# Jerusalem Mathematics Colloquium

Thursday, 19th June 2008, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Guy Kindler

(Weizmann)

"Can cubic tiles be sphere-like?"

** Abstract: ** The unit cube tiles R^d by Z^d, in the sense that its translations by
vectors from Z^d cover R^d. It is natural to ask what is the minimal
surface area of a body that tiles R^d by Z^d. The volume of any such
body should clearly be at least 1, and therefore its surface area must
be at least that of a unit volume ball, which of order sqrt(d). The
surface area of the cube, however, is of order d, and no better tiling
was known. In this work we use a random construction to show that the
optimal surface area is indeed of order sqrt(d), namely there exist
bodies that tile R^d as a cube would, but have sphere-like surface
areas.

Tiling problems were considered for well over a century, but this
particular tiling problem was also recently considered in computer
science because of its relations with the Unique Games conjecture and
with the Parallel Repetition problem. Indeed, our current result
follows from the same idea that was used recently by Raz in his
counter example for the strong Parallel Repetition conjecture.

Joint work with Ryan O'Donnell, Anup Rao, and Avi Wigderson

Light refreshments will be served in the faculty lounge at
3:30.

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