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Mathematics Colloquium

1. Owen Lewis, and 2. Oleksandr Vlasiuk

1. Trust your gut: the physics and mathematics behind maintenance of the gastric mucus barrier
2. Minimizing $p$-frame energies

Date: Friday, January 11, 2019
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstracts. 1. The gastric mucus layer is widely recognized to serve a protective function, shielding your stomach wall from the extremely low pH and digestive enzymes present in the stomach lumen. However, there is no consensus on the mechanism by which the mucus layer hinders the flux of hydrogen protons from lumen to wall, while allowing acid and enzymes secreted at the wall unimpeded transport to the lumen. We will discuss a mathematical description of electro-diffusion within the mucus layer and use it to test physiological hypotheses that are beyond current experimental techniques. Time permitting, we will discuss the influence that the electro-chemistry of diffusive ionic species has on swelling behavior in a two-phase gel model of freshly secreted mucus.

2. Consider the problem of finding probability measures $ \mu $ on the unit sphere that minimize \[ \iint\limits_{\mathbb S^d} |\bs x\cdot \bs y |^p\, d\mu(\bs x) d\mu(\bs y) \longrightarrow \min. \] A variant of this question arises naturally in signal processing: when $p=2$, the discrete minimizers of such energies are unit norm tight frames (generalizations of the notion of orthonormal basis). $p$-frame energies for $p \notin 2\mathbb N $ have continuous non-positive definite kernels, representatives of which also appear in quantum-mechanical models, the problem of moments, and other areas of mathematical physics and functional analysis. For even values of $ p $, the uniform surface measure on $ \mathbb S^d $ is a minimizer, and spherical designs (polynomial quadrature rules on the sphere) of order $ 2p$ are discrete minimizers. For non-even $p$, the situation is much more involved. We have established minimizers for some ranges of non-even $ p $ and certain dimensions $ d $. For example, for $p\in (2,4)$ and $d=2$, the minimizer is supported on the regular icosahedron, while the shortest vectors of the Leech lattice provide the minimizer on $\mathbb S^{23}$ for $p\in (8,10)$.