A group-theoretic approach to fast matrix multiplication

We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n^(2 + o(1)) support n-by-n matrix multiplication, a necessary condition for the approach to yield exponent 2. Although we cannot yet completely achieve both (1) and (2), we hope that it may be possible, and we suggest potential routes to that result using the constructions in this paper.Comment: 12 pages, 1 figure, only updates from previous version are page numbers and copyright informatio

New upper bounds on sphere packings I

We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24.Comment: 26 pages, 1 figur

Algorithmic design of self-assembling structures

We study inverse statistical mechanics: how can one design a potential function so as to produce a specified ground state? In this paper, we show that unexpectedly simple potential functions suffice for certain symmetrical configurations, and we apply techniques from coding and information theory to provide mathematical proof that the ground state has been achieved. These potential functions are required to be decreasing and convex, which rules out the use of potential wells. Furthermore, we give an algorithm for constructing a potential function with a desired ground state.Comment: 8 pages, 5 figure