Gaze-dependent topography in human posterior parietal cortex.

The brain must convert retinal coordinates into those required for directing an effector. One prominent theory holds that, through a combination of visual and motor/proprioceptive information, head-/body-centered representations are computed within the posterior parietal cortex (PPC). An alternative theory, supported by recent visual and saccade functional magnetic resonance imaging (fMRI) topographic mapping studies, suggests that PPC neurons provide a retinal/eye-centered coordinate system, in which the coding of a visual stimulus location and/or intended saccade endpoints should remain unaffected by changes in gaze position. To distinguish between a retinal/eye-centered and a head-/body-centered coordinate system, we measured how gaze direction affected the representation of visual space in the parietal cortex using fMRI. Subjects performed memory-guided saccades from a central starting point to locations “around the clock.” Starting points varied between left, central, and right gaze relative to the head-/body midline. We found that memory-guided saccadotopic maps throughout the PPC showed spatial reorganization with very subtle changes in starting gaze position, despite constant retinal input and eye movement metrics. Such a systematic shift is inconsistent with models arguing for a retinal/eye-centered coordinate system in the PPC, but it is consistent with head-/body-centered coordinate representations

Semidefinite code bounds based on quadruple distances

Let $A(n,d)$ be the maximum number of $0,1$ words of length $n$, any two having Hamming distance at least $d$. We prove $A(20,8)=256$, which implies that the quadruply shortened Golay code is optimal. Moreover, we show $A(18,6)\leq 673$, $A(19,6)\leq 1237$, $A(20,6)\leq 2279$, $A(23,6)\leq 13674$, $A(19,8)\leq 135$, $A(25,8)\leq 5421$, $A(26,8)\leq 9275$, $A(21,10)\leq 47$, $A(22,10)\leq 84$, $A(24,10)\leq 268$, $A(25,10)\leq 466$, $A(26,10)\leq 836$, $A(27,10)\leq 1585$, $A(25,12)\leq 55$, and $A(26,12)\leq 96$. The method is based on the positive semidefiniteness of matrices derived from quadruples of words. This can be put as constraint in a semidefinite program, whose optimum value is an upper bound for $A(n,d)$. The order of the matrices involved is huge. However, the semidefinite program is highly symmetric, by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry. By block diagonalizing this algebra, the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of $n$ and $d$.Comment: 15 page

Higher-Dimensional Algebra VII: Groupoidification

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify - or more precisely, groupoidify - the positive part of the quantum group associated to the quiver.Comment: 67 pages, 14 eps figures; uses undertilde.sty. This is an expanded version of arXiv:0812.486