UIUC Dept. of Mathematics Seminar Calendar

    February 2011            March 2011             April 2011     
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Thursday, March 31, 2011

Mathematical Biology Seminar
1:00 pm in 345 Altgeld Hall, Thursday, March 31, 2011

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Submitted by rdeville.

Tom Butler (MIT)
Evolutionary constraints on visual cortex architecture from the dynamics of hallucinations
Abstract: In the cat or primate visual cortex (V1), normal vision corresponds to a state where neural excitation patterns are driven by external visual stimuli. A spectacular failure mode of V1 occurs when such patterns are overwhelmed by spontaneously generated spatially self organized patterns of neural excitation. These are experienced as geometric visual hallucinations. The problem of identifying the mechanisms by which V1 avoids this failure is made acute by recent advances in the statistical mechanics of pattern formation which suggest that the hallucinatory state should be very robust. Here, we report how incorporating physiologically realistic long range connections between inhibitory neurons changes the behavior of a model of V1. We find that sparse long range inhibition in V1 plays a previously unrecognised but key functional role in preserving the normal vision state. Surprisingly, it also contributes to the observed regularity of geometric visual hallucinations. Our results provide an explanation for the observed sparsity of long range inhibition in V1---this generic architectural feature is an evolutionary adaptation that tunes V1 to the normal vision state. In addition it has been shown that exactly the same long range connections play a key role in the development of orientation preference maps. Thus V1's most striking long-range features---patchy excitatory connections and sparse inhibitory connections---are completely constrained by two considerations: the need for the visual state to be robust, and the developmental requirements of the orientational preference map.

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, March 31, 2011

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Submitted by kapovich.

Brent Solie (UIUC Math)
Quantifying the Residual Property of Gamma-Limit Groups
Abstract: Let Gamma be a fixed hyperbolic group. A Gamma-limit group is a finitely generated, fully residually Gamma group; informally, we think of G as admitting many different maps into Gamma. In order to study these maps, we introduce a new invariant of limit groups called discriminating complexity. Discriminating complexity measures how difficult it is to map a Gamma-limit group into Gamma while remaining injective in a neighborhood of the identity. We will outline a proof that the discriminating complexity of any Gamma-limit group is polynomial, utilizing the geometry of relatively hyperbolic groups and an embedding theorem of Kharlampovich-Myasnikov.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, March 31, 2011

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Submitted by aimo.

Kirsi Peltonen (Aalto University, Helsinki, Finland)
Quasiregular dynamics
Abstract: Joint work with Gaven Martin, Massey University, New Zealand. We report on recent developments of a higher dimensional real counterpart of the iteration theory of rational functions in the extended complex plane. We study a subclass of quasiregular mappings, called uniformly quasiregular mappings (UQR), acting on a compact Riemannian manifold so that all the iterates of the mapping are $K$-quasiregular for fixed distortion $K$, independently of the number of iterates. Surprisingly, many manifolds in addition to the n-sphere support nontrivial UQR mappings. We describe some basic constructions and obstructions for UQR mappings and discuss on some dynamical aspects of the theory.