2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019
Current contact: Brandi Henry and Rebekah Palmer
The seminar takes place on Fridays at 1:00 pm in Room 617 on the sixth floor of Wachman Hall. Pizza and refreshments are available beforehand in the lounge next door.
Narek Hovsepyan, Temple University
Abstract: The need for analytic continuation arises frequently in the context of inverse problems. We consider several such problems and show that they exhibit a power law precision deterioration as one moves away from the source of data. We introduce a general Hilbert space-based approach for determining these exponents. The method identifies the "worst case" function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus, an ellipse or an upper half-plane the solution of the integral equation and the corresponding exponent can be found explicitly.
This is a joint work with Yury Grabovsky
Dong Bin Choi, Temple University
Abstract: In 1952, Kurt Heegner proved (up to minor gaps) that the imaginary quadratic fields $K$ with class number $h(K) = 1$ are $\mathbb{Q}(\sqrt{d}$ with $d = -1, -2, -3, -7, -11, -19, -43, -67, -163$. To understand the problem in context, we begin from integer quadratic forms $Q(x,y) = ax^2 + bxy + cy^2$ and the question of what numbers $n$ occur as solutions to $ax^2 + bxy + cy^2 = n$ for a given $a, b, c$. This depends on the equivalence class of the quadratic form, which turns out to be closely related to ideal classes of the quadratic integer ring with discriminant $b^2 - 4ac$. We touch on some number-theoretic notions used in Heegner's proof, such as the genus of a quadratic form and the $j$-invariant of lattices. We sketch Heegner's proof (following Kezuka 2012), and conclude with solutions to some generalizations of the problem.
Najmej Salehi, Temple University
Delaney Aydel, Temple University
Abstract: Let $T_n$ denote the $n$th Taft algebra. We fully classify inner-faithful actions of $T_n \otimes T_n$ on four-vertex Schurian quivers as extensions of the actions of $\mathbb{Z}_n \times \mathbb{Z}_n$. One example will be presented in full, with the remaining results briefly given.
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.
Rebekah Palmer, Temple University
Abstract: To study structures that are not quite smooth, it is convenient to transform them so that we can apply the many tools developed for smooth structures. Given a variety with singularities, we ask if there is a consistent method of transformation into a non-singular variety. In this talk, we will introduce ourselves to some varieties with singularities, demonstrate how to uniquely resolve algebraic curves, and further discuss approaches to resolving higher-dimensional varieties.
Thomas Ng, Temple University
Abstract: The class of all groups (even finitely generated ones) are known to have wildly inefficient, in fact often unsolvable, algorithmic properties. It is thus helpful to specialize to well-behaved subclasses. In the early 1900s Dehn studied groups as geometric objects and showed that finitely generated groups arising as the fundamental group of manifolds in dimensions less that 4 are often very algorithmically efficient. Given the existence of these well-behaved groups, one is naturally led to ask how unique they are or under what conditions are they uniquely determined. Following Gromov's notion of boundary for a hyperbolic group introduced in 1987, Cannon conjectured that for hyperbolic groups the condition that its boundary is a 2-sphere uniquely determines 3-manifold fundamental groups. In this talk we will briefly review the notions of Cayley graph, hyperbolicity, and boundary of a group. We will then survey some of the recent results that inspired and attempt to solve the Cannon conjecture.