Differential Geometry seminars

Functional connectivity has been the focus of many research groups aiming to study the interaction between cells and brain regions. A standard method for analyzing connectivity is to statistically compare pairwise interactions between cells or brain regions across behavioral states or conditions. This methodology ignores the intrinsic properties of functional connectivity as a multivariate and dynamic signal, expressing the correlational configuration of the network. In this talk, I will present a geometric approach, combining Graph Theory and Riemannian Geometry to build "a graph of graphs" and extract the latent dynamics of the overall correlational structure. Using this approach, we formulate the statistical relations between network dynamics and spontaneous behavior as a second-order Taylor’s expansion. Our analysis shows that fast fluctuations in functional connectivity of large-scale cortical networks are closely linked to variations in behavioral metrics related to the arousal state. We further expand this methodology to longer time scales to study the effect of dopamine on network dynamics in the primary motor cortex (M1) during learning. We developed a series of analysis methods indicating that as animals learn to perform a motor task, the network of pyramidal neurons in layer 2-3 gradually and monotonically reorganizes toward an "expert" configuration. Our results highlight the critical role of dopamine in driving synaptic plasticity: Blocking dopaminergic neurotransmission locally in M1 prevented motor learning at the behavioral level and concomitantly halted plasticity changes in network activity and in functional connectivity. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2025-01-29. Recording duration: 00:27:08.

Egor Shelukhin presents cases of the Hilbert–Smith conjecture for group actions by homeomorphisms with symplectic structure. The results exclude faithful actions of the additive p-adic group in this setting and give additional restrictions on group actions in symplectic topology. The argument combines a new approach to these action problems with power operations in Floer cohomology and quantitative methods in symplectic topology.

Changes in the geometry and topology of self-assembled membranes underlie diverse processes across cellular biology and engineering. Similar to lipid bilayers, monolayer colloidal membranes studied by the Sharma (IISc Bangalore) and Dogic (UCSB) Labs have in-plane fluid-like dynamics and out-of-plane bending elasticity, but their open edges and micron length scale provide a tractable system to study the equilibrium energetics and dynamic pathways of membrane assembly and reconfiguration. First, we discuss how doping colloidal membranes with short miscible rods transforms disk-shaped membranes into saddle-shaped minimal surfaces with complex edge structures. Theoretical modeling demonstrates that their formation is driven by increasing positive Gaussian modulus, which in turn is controlled by the fraction of short rods. Further coalescence of saddle-shaped surfaces leads to exotic topologically distinct structures, including shapes similar to catenoids, tri-noids, four-noids, and higher order structures. We then mathematically explore the mechanics of these catenoid-like structures subject to an external axial force and elucidate their intimate connection to two problems whose solutions date back to Euler: the shape of an area-minimizing soap film and the buckling of a slender rod under compression. A perturbation theory argument directly relates the tensions of membranes to the stability properties of minimal surfaces. We also investigate the effects of including a Gaussian curvature modulus, which, for small enough membranes, causes the axial force to diverge as the ring separation approaches its maximal value.