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): Continued fractions and the geometry of conformally invariant sets

Abstract: A well-known family of sets with non-integer dimension is obtained as follows: consider points in the interval [0,1] whose base-b expansion only contains digits in some (non-empty, proper) subset of {0, ... , b-1}. The prototypical example of such a set is the well-known middle-thirds Cantor set. While such sets are pathological in a classical sense, in the grand scheme of fractal sets, they are very nice sets. As we will see, one explanation for their nice properties is that these 'missing digit' sets are invariant for a well-behaved dynamical system (integer multiplication modulo 1) on a compact manifold (the torus). But what happens if we remove compactness? In this case, the canonical example is analogous to the above construction, except with the continued fraction expansion in place of the base-b expansion. The dynamics are still very well-behaved, but we lose compactness.
In this talk I will give a gentle introduction to the theory of conformally invariant sets; and to discuss the myriad of ways in which things go wrong without compactness. Most results are old; any new results are based on joint work with Amlan Banaji (University of Loughborough).
Alex Rutar's Website


William Dugan (University of Massachusetts Amherst): On the f-vector of flow polytopes for complete graphs

Abstract: The Chan-Robbins-Yuen polytope (CRYn) of order n is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph Kn+1 with netflow (1,0,0, ... , 0, -1). The volume and lattice points of this polytope have been actively studied, however its face structure has been studied less. We give explicit formulas and generating functions for the f vector of CRYn by using Hille's (2007) result bijecting faces of a flow polytope to certain graphs, as well as Andresen-Kjeldsen's (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes over the complete graph having arbitrary (non-negative) netflow vectors and study the face lattice of CRYn.
William Dugan's Website


Zhanar Berikkyzy (Fairfield University): Rainbow numbers of equations and graphs

Abstract: The rainbow number of a set of integers X for a given equation eq is the smallest number of colors r such that every exact r-coloring of X admits a rainbow solution to the equation eq. During this talk, we will discuss this number for several important equations, including Schur's and Sidon equations. We will then consider this parameter in graph setting and connect it to the anti-van der Waerden number of a graph, the fewest number of colors needed to guarantee a rainbow 3-term arithmetic progression. We will survey recent results for various classes of graphs, including trees and products of trees.
Zhanar Berikkyzy's Website


Ilya Kachkovskiy (Michigan State University): Quasiperiodic operators with monotone potentials: robust localization and Cantor spectra.

Abstract: The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on $d$-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson localization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types.
We will consider a class of quasiperiodic operators $$ (H(x)\psi)_n=\epsilon(\Delta\psi)_n+f(x+n\cdot\omega)\psi_n,\quad n\in \mathbb Z^d, $$ where $\Delta$ is the discrete Laplacian, $\omega$ is a vector with rationally independent components, and $f$ is a $1$-periodic function on $\mathbb R$, monotone on $(0,1)$ with a positive lower bound on the derivative.
The above condition of monotonicity, as it turns out, implies several unique phenomena, including a robust version of uniform Anderson localization and a new connection between spectral gaps, vanishing of eigenfunctions, and rank one perturbations. In particular, we show that a large class of such operators in 1D has Cantor spectrum.
The talk is based on joint works with S. Jitomirskaya, L. Parnovski, and R. Shterenberg.
Ilya Kachkovskiy's Website

View past events

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