bio  website  math.ucla.edu/~tao 

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Professor of Mathematics at UCLA
1d

revised 
Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
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1d

awarded  Nice Answer 
1d

answered  Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$? 
2d

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Semiprime number theorem with small prime factor
More precisely, the number of semiprimes $p_1 p_2$ up to $N$ with $p_2 \geq p_1 > N^\theta$ is asymptotic to $\int_\theta^{1/2} \frac{dt}{t(1t)} \frac{N}{\log N}$ (this follows from the prime number theorem, and I recommend it is as an exercise). The number of semiprimes as a whole grows like $N \log\log N /\log N$, so the set of exceptions to $p_1 \leq N^\theta$ are very small. 
Mar 24 
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Which way for reading the proofs?
I have some advice on reading papers at terrytao.wordpress.com/adviceonwritingpapers 
Mar 24 
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Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?
If one is willing to consider the prime factorisation coming from the fundamental theorem of arithmetic as "natural", then the problem reduces to finding a "natural" bijection between the nonnegative integers and the integers, which maps 0 to 0. 
Mar 14 
awarded  Favorite Question 
Mar 4 
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Why are 1 and 1 eigenvalues of this matrix?
Ah, fair point: a dimension count shows that this is true for an open set of P, but not necessarily for all P. Nevertheless, this is still enough to demonstrate that spectrum at +1 or 1 occurs on a Zariski dense set of parameters, and hence for all choices of parameters (as this is a Zariski closed condition). (What appears to happen is that as P transitions outside of the range where the reflection can be conjugated to be orthogonal, some eigenvalues on the unit circle collide and become a pair of real eigenvalues, but one still keeps an eigenvalue at +1 and 1 throughout the process.) 
Mar 3 
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Strichartz estimates for the wave equation
Yes, under various conditions on the exponents: the classic reference is Ginibre and Velo, ams.org/mathscinetgetitem?mr=1351643 , and there are many subsequent references also. 
Mar 3 
revised 
Zeta functions versus Cramer's conjecture
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Mar 3 
answered  Zeta functions versus Cramer's conjecture 
Mar 2 
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Why are 1 and 1 eigenvalues of this matrix?
If $n$ is even and $t_n=\dots=t_2=0$ then $A(t_n) \dots A(t_1) A(t) = S(t)$ will generically have no spectrum at +1 or 1. The property of having no spectrum at +1 or 1 is a Zariski open condition, and so $A(t_n) \dots A(t)$ will have no spectrum at +1 or 1 generically (that is, outside of an algebraic set of positive codimension). 
Mar 2 
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Density of numbers whose prime factors all come from a fixed congruence class
I would recommend looking at the paper of Granville, Koukoulopoulos, and Matomaki at arxiv.org/abs/1205.0413 for the state of the art on the latter question. It looks like $D(a,q;x)$ should be something like $x \log^{1/\phi(q)  1} x$. 
Mar 2 
revised 
Why are 1 and 1 eigenvalues of this matrix?
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Mar 2 
answered  Why are 1 and 1 eigenvalues of this matrix? 
Feb 26 
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Polynomial growth without Gromov's theorem
My feeling is that any structural analysis of groups (or other grouplike objects) of polynomial growth that is refined enough to give this claim would also be powerful enough to prove Gromov's theorem. For your application, there may be some recent work of Hrushovski on the structural theory of approximate equivalence relations that may be relevant (see ma.huji.ac.il/~ehud/approxeq.pdf ), though it may not be so easy to translate his general formalism to a specific concrete setting such as yours. 
Feb 25 
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Proof of Pollard's inequality
One can also prove Pollard's inequality by submodularity arguments: terrytao.wordpress.com/2011/12/26/… , although ultimately this is again basically an induction on the measure of one of the sets $A,B$. 
Feb 24 
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injective homogeneous polynomial functions $p(x,y) \in \mathbb{Z}[x,y]:{\mathbb{N}}^2 \to \mathbb{N}$
As with your preceding question, probabilistic heuristics (see e.g. terrytao.wordpress.com/2012/09/18/… ) suggest such polynomials should be generically injective in degree 5 and higher but not in degree 4 or lower, at least for "typical" choices of polynomial $p$. It may be possible to use the BombieriLang conjecture to make the notion of "typical" precise (probably has to do with the the nontrivial component of the variety $p(x,y)=p(z,w)$ being of general type). 
Feb 24 
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Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
Probabilistic heuristics suggest that the set of such Fourier coefficients (or equivalently, the set of nontrivial integer solutions to $m_1^4 + n m_2^4 = m_3^4 + n m_4^4$) is very sparse (only about $O(\log X)$ such solutions up to height $X$). So it is unlikely that analytic methods will be of much help here. Maybe there is some algebraic number theory approach but it doesn't look too promising (e.g. I don't see a norm form or other obviously algebraic structure here). 
Feb 23 
awarded  Good Answer 