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Terence Tao
Works at UCLA
Attended Princeton University
Lives in Los Angeles
28,000 followers|2,109,481 views

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### Terence Tao

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It's unusual to see a playable game with what is essentially a Turing-complete user interface.  My son and I had a lot of fun getting through the first dozen or so levels so far.

Truly geeky fun.﻿
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That was a very clever game.﻿

### Terence Tao

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The American Mathematical Society has issued a call for proposals for the von Neumann symposium for 2017 (a week long summer conference on a mathematical topic of current interest).  [I'm serving on the selection committee for this symposium.]﻿
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Really?﻿

### Terence Tao

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This paper clears up what was an odd gap between the blowup and global regularity theory for certain simplified toy models of the Navier-Stokes equation.  To vary the comparative strength between the nonlinear and dissipative components of Navier-Stokes, one can replace the dissipative Laplacian term in the true Navier-Stokes equation with a hyperdissipative term (a power of the Laplacian with exponent alpha larger than one) or a hypodissipative term (a power with exponent alpha less than one).  The larger alpha is, the more powerful the dissipation, and the more likely one believes global regularity holds.

In three spatial dimensions, the critical exponent is alpha=5/4, and it is known that for alpha at or above this level one has global regularity.  A few years ago, I observed that one could shave a small number of logarithms from the critical dissipation (which roughly corresponds to setting alpha to be "infinitesimally" below 5/4) and still have global regularity.  On the other hand, if one shaves off too many logarithms, then an argument in a more recent paper of mine shows that one can construct a toy (non-autonomous) dyadic model of Navier-Stokes which exhibits finite time blowup.  However, there was a puzzling intermediate region in which neither global regularity nor finite time blowup was clear.

What these authors have done is performed a finer analysis of the energy flow between dyadic scales to show that (a simplified dyadic model of) the Navier-Stokes equation exhibits global regularity in this intermediate regime; they are currently working on extending these results to the non-dyadic Navier-Stokes equation (with slightly supercritical hyperdissipation).

Unfortunately, this work does not directly impact the true Navier-Stokes equations (in which alpha=1), but it does improve our understanding of where the threshold between critical and genuinely supercritical behaviour lies.

#spnetwork #recommend arXiv:1403.2852﻿
Selected Papers Network
Use your Google sign-in to tag, share, discuss and recommend papers: Share This Paper. Share on Google+. Global regularity for a logarithmically supercritical hyperdissipative dyadic equation. David Barbato, Francesco Morandin, Marco Romito. We prove global existence of smooth solutions for a ...
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### Terence Tao

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Somewhat late on this, but: the 2014 Wolf prize in mathematics is awarded to Peter Sarnak.﻿
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### Terence Tao

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There is a lot of discussion in various online mathematical forums currently about the interpretation, derivation, and significance of Ramanujan's famous (but extremely unintuitive) formula

1+2+3+4+... = -1/12   (1)

or similar divergent series formulae such as

1-1+1-1+... = 1/2 (2)

or

1+2+4+8+... = -1. (3)

One can view this topic from either a pre-rigorous, rigorous, or post-rigorous perspective (see this page of mine for a description of these three terms: http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/  ).  The pre-rigorous approach is not particularly satisfactory: here one is taught the basic rules for manipulating finite sums (e.g. how to add or subtract one finite sum from another), and one is permitted to blindly apply these rules to infinite sums.  This approach can give derivations of identities such as (1), but can also lead to derivations of even more blatant absurdities such as 0=1, which of course makes any similar derivation of (1) look quite suspicious.

From a rigorous perspective, one learns in undergraduate analysis classes the notion of a convergent series and a divergent series, with the former having a well defined limit, which enjoys most of the same laws of series that finite series do (particularly if one restricts attention to absolutely convergent series).  In more advanced courses, one can then learn of more exotic summation methods (e.g. Cesaro summation, p-adic summation or Ramanujan summation) which can sometimes (but not always) be applied to certain divergent series, and which obey some (but not all) of the rules that finite series or absolutely convergent series do.  One can then carefully derive, manipulate, and use identities such as (1), so long as it is made precise at any given time what notion of summation is in force.  For instance, (1) is not true if summation is interpreted in the classical sense of convergent series, but it is true for some other notions of summation, such as Ramanujan summation, or a real-variable analogue of that summation that I describe in this post: http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/#comment-265849

From a post-rigorous perspective, I believe that an equation such as (1) should more accurately be rendered as

