Archive for the ‘Uncategorized’ Category

FCRC talks

July 10, 2015

Mental Wilderness

I was at FCRC (the CS conference conglomerate that happens once every 4 years), June 13-19. Here are some of the talks I found particularly memorable.

Personal notes on FCRC talks are at https://workflowy.com/s/wkI79JfN0N and on STOC/CCC/EC talks (very rough) are at https://dl.dropboxusercontent.com/u/27883775/wiki/math/pdfs/stoc.pdf. Note that neither has been edited.

FCRC award/plenary talks

  • Turing award lecture (The Land sharks are on the squawk box), Michael Stonebraker (https://www.youtube.com/watch?v=BbGeKi6T6QI): Stonebraker draws parallels between his work in building Postgres, a relational database system, and his cross-country bike trip. He described numerous challenges and how they were overcome in both situations, concluding that they were both about “making it happen”.
  • Interdisciplinary research from the view of theory, Andrew Yao: Yao comes from the viewpoint of theoretical computer science but has then worked on connections to physics (quantum computation), economics (auction theory), and cryptography (certifiable randomness). Theoretical computer science started with computability…

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Paul Erdös’s 102-ennial

February 27, 2015

in theory

Paul Erdös would be 102 year old this year, and in celebration of this the Notices of the AMS have published a two-part series of essays on his life and his work: [part 1] and [part 2].

Of particular interest to me is the story of the problem of finding large gaps between primes; recently Maynard, Ford, Green, Konyagin, and Tao solved an Erdös $10,000 question in this direction. It is probably the Erdös open question with the highest associated reward ever solved (I don’t know where to look up this information — for comparison, Szemeredi’s theorem was a $1,000 question), and it is certainly the question whose statement involves the most occurrences of “$latex log$”.

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SUIF publications

January 15, 2015

Papers from the Stanford Compiler Group

http://suif.stanford.edu/papers

Actions do change the world.

January 11, 2015

Quantum Frontiers

I heard it in a college lecture about Haskell.

Haskell is a programming language akin to Latin: Learning either language expands your vocabulary and technical skills. But programmers use Haskell as often as slam poets compose dactylic hexameter.*

My professor could have understudied for the archetypal wise man: He had snowy hair, a beard, and glasses that begged to be called “spectacles.” Pointing at the code he’d projected onto a screen, he was lecturing about input/output, or I/O. The user inputs a request, and the program outputs a response.

That autumn was consuming me. Computer-science and physics courses had filled my plate. Atop the plate, I had thunked the soup tureen known as “XKCD Comes to Dartmouth”: I was coordinating a visit by Randall Munroe, creator of the science webcomic xkcd, to my college. The visit was to include a cake shaped like the Internet, a robotic velociraptor, and

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Dual spaces

January 11, 2015

lim Practice= Perfect

Suppose $latex {(A,||cdot||_A)}&fg=000000$ and $latex {(B,||cdot||_B)}&fg=000000$ are Banach space, $latex {A^*}&fg=000000$ and $latex {B^*}&fg=000000$ are their dual spaces. If $latex {Asubset B}&fg=000000$ with $latex {||cdot||_Bleq C||cdot||_A}&fg=000000$, then

$latex displaystyle i:Amapsto B&fg=000000$

$latex displaystyle quad xrightarrow x&fg=000000$

is an embedding. Let us consider the relation of two dual spaces. For any $latex {fin B^*}&fg=000000$

$latex displaystyle |langle f,xrangle|=|f(x)|leq ||f||_{B^*}||x||_Bleq C||f||_{B^*}||x||_Aquad forall, xin A&fg=000000$

Then $latex {f|_{A}}&fg=000000$ will be a bounded linear functional on $latex {A}&fg=000000$

$latex displaystyle i^*:B^*mapsto A^*&fg=000000$

$latex displaystyle qquad frightarrow f|_A&fg=000000$

is a bounded linear operator.

In a very special case that $latex {A}&fg=000000$ is a closed subset of $latex {B}&fg=000000$ under the norm $latex {||cdot||_B}&fg=000000$, one can prove $latex {i^*}&fg=000000$ is surjective. In fact $latex {forall,gin A^*}&fg=000000$ can be extended to $latex {bar{g}}&fg=000000$ on $latex {B}&fg=000000$ by Hahn-Banach thm such that $latex {i^*bar{g}=g}&fg=000000$. Then

$latex displaystyle A^*=B^*/ker i^*.&fg=000000$

Let us take $latex {A=H^1_0(Omega)}&fg=000000$ and $latex displaystyle B=H^1(Omega)&fg=000000$…

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How many rational distances can there be between N points in the plane?

