Terry Tao
Reputation
39,096
242/100 score
 Nov 27 comment Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$ In more physical terms, one can place an arbitrary heat source outside of $D$, and this will surely affect the temperature inside of $D$ at later times, leading to non-uniqueness for the linear heat equation $\partial_t u - \Delta u = 0$ in the absence of a boundary condition, and your quasilinear equation $\partial_t u - u \Delta u = 0$ for $u$ near $1$ will surely have very similar behaviour. Nov 27 comment Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$ It's highly unlikely that there would be uniqueness without a boundary condition. After all, if $D'$ is a small neighbourhood of $D$, one can non-uniquely smoothly continue the initial data $u(0,\cdot)$ from $D$ to $D'$. Any classical solution to your PDE in $[0,T] \times D'$ would restrict to a classical solution in $[0,T] \times D$, and given the infinite speed of propagation for parabolic equations, this should give non-uniqueness in $[0,T] \times D$ (assuming existence of smooth solutions, of course). Nov 25 comment Are there unconditional results for boundedness of finitely many rational points on $f(x,y)=n$ for all $n$? Conditionally on the Bombieri-Lang conjecture, this should be true for all $f$ that are not degenerate in the sense that the curves $f(x,y)=n$ are reducible or have genus at most 1, from the work of Caporaso, Harris, and Mazur: ams.org/mathscinet-getitem?mr=1325796 . But this type of result is unlikely to be made unconditional in the near future. Nov 25 comment Horn's spectrum problem with random Hermitian matrices Actually, the inequalities in the compact operator case are simply the union of the inequalities in the finite dimensional cases: see for instance the paper of Bercovici, Li, and Timotin in ams.org/mathscinet-getitem?mr=2567501 , together with the references therein. Nov 25 comment Horn's spectrum problem with random Hermitian matrices Ronald Speicher has a number of nice surveys on free probability which in particular address the asymptotic version of the original question: mast.queensu.ca/~speicher/survey.html Nov 25 comment Horn's spectrum problem with random Hermitian matrices I'd like to mention that the initial solution to Horn's conjecture also relies heavily on Klyachko's paper "Stable bundles, representation theory and Hermitian operators", Selecta Math. (N.S.) 4 (1998), no. 3, 419–445, in addition to my paper with Knutson. (There have since been subsequent proofs of Horn's conjecture that avoid the difficult GIT machinery of Klyachko, though.) Nov 18 awarded Enlightened Nov 18 awarded Nice Answer Nov 18 revised Where did the term “additive energy” originate? added 4 characters in body Nov 18 answered Where did the term “additive energy” originate? Oct 27 answered Maximal $L_1$ norm of Fourier Transform of a Subset Oct 24 comment Real-world applications of mathematics, by arxiv subject area? Wasn't my original contribution (I merged it from another, now-deleted, answer), but I updated the link anyway. Oct 24 revised Real-world applications of mathematics, by arxiv subject area? deleted 24 characters in body Oct 21 comment when the composition of two ergodic maps is ergodic? For commuting circle shifts, your question boils down to asking for sufficient criteria for the sum of two irrational numbers to be irrational. It seems difficult to come up with any usable criteria other than the tautological "the sum is not rational". Oct 19 awarded Yearling Oct 8 comment What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$? I'm accepting this answer as being (a) one with a little bit of precedent (from the n-lab), (b) a term which does not require specialised knowledge in order to understand, and (c) the most popular of the alternatives suggested here. I also like the fine distinction between the tuple $(x \mapsto f_1(x),\dots,x \mapsto f_n(x))$ of $f_1,\dots,f_n$ and the tupling $x \mapsto (f_1(x),\dots,f_n(x))$ of $f_1,\dots,f_n$; these two concepts are canonically and naturally equivalent and it is a fairly safe abuse of notation to call them both $(f_1,\dots,f_n)$, and the similar names support this. Oct 8 accepted What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$? Oct 6 awarded Nice Question Oct 5 asked What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$? Oct 2 awarded Enlightened