1+2+3+4+... = -1/12 + ...

where the "..." on the right-hand side denotes terms which could be infinitely large (or divergent) when interpreted classically, but which one wishes to view as "negligible" for one's intended application (or at least "orthogonal" to that application).  For instance, as a rough first approximation (and assuming implicitly that the summation index in these series starts from n=1 rather than n=0), (1), (2), (3) should actually be written as

1+2+3+4+... = -1/12  + 1/2 infinity^2   (1)'

1-1+1-1+... = 1/2 - (-1)^{infinity} /2 (2)'

or

1+2+4+8+... = -1 + 2^{infinity}  (3)'

and more generally

1+x+x^2+x^3+... = 1/(1-x) + x^{infinity}/(x-1)

where the terms involving infinity do not make particularly rigorous sense, but would be considered orthogonal to the application at hand (a physicist would call these quantities unphysical) and so can often be neglected in one's manipulations.  (If one wanted to be even more accurate here, the 1/2 infinity^2 term should really be the integral of x dx from 0 to infinity.)  To rigorously formalise the notion of ignoring certain types of infinite expressions, one needs to use one of the summation methods mentioned above (with different summation methods corresponding to different classes of infinite terms that one is permitted to delete); but the above post-rigorous formulae can still provide clarifying intuition, once one has understood their rigorous counterparts.  For instance, the formulae (1)' and (3)' are now consistent with the left-hand side being positive and diverging to infinity, and the formula (2)' is consistent with the left-hand side being indeterminate in limit, with both 0 and 1 as limit points.  The fact that divergent series often do not behave well with respect to shifting the series can now be traced back to the fact that the infinite terms in the above identities produce some finite remainders when the infinity in those terms is shifted, say to infinity+1.

For a more advanced example, I believe that the "field of one element" should really be called "the field of 1+... elements", where the ... denotes an expression which one believes to be orthogonal to one's application.﻿
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It seems physicist regard it correct somehow but mathematician do the opposite. It is very strange.﻿
Have him in circles
28,000 people

### Terence Tao

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A picture is worth up to 64K bytes.

[It makes me wonder if there are any difficult mathematical topics of public interest which would also be worth representing in comic form.  I know a few examples, e.g. Lob's theorem http://lesswrong.com/lw/t6/the_cartoon_guide_to_l%C3%B6bs_theorem/ , but perhaps there could be others.]﻿
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Though not a consequence of the theorem itself, here is an interesting exploration that turns it into a kind of spreadsheet evaluator - http://blog.sigfpe.com/2006/11/from-l-theorem-to-spreadsheet.html﻿

### Terence Tao

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I ended my term on the Abel committee last year and was not directly involved in the decision this time around, but Sinai is certainly a worthy choice for this award.﻿
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excellent!!! ﻿

### Terence Tao

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The NSF is calling for proposals for week-long CBMS regional research conferences (based around a single lecturer giving an intensive series of lectures on one topic, with the notes to be converted into a book).  I gave one of these in Park City (on dispersive PDE) back in 2003; it was extremely work-intensive (two lectures daily for five days), but very productive and enjoyable.﻿
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I just ordered your book "Solving Mathematical Problems: A Personal Perspective". No doubt I would grasp only a fraction of it, but looking forward to reading it.﻿

### Terence Tao

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The IHES summer school (July 9-23 2014) on analytic number theory (covering, among other things, the recent progress of Zhang and others on prime gaps) is now accepting applications.﻿
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### Terence Tao

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One of the most basic classical problems in design theory - the existence of designs generalising the famous "school girls problem" of Kirkman - has now been solved.  Van Vu has a nice blog post on the problem and the proof at http://vuhavan.wordpress.com/2014/01/14/existence-of-designs/ ; see also Gil Kalai at http://gilkalai.wordpress.com/2014/01/16/amazing-peter-keevash-constructed-general-steiner-systems-and-designs/

#spnetwork arXiv:1401.3665﻿
Selected Papers Network
Use your Google sign-in to tag, share, discuss and recommend papers: Share This Paper. Share on Google+. The existence of designs. Peter Keevash. We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural ...
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So happy that design theory is getting the respect and recognition that it deserves.﻿
People
Have him in circles
28,000 people
Education
• Princeton University
Mathematics, 1992 - 1996
• Flinders University
Mathematics, 1989 - 1992
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Occupation
Mathematician
Employment
• UCLA
Mathematician, present
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