Title: Covering properties of configurations optimal for discrete Chebyshev constants

Date: 11/07/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

Speaker: Alexander Reznikov, Vanderbilt University

It is well known that, for a fixed integer N and as s goes to infinity, the optimal configurations for minimal discrete N-point s-Riesz energy on a compact set A converge to N-point configurations that solves the best-packing problem on A. We present a max-min problem that, in the limit, solves the best-covering problem, which is somewhat dual to best-packing. We will discuss other distributional properties of optimal configurations for this new problem, as well as applications to numerical integration and convex geometry.

Our idea of constructing a cofinite graph starts by defining a uniform topological graph Gamma. This idea is motivated by cofinite groups structure, due to B. Hartley.
We define and establish a theory of cofinite connectedness of a cofinite graph. We found that if G is a cofinite group and Gamma = Gamma(G;X) is the Cayley graph with respect to a generating set X of G, then Gamma can be given a suitable cofinite uniform topological structure so that X generates G topologically if and only if Gamma is cofinitely connected.
Next we develop group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the aforesaid action becomes uniformly continuous. We then apply the general theory to additional structure such as groupoids,
thus leading to the notions of cofinite groupoids.
This is a joint work with Dr. J.M.Corson (University of Alabama, Department of Mathematics) and Dr. B. Das (University of North Georgia, Department of Mathematics).

Speaker: Gabriel Angelini-Knoll, Wayne State University

Periodicity is a highly studied phenomenon in homotopy theory. For example, R. Bott showed that the homotopy groups of the classifying space of the infinite orthogonal group are periodic with period eight. This periodicity is also reflected in the stable homotopy groups of spheres using the J homomorphism, a map from the homotopy groups of the infinite orthogonal group to the stable homotopy groups of spheres. Using this map J.F. Adams produced a “height one” periodic family in the homotopy groups of spheres.
The image of J can be realized as a topological space and at odd primes this space is equivalent to algebraic K-theory of certain finite fields after p-completion. Therefore, a periodic family of height one in the language of chromatic homotopy theory is detected by algebraic K-theory of finite fields. In my talk, I will describe a higher height version of this phenomenon that I prove in my thesis. In particular, I demonstrate that a periodic family of height two is detected in mod (p, v_1)-homotopy of iterated algebraic K-theory of a finite field of order q, where q is a generator of the units in the p-adic integers. This result gives some evidence for a red-shift type phenomenon, which loosely states that algebraic K-theory increases the wavelength of periodicity.

In many practical imaging scenarios, including computed tomography and magnetic resonance imaging (MRI), the goal is to reconstruct an image from few of its Fourier domain samples. Many state-of-the-art reconstruction techniques, such as total variation minimization, focus on discrete 'on-the-grid' modelling of the problem both in spatial domain and Fourier domain. While such discrete-to-discrete models allow for fast algorithms, they can also result in sub-optimal sampling rates and reconstruction artifacts due to model mismatch. Instead, we present a framework that allows for the recovery of a continuous domain 'off-the-grid' representation of piecewise constant images from the optimal number of Fourier samples. The main idea is to model the edge set of the image as the level-set curve of a continuous domain band-limited function. Sampling guarantees can be derived for this framework by investigating the algebraic geometry of these curves. Finally, we show how this model can be put into a robust and efficient optimization framework by posing signal recovery entirely in Fourier domain as a structured low-rank matrix completion problem, and demonstrate the benefits of this approach over standard discrete methods in the context of undersampled MRI reconstruction.