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Cetin Kaya Koc
Works at University of California Santa Barbara
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Cetin Kaya Koc

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I'm on the editorial board of Combinatorica. Whether I should be is another matter, since it is a journal owned by Springer, one of the big commercial publishers. But I am, and as a result I have a free subscription to the journal. Today I found the latest issue in my pigeonhole, and the last paper in the issue was a paper by Csaba Tóth, entitled, "The Szemerédi-Trotter theorem in the complex plane." 

This paper is remarkable for two reasons. One, which provokes this post, is that at the beginning of the paper it says, "Received December 1999, Revised May 16 2014." So the paper is coming out over 15 years after it was submitted. Doubtless this isn't a record, but it's still a pretty big gap. I noticed it because my first reaction on seeing the title was, "But I thought this had been done a long time ago."

The other reason is the result itself. The Szemerédi-Trotter theorem states that if you have n points and m lines in the plane, then the number of incidences (that is, pairs (P,L) where P is a point in your collection, L is a line in your collection, and P is a point in L) is at most C(n + m + n^{2/3}m^{2/3}). This slightly curious looking bound is best possible up to the constant C and is more natural than it looks.

The known proofs of the theorem relied heavily on the topological properties of the plane, which meant that it was far from straightforward to generalize the result to lines and points in the complex plane (by which I mean C^2 and not C). Indeed, it was an open problem to do so, and that was what Tóth solved.

If you're feeling ambitious, there is also a lovely conjecture in the paper. Define a d-flat in R^{2d} to be an affine subspace of dimension d. Suppose now that you have n points and m d-flats with the property that no two of the d-flats intersect in more than a point. Is it the case that the number of incidences is at most C(n + m + n^{2/3}m^{2/3})? The constant C is allowed to depend on the dimension d but not on anything else. Note that even for d=2 this would be a new result, since Tóth's theorem is the special case where the d-flats are complex lines.

I should say that I haven't checked whether there has been any progress on this conjecture, so I don't guarantee that it is open. If anyone knows about its status, it would be great if you could comment below.

#spnetwork  DOI: 10.1007/s00493-014-2686-2
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The Riemann Hypothesis is one of the great unsolved problems of mathematics and the reward of $1000000 of Clay Mathematics Institute prize money awaits the person who solves it. But-with or without money-its resolution is crucial for our understanding of the nature of numbers.
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A few hundred meters before the Mount Ararat proper summit ..
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Langston Hughes’ 113th Birthday #GoogleDoodle
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Hayran olunmayacak gibi degil .. ELINE SAGLIK.
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I didn't know this (didn't care to check before) but in Gmail ignores dots (.) in your email address. So, abc.def@gmail.com is actually the same as abcdef@gmail.com or even a.bcdef@gmail.com.. Try to send emails to these addresses from another account or t login to any of them with the same password! :)
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Nice!
 
Here is the kit for my recent graph theory project:

Math for eight-year-olds: graph theory for kids! 
http://jdh.hamkins.org/math-for-eight-year-olds/

Print out to double-sided, and then fold each page in half. Place the folded pages one after the other (not nested) inside the cover page.
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Muberra hanim, hem sagolun ve hem de verem olursam, sebebi sizsiniz.
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Unable to get an academic position, Zhang kept the books for a Subway franchise. Credit Photograph by Peter Bohler
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This week, +Kester Tong and I gave a talk at #SciPy201  about PNaCl, Python, scientific Python, why we'd want to do this, and the amazing work on coLaboratory that he, +Kayur Patel and their team have worked on with +Fernando Perez , +Brian Granger , +Min RK and the rest of the IPython team.  I was pretty happy with the talk, and pretty excited about the possibilities, but more than anything else, I'm honored to have worked with such an amazing group of people -- both the ones I've named here, and the +Portable Native Client (PNaCl) team, especially the ever-patient +Sam Clegg.
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Have him in circles
229 people
Aysen Koc's profile photo
Andy Lau's profile photo
Jamyanjav Zundui-Yondon's profile photo
berat BİBER's profile photo
Aysenur Eser's profile photo
Ümit Şenses's profile photo
Ihsan Cicek's profile photo
Evan A.'s profile photo
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Professor
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  • University of California Santa Barbara
    Research Professor, 2008 - present
  • Claveo Software
    CEO, 2010 - 2013
  • Cryptocode
    CEO, 2003
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Istanbul, Turkey
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