Spring 2025

Blake Jackson (University of Connecticut): An intersection-dimension formula for non-rigid modules of type affine Dn

Abstract: We give a geometric model for the non-rigid modules over acyclic path algebras of type affine Dn. Similar models have been provided for module categories over path algebras of types An, Dn, and affine An as well as the rigid modules of type affine Dn. A major draw of these geometric models is the "intersection-dimension formulas" they often come with. These formulas give an equality between the intersection number of the curves representing the modules in the geometric model and the dimension of the extension spaces between the two modules. Essentially, these formulas allow us to calculate the homological data between two modules combinatorially. Previous work has failed to cover the case of non-rigid modules since they are not tilting objects; thus, the tilting theory tools cannot touch them. Another confounding issue is that while rigid modules have no self-extensions, non-rigid modules can have arbitrarily many distinct (up to isomorphism) self-extensions. Therefore, the curves representing these modules can have an arbitrarily high number of self-intersections.
Blake Jackson's Website


David Svintradze (New Vision University): Manifold solutions to Navier-Stokes equations

Abstract: We have developed dynamic manifold solutions for the Navier-Stokes equations using an extension of differential geometry called the calculus for moving surfaces. Specifically, we have shown that the geometric solutions to the Navier-Stokes equations can take the form of fluctuating spheres, constant mean curvature surfaces, generic wave equations for compressible systems, and arbitrarily curved shapes for incompressible systems in various scenarios. These solutions apply to predominantly incompressible and compressible systems for the equations in any dimension, while the remaining cases are yet to be solved. We have demonstrated that for incompressible Navier-Stokes equations, geometric solutions are always bound by the curvature tensor of the closed smooth manifold for every smooth velocity field. As a result, solutions always converge for systems with constant volumes.
David Svintradze's Website


Alex Rutar (University of Jyväskylä)

Abstract: tba.
Alex Rutar's Website


William Dugan (University of Massachusetts Amherst)

Abstract: tba.
William Dugan's Website


Zhanar Berikkyzy (Fairfield University)

Abstract: tba.
Zhanar Berikkyzy's Website


Ilya Kachkovskiy (Michigan State University)

Abstract: tba.
Ilya Kachkovskiy's Website

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This colloquium is coordinated by: Daniel Glasscock, Amanda Redlich, Joris Roos, Bobbie Wu