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Edward Letzter, Temple University
A discussion of more recent results, and still-open questions, on free subalgebras and free multiplicative subsemigroups of associative algebras.
Atilla Yilmaz, Temple University
I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.
Andrew Cooper, NC State
Given a space \(X\), the configuration space \(F(X,n)\) is the space of possible ways to place \(n\) points on \(X\), so that no two occupy the same position. But what if we allow some of the points to coincide?
The natural way to encode the allowed coincidences is as a simplicial complex \(S\). I will describe how the configuration space \(M(S,X)\) obtained in this way gives rise to polynomial and homological invariants of \(S\), how those invariants are related to the cohomology ring \(H^*(X)\), and what this has to do with the topology of spaces of maps into \(X\).
I will also mention some potential applications of this structure to problems arising from international relations and economics.
This is joint work with Vin de Silva, Radmila Sazdanovic, and Robert J Carroll.
Benjamin Seibold, Temple University
Brandi Henry, Temple University
Abstract: Microbial cells form communities, called biofilms, by producing an extracellular adhesive. By tracking the movement of 1µm glyoxylate beads in biofilms through the use of laser-scanning confocal microscopy and image-processing software, we can study the properties of the biofilm that may affect interactions of other cells with the microbiota. We developed a tool that can analyze the distance the beads travel, the volume of the region within which the beads travel, the time for which the bead is associated with the biofilm, the velocity with which the bead travels, and density of the region within which the bead travelled. Bead movement was studied for Enterococcus faecalis, Salmonella Typhimurium, Escherichia coli biofilms, and their isogenic curli mutants. Consistent with visual observations, our statistical analysis showed that the presence of curli in biofilms introduces a rigidity to the biofilm structure. Conversely, the lack thereof correlates to more bead movement suggesting less rigidity. In biofilms lacking curli where more free movement occurred, we analyzed the dependency of bead movement on the local density. While greater movement occurred in less dense environments, bead movement is not strictly dependent on density, suggesting other material properties of the biofilm influence bead movement.
Oleg Lazarev, Columbia University
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Francesco Lin, Princeton University PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
There are no conferences this week.