bio  website  math.ucla.edu/~tao 

location  Los Angeles  
age  39  
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Professor of Mathematics at UCLA
1d

comment 
Intuition for Integral Transforms
en.wikipedia.org/wiki/… 
Jul 18 
awarded  Nice Answer 
Jul 10 
comment 
Kloostermanlike sum with inverse to different moduli
The expressions you write are currently ambiguous. For instance, $\bar{s}^{(r)}$ is only determined up to a multiple of $r$, and this affects the first exponential. Are you going to take the representative of this residue class in $\{0,1,\dots,r1\}$? If so, the sum is going to get rather nonalgebraic, and it will be significantly more difficult to use the usual methods to handle these sums. 
Jul 2 
revised 
Polynomial representing all nonnegative integers
added 405 characters in body 
Jul 2 
revised 
Polynomial representing all nonnegative integers
added 2399 characters in body 
Jul 2 
comment 
$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
It looks like the $b_n$ are in fact bounded in $L^\infty$, because the $u_n$ are. In any case, from the uniform boundedness principle, the $b_n(t)$ have to be bounded in $L^q$ if one is to have weak convergence, so such a bound must already appear somewhere in the argument. 
Jul 2 
awarded  Curious 
Jun 30 
comment 
$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
I think you are omitting an important hypothesis (mentioned in the paper), namely that the $b_n(t)$ are uniformly bounded in $L^q(\Omega)$. With this uniform bound, one can obtain weak convergence through testing against $C^\infty_c(\Omega)$ functions, at which point one can use the $C^0$ strong convergence to conclude. 
Jun 26 
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Are there any nontrivial nearisometries of the $n$dimensional cube?
It occurs to me that inverse LittlewoodOfford theory (see e.g. Ch7 of my book with Van Vu) is likely to be helpful. Note that for most i, the distribution of $(Ox)_i$, with $x$ drawn uniformly from the discrete cube, is concentrated at $\pm 1$, which by ILO theory gives strong constraints on the coefficients of the i^th row of O (most of the entries must lie very close to a generalised arithmetic progression). By using multidimensional ILO and working with several i simultaneously, one should be able to cut down the possibilities further. 
Jun 26 
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Are there any nontrivial nearisometries of the $n$dimensional cube?
Nice question! A model question: if $Ox$ lies exactly in the unit cube $\{1,+1\}^n$ for $n^{100}$ of the $x$'s, does this force $O$ to be an approximate cube symmetry? Your situation is close to this, except that rather than exactly lying in the unit cube, all but $O(\log n)$ of the coefficients of $Ox$ lie within $0.0001$ of $\{+1,1\}$ (together with some similar assertions of this nature). It's possible that some additive combinatorics or some BlumLubyRubinfeld type arguments can be used to improve this structure, though $\{1,+1\}^n$ isn't additively closed, which is a problem. 
Jun 19 
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In what ways is physical intuition about mathematical objects nonrigorous?
In any event, how certain could one be that failure of claim i and failure of claim j could not possibly be (completely or partially) statistically independent? I doubt that one's certainty on this issue could be anywhere close to $1  10^{10^{100}}$. 
Jun 19 
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In what ways is physical intuition about mathematical objects nonrigorous?
One does not need full statistical independence here; even if only a tiny portion (say $10^{10^{100}}$) of the event of failure for $i$ is independent of the event of failure for $j$, then there is still a strong likelihood of breakdown of the argument once $n$ exceeds, say, $10^{2 \times 10^{100}}$. For instance, an argument that somehow proceeds through an induction over all possible ensembles of an Nparticle system, appealing to the second law of thermodynamics for each such ensemble, would be subject to this problem and be mathematically dubious. 
Jun 18 
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In what ways is physical intuition about mathematical objects nonrigorous?
If $n$ is large (e.g. larger than $10^{10^{100}}$), then it becomes essential in mathematics that each of the claims $P(i) \implies P(i+1)$ are known with $100\%$ certainty (again, in the internal sense of the mathematical reasoning model), as even a $10^{10^{100}}$ failure rate for each of these steps will throw the derivation of $P(n)$ into serious doubt. This requirement of absolute certainty is particularly in effect for mathematical arguments involving countable or uncountable infinities; one simply cannot afford any failure probability at all, no matter how small. 
Jun 18 
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In what ways is physical intuition about mathematical objects nonrigorous?
I should perhaps clarify that in my post above, terms such as "certainty" and "probability" were meant to be interpreted internally within one's accepted mode of reasoning (whether physical or mathematical), and not with regards to the external degree of accuracy to which one can actually apply this reasoning. For instance, a common technique in mathematical reasoning is mathematical induction, in which a claim P(n) is deduced from an initial claim P(0) and the claims $P(i) \implies P(i+1)$ for $i=0,\dots,n1$. (cont.) 
Jun 17 
revised 
Why differential forms are important?
corrected spelling of "Stokes" 
Jun 16 
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References for LWP of a NLS Equation
"NLS" and "inverse square potential". (I had also tried NLS and "Coulomb potential" but this gave less relevant hits.) 
Jun 16 
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References for LWP of a NLS Equation
A google search turns up this article tandfonline.com/doi/full/10.1080/… , although I don't have full access to it at present. 
Jun 16 
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A question on the proof of CheegerColding's second paper
From a quick glance at the paper, it appears that the second inequality is not a consequence of the first, but is simply a direct consequence of (the contrapositive of) Lemma 3.1, applied to the $\epsilon+\psi$neighbourhood of $E_{\eta,i}$. 
Jun 15 
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A sumset inequality
I believe Ruzsa's survey article link.springer.com/chapter/10.1007%2F9781461224181_21 should contain a similar counterexample, though I can't access it currently. (The existence of such a counterexample is Exercise 2.4.9 of my book with Van, but we didn't give a hint.) 
Jun 14 
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A question on the proof of CheegerColding's second paper
The paper in your link does not have a Theorem 3.7 or a Lemma 3.1. I suggest reediting your question to add significantly more detail, and to quote relevant parts the paper (by screenshot if necessary). 