Evidence for reciprocal antagonism between motion sensors tuned to coarse and fine features

Early visual processing analyses fine and coarse image features separately. Here we show that motion signals derived from fine and coarse analyses are combined in rather a surprising way: Coarse and fine motion sensors representing the same direction of motion inhibit one another and an imbalance can reverse the motion perceived. Observers judged the direction of motion of patches of filtered two-dimensional noise, centered on 1 and 3 cycles/deg. When both sets of noise were present and only the 3 cycles/deg noise moved, judgments were reversed at short durations. When both sets of noise moved, judgments were correct but sensitivity was impaired. Reversals and impairments occurred both with isotropic noise and with orientation-filtered noise. The reversals and impairments could be simulated in a model of motion sensing by adding a stage in which the outputs of motion sensors tuned to 1 and 3 cycles/deg and the same direction of motion were subtracted from one another. The subtraction model predicted and we confirmed in experiments with orientation-filtered noise that if the 1 cycle/deg noise flickered and the 3 cycles/deg noise moved, the 1 cycle/deg noise appeared to move in the opposite direction to the 3 cycles/deg noise even at long durations.

Geometry of the SL(3,ℂ)-character variety of torus knots.

Let G be the fundamental group of the complement of the torus knot of type (m, n). It has a presentation G = . We find a geometric description of the character variety X(G) of characters of representations of G into SL(3,ℂ), GL(3,ℂ) and PGL(3,ℂ).

E-polynomial of the SL(3,C)-character variety of free groups.

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3,C). Expanding upon techniques of Logares, Muñoz and Newstead (Rev. Mat. Complut. 26:2 (2013), 635-703), we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give a new proof of results of Cavazos and Lawton (Int. J. Math. 25:6 (2014), 1450058).

E-polynomials of the SL(2, C)-character varieties of surface groups

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2, C), for any possible holonomy around the puncture. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behavior of the E-polynomial under fibrations.