December 30, 2014

Quomodocumque

Terry has a nice post up bout the Erdös-Ulam problem, which was unfamiliar to me.  Here’s the problem:

Let S be a subset of R^2 such that the distance between any two points in S is a rational number.  Can we conclude that S is not topologically dense?

S doesn’t have to be finite; one could have S be the set of rational points on a line, for instance.  But this appears to be almost the only screwy case.  One can ask, more ambitiously:

Is it the case that there exists a curve X of degree <= 2 containing all but 4 points of S?

Terry explains in his post how to show something like this conditional on the Bombieri-Lang conjecture.  The idea:  lay down 4 points in general position.  Then the condition that the 5th point has rational distances from x1,x2,x3, and x4 means that point lifts to a rational point on…

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How NOT to reference papers

September 16, 2014

Igor Pak's blog

In this post, I am going to tell a story of one paper and its authors which misrepresented my paper and refused to acknowledge the fact. It’s also a story about the section editor of Journal of Algebra which published that paper and then ignored my complaints. In my usual wordy manner, I do not get to the point right away, and cover some basics first. If you want to read only the juicy parts, just scroll down…

What’s the deal with the references?

First, let’s talk about something obvious. Why do we do what we do? I mean, why do we study for many years how to do research in mathematics, read dozens or hundreds of papers, think long thoughts until we eventually figure out a good question. We then work hard, trial-and-error, to eventually figure out a solution. Sometimes we do this in a matter of hours and sometimes…

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Some Strange Math Facts

September 15, 2014

Gödel's Lost Letter and P=NP

250px-STAN_ULAM_HOLDING_THE_FERMIAC

Stanislaw Ulam was a Polish-American mathematician whose work spanned many areas of both continuous and discrete mathematics. He did pioneering research in chaos theory and Monte Carlo algorithms, and also invented the concept of a measurable cardinal in set theory. His essential modification of Edward Teller’s original H-bomb design is used by nearly all the world’s thermonuclear weapons, while he co-originated the Graph Reconstruction conjecture. His name is also associated with the equally notorious 3n+1 conjecture. Thus he was involved in some strange corners of math.

Today Ken and I want to talk about some strange facts observed by Ulam and others that we did not know or fully appreciate.

Perhaps you can use them, perhaps you may enjoy them, but they are all kind of fun. At least we think so. Ulam’s autobiographyAdventures of a Mathematician shows his sense of fun, and he was…

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ICM2014 — Kollár, Conlon, Katz, Krivelevich, Milnor

September 4, 2014

Gowers's Weblog

As the ICM recedes further into the past, these posts start to feel less and less fresh. I’ve had an enforced break from them as over the course of three days I drove my family from the south of France back to Cambridge. So I think I’ll try to do what I originally intended to do with all these posts, and be quite a lot briefer about each talk.

As I’ve already mentioned, Day 3 started with Jim Arthur’s excellent lecture on the Langlands programme. (In a comment on that post, somebody questioned my use of “Jim” rather than “James”. I’m pretty sure that’s how he likes to be known, but I can’t find any evidence of that on the web.) The next talk was by Demetrios Christodoulou, famous for some extraordinarily difficult results he has proved in general relativity. I’m not going to say anything about the talk, other…

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Hales-Jewett and a generalized van der Warden Theorems

August 27, 2014

I Can't Believe It's Not Random!

The Hales-Jewett theorem is one of the most fundamental results in Ramsey theory, implying the celebrated van der Waerden theorem on arithmetic progressions, as well an its multidimensional and IP versions. One interesting property of the Hales-Jewett’s theorem is that it is a set theoretical statement, having no structure, and hence making it versatile for applications. Recently I realized that there exists an equivalent formulation of this theorem using some algebraic structure, and indeed it can be seen as an analogue of van der Waerden’s theorem. The main purpose of this post is to establish that equivalence. In an initial section I present the deduction of the multidimensional van der Waerden from the Hales-Jewett theorem, both to set the mood and to set the stage to later establish the analogy between van der Waerden’s theorem and the equivalent formulation of the Hales-Jewett theorem.

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