12,789 reputation
22765
bio website math.princeton.edu/directory/…
location Princeton, NJ
age 19
visits member for 3 years, 7 months
seen 24 mins 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


2d
awarded  Constituent
Dec
18
awarded  Nice Answer
Dec
17
awarded  Nice Answer
Dec
10
comment Is every non-trivial ideal in a commutative ring is a principal ideal?
Dear @athos, $\mathbb{Z}[X]$ is the polynomial ring over $\mathbb{Z}$ in one variable $X$. Also, $(2,X)$ is the ideal generated by $2,X\in \mathbb{Z}[X]$, that is, the set of all elements of $\mathbb{Z}[X]$ of the form $2p + Xq$ for $p,q\in \mathbb{Z}[X]$.
Dec
9
comment Is every non-trivial ideal in a commutative ring is a principal ideal?
In addition to @Qiaochu's example, you could also take $(2,X)\subseteq \mathbb{Z}[X]$.
Dec
8
awarded  Caucus
Oct
21
comment Two continuous functions with connected images
Dear @Alfred, one should check that that the preimage under $h$ of any closed set is closed. Now, if $C$ is a closed set, then $h^{-1}(C)\cap [0,1]=f^{-1}(C)$ and $h^{-1}(C)\cap [1,2]=g^{-1}(C)$. Does that help?
Oct
20
comment Definition question in algebraic topology.
Dear @jip, a short exact sequence usually refers to a sequence of the form $0\to A\to B\to C\to 0$ (where none, some, or all of $A,B,C$ could be non-zero); this is simply a matter of terminology, though. In this terminology, the Mayer-Vietoris sequence is not short exact in general, but it is an exact sequence so we refer to it as "long exact".
Oct
20
answered Definition question in algebraic topology.
Oct
7
comment Bound of the gradient of a $C^{m}$ function.
Dear @Taggy, I might be missing something but if $n=1$, then this states that $\left|f'(x)\right|\leq \frac{C}{\left|x\right|}$ for all $f\in C^1(\mathbb{R})$, all $x\in\mathbb{R}$, and some constant $C$ (depending on $f$). In particular, $\lim_{\left|x\right|\to \infty} \left|f'(x)\right|=0$ but I don't think this is true for all $C^1$ functions - e.g., $f(x)=x^2$. or mm-aops great counterexample.
Oct
7
comment Proving a function is an open map
Dear @Eric_, does $g:U\to\mathbb{R}^2$ necessarily have to be an open map? What if $g$ is constant? Do you want to have further assumptions on $g$?
Sep
30
awarded  Explainer
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!