thesis progressing? (I know this is off-topic but maybe we can exchange emails if you aren't too busy and I can delete this comment once you see it.) It's great to see you back on stackexchange again.
Sep
5
comment Can essentially bounded function take infinite value on measure zero set?
+1 for the excellent examples. I know this is nitpicky and only correcting a typo, but just to make your answer perfect, shouldn't $\left\|f\right\|_{\infty}=3$ in the first example?
Sep
3
comment A Problem from Docarmo's Differential Geometry
Yes, @Groups, you are correct - this follows from the fact that sums and products of differentiable functions are differentiable, as well as the relevant rules for differentiating such sums and products!
Sep
3
answered A Problem from Docarmo's Differential Geometry
Jul
13
comment Prerequisites for Atiyah Macdonald
Hi @Ben, thank you so much for your comment! I didn't notice it earlier, and I apologise for my slow reply. You are absolutely right that a knowledge of the Tor and Ext functors is required for some of the exercises, especially on flatness. I didn't know any homological algebra when I first read the textbook, so I just ignored those exercises, but I now know that it is not that hard to pick up what is required (and it is well worth it in the long run). I recall Matsumura's Commutative Ring Theory furnishes a quick introduction to the required homological algebra in one of the appendices.
Jul
13
comment Symplectomorphism Preserves Cotangent fibrations
Thanks for your reply, @user54440. You can prove $g(p,\sigma_p)=(q,\eta_q)$ $\implies$ $g(p,\lambda\sigma_p)=(q,\lambda\eta_q)$ for all negative $\lambda < 0$ as follows: assume $g(p,-\sigma_p)=(q,\eta_q')$, in which case the claim for $\lambda>0$ implies $g(p,\lambda\sigma_p)=(q,\lambda\eta_q')$ for negative $\lambda<0$. However, $g$ is differentiable at $(p,0)$, and thus $\lambda\to g(p,\lambda\sigma_p)$ is differentiable at $\lambda=0$ (by the chain rule). Now, compare the left-hand and right-hand derivatives of this function at $\lambda=0$ to deduce that $\eta_p'=-\eta_p$. QED
Jul
12
comment Symplectomorphism Preserves Cotangent fibrations
BTW, the fact that (b) is true for $\lambda=0$ can be seen as follows (without a continuity argument): if $g(p,\sigma_p)=(q,\eta_q)$, then we wish to show that $g(p,0)=(q,0)$. Indeed, if $g(p,0)=(q,\eta_q')$, then applying (a) yields $g(p,0)=(q,\lambda\eta_q')$ for all positive $\lambda>0$. However, this forces $\eta_q'=0$ (just apply the claim of the last sentence for two different (positive) values of $\lambda$).
Jul
12
comment Symplectomorphism Preserves Cotangent fibrations
... If you know (b) is true for $\lambda=0$ (which you observed is a continuity argument), then you can conclude that $g(p,\sigma_p)=(q,\eta_q)$ $\implies$ $g(p,0)=(q,0)$, and since this is true for all $\sigma_p\in T_p^{\ast} M$, it follows that $g$ is fiber-preserving (i.e., $g(T_p^{\ast} M)=T_q^{\ast}M$). I guess this is essentially what Robert Bryant stated in his answer, but this is the way I thought about (and expressed) the solution. (Also, the linearity of $g$ on fibers follows once you've proven that $g=f^{\#}$ so this also proves (b).) Hope it isn't redundant and helps in some way!
Jul
12
comment Symplectomorphism Preserves Cotangent fibrations
I noticed this question and I was going to answer (and I noticed your two questions on mathoverflow). I think essentially the key points are: (a) $g$ commutes with $\theta$ so $g(p,\sigma_p)=(q,\eta_q)$ $\implies$ $g(p,\lambda\sigma_p) = (q,\lambda\eta_q)$ for positive $\lambda>0$ and (b) by the differentiability of $g$ at $(p,0)$, you can conclude that $g(p,\lambda\sigma_p)=(q,\lambda\eta_q)$ for all $\lambda$. I find (b) unnecessary though (I think the hint in da Silva's textbook is misleading in this regard?) ...
Jul
11
comment Prime radical that is nil but not nilpotent
What are your thoughts on the problem? Can you see that the prime radicals of the finite products $\prod_{n=1}^{N} \mathbb{Z}/2^n\mathbb{Z}$ are nilpotent (but with nilpotent power depending on $N$)? I think that would help. Of course, that the prime radical is nil is an elementary result in commutative algebra.
Jul
6
comment basic exercise about Schwartz spaces
I might be missing something but I'm unsure about this answer. The boundedness of $\left|x^{k}f(x)\right|$ is only in issue if $k<0$ by the definition of $f\in {\cal S}(\mathbb{R})$. So, assume henceforth that $k<0$. In this case, the boundedness of $\left|x^kf(x)\right|$ away from $0$ is evident. Unfortunately, near $x=0$, there seems to be an issue. If $f(0)=0$ as assumed, then there is no issue if $k=-1$, i.e., $\left|\frac{f(x)}{x}\right|$ is bounded near $0$. However, how is $\left|x^kf(x)\right|$ even defined at $0$ if $k<-2$ and the order of vanishing of $f$ at $0$ is $1$?
Jul
6
comment Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?
Hi @DivyangBhimani, sorry, I think it should be $\left\|f\right|_{L^1(\mathbb{R})}\geq 1$ in my second comment above - I got the direction of the inequality estimate incorrect! Sorry! I think this is still useful to know though, a lower bound on $\left\|f\right\|_{L^1(\mathbb{R})}$, but I do wonder whether we could obtain a nice upper bound as well.
Jul
6
comment Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?
... in the meantime, please let me know if you think of anything yourself; I think this is an interesting problem!
Jul
6
comment Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?
Hi @Divyang, it might be easy to answer this question but I don't think I will contemplate it right now in too much detail (but I will try to do so later on). However, I will quickly post one observation: by the boundedness of the Fourier transform operator (which, in fact, has norm $1$), we know that $\left\|f\right|_{L^1(\mathbb{R})}\leq 1$ (since $\widehat{f}$ is bounded in $L^{\infty}$ norm by $1$). (You might have already observed this.) That's what immediately comes to my mind but there might be other avenues of obtaining better bounds which I will think about ...
Jul
6
answered finding $ \int_{C(0,2)^+} \frac{z^3}{z^3+z^2-z-1} $
Jul
6
comment Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?
Also, if you want to allow $W$ to be arbitrary, then simply note that there exists $\epsilon>0$ such that $\{x\in\mathbb{R}: d(x,C)<\epsilon\}\subseteq W$ for any open $W\supseteq C$ and you can apply the reasoning above with "$\{x\in\mathbb{R}:d(x,C)<\epsilon\}$" in place of "$W$".
Jul
6
answered Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?
Jul
6
revised Show that $z^n$ is analytic. Hence find its derivative.
added 462 characters in body
Jul
6
answered Show that $z^n$ is analytic. Hence find its derivative.
Jul
5
revised Proving linear independence of matrices
added 24 characters in body