arXiv subjects

Jack A. Cook

Publications and source records attributed to Jack A. Cook.

4 recordsLinked to original sources

arXiv2026

Computing the Cousin-Zuckerman Resolution and the Lusztig-Vogan Bijection

The goal of this article is to give a proof of a result seemingly absent from the literature characterizing global sections of standard $\mathcal{D}$-modules on the flag variety. This characterization yields a mixture of the Langlands Classification of admissible representations with the Knapp-Zuckerman classification of tempered representations of a real reductive group. We use this result to compute the Cousin-Zuckerman resolution of the trivial representation in terms of standard $(\mathfrak{g},K)$-modules. Further, in the case of $GL(n,\mathbb{H})$ we use this to prove the Lusztig-Vogan bijection for $n=2,3$ and compute the lowest $K$-type map for the zero and principal orbits for general $n$ as well as the image of the trivial representation for even orbits.

math.RT↗

arXiv2025

Vanishing Cohomology of Dominant Line Bundles for Real Groups

In \cite{Broer1993}, it was shown that certain line bundles on $\widetilde{\mathcal{N}}=T^*G/B$ have vanishing higher cohomology. We prove a generalization of this theorem for real reductive algebraic groups. More specifically, if $\mathcal{N}_\theta$ denotes the cone of nilpotent elements in a Cartan subspace $\mathfrak{p},$ we have a similar construction of a resolution of singularities $\widetilde{\mathcal{N}_\theta}.$ We prove that for a certain cone of weights $H^i(\widetilde{\mathcal{N}_\theta},\mathcal{O}_{\widetilde{\mathcal{N}_\theta}}(\lambda))=0$ for $i> 0.$ This follows by combining a simple calculation of the canonical bundle for $\widetilde{\mathcal{N}_\theta}$ with Grauert-Riemenschneider vanishing. Restricting to the structure sheaf, we get a characterization of the singularities of the normalization of $\mathcal{N}_\theta.$ We use this to show that for groups of QCT (Definition 2), $\mathbb{C}[\mathcal{N}_\theta]$ is equivalent as a $K$-representation to a certain cohomologically induced module giving a new proof of a result in \cite{KostantRallis1971}.

math.RT↗

arXiv2023

A Geometric Framework for Odor Representation

We present a generalized theoretical framework for olfactory representation and plasticity, using the theory of smooth manifolds and sheaves to depict categorical odor learning via distributed neural computation. Beginning with the space of all possible inputs to the olfactory system, we develop a dynamic model for odor learning that culminates in a perceptual space in which categorical odor representations are hierarchically constructed through experience, exhibiting statistically appropriate consequential regions and clear relationships between the broader and narrower identities to which a given odor might be assigned. The model reflects both the sampling-based physical similarity relationships among odorants, as observed in physiological receptor response profiles, and the acquired, learning-dependent perceptual similarity relationships among odors that can be measured behaviorally, and defines the relationship between them. Individual training and experience generates correspondingly more sophisticated odor identification capabilities. Because these odor representations are constructed from experience and depend on local, distributed plasticity mechanisms, geometries that fix curvature are insufficient to describe the capabilities of the system. This generative framework also encompasses hypotheses explaining representational drift in postbulbar circuits and the context-dependent remapping of perceptual similarity relationships.

q-bio.NC↗

arXiv2020

A Differential Topological Model for Olfactory Learning and Representation

This thesis is designed to be a self-contained exposition of the neurobiological and mathematical aspects of sensory perception, memory, and learning with a bias towards olfaction. The final chapters introduce a new approach to modeling focusing more on the geometry of the system as opposed to element wise dynamics. Additionally, we construct an organism independent model for olfactory processing: something which is currently missing from the literature.

q-bio.NC↗