Understanding of the wound healing process can be used to make more tailor-made medicine and to determine the nature of this healing process. In this research we use MATLAB software along with the ADI method to solve a partial differential equation that models wound healing by treating keratin as a diffusion process. A significant hurdle to overcome is finding the appropriate initial conditions, that is to accurately extract boundary data from photos taken with different equipment, lighting, or resolution.

An algebraic curve is a one-dimensional set defined by polynomial equations, such as a parabola in the plane (given by y-x^2=0) or the z-axis in the space (given by x=y=0). Let Y be an algebraic curve. Then a multiplicity structure on Y is another curve Z, which as a set has the same points as Y but with a higher and fixed multiplicity at each point. For example, the y-axis in the plane is given by the equation x=0 and if we intersect it with horizontal lines, say with y-b=0, we get the points (0,b) on the y-axis. Now if we take the line given by x^2=0 and intersect it with the horizontal lines as above we get the points (0,b) with multiplicity 2. Hence we call the later curve a double structure on the previous one. Similarly the equation x^3=0 gives a triple structure on the y-axis in the plane and so on. Structures like these might sound naive but they are crucial to understand the behaviors of families of curves. For example, the family of parabolas ty-x^2=0 deforms into the double line x^2=0 as t approaches 0. Although the notion of multiplicity is pretty geometric, we can use tools from abstract algebra to make it rigorous. This makes the subject challenging and yet very interesting at the same time. Classifying the multiplicity structures on a curve is still a wide open field in algebraic geometry. It is now well understood how the double and triple structures on a line look. A natural question then arises how do the double and triple structures look on conics? It turns out that the answers are much more complicated than for lines. In this poster I am going to show some of my research in that direction.

An algebraic integer is a complex number that is a root of a polynomial with integer coefficients and a leading coefficient of 1. This includes numbers like the square root of 2 and the cube root of 10, for example. A field is a set in which we can add, subtract, multiply, and divide (among other details). Consider the set of all algebraic integers in any given field containing the rational numbers. The index of an algebraic integer in this set is a natural number that measures how close the algebraic integer is to generating the set. For instance, the imaginary number i (the square root of -1) is an algebraic integer which generates the set of all complex numbers of the form a + bi where a and b are integers, and so has index 1. The closer the index is to 1, the closer the algebraic integer is to generating the set. We investigate these indices in cubic fields, determining not only which numbers occur as indices in given families, but also that the minimal index is unbounded as one traverses the set of all cubic fields in those families.

Nearby, the Large Magellanic Cloud galaxy (LMC), has ejected massive amounts of gaseous material, some of which is headed toward the Milky Way. The material consists of ionized hydrogen gas which is a consequence of significantly energetic events that have occurred in the LMC. Such events are not only the cause of the ionized material, but also the immense amount of material being thrown out. This ejected wind holds a substantial amount of information regarding both galaxies in general and the LMC’s physical processes. Studying this ionized outflow will reveal new details concerning the internal processes that produce such massive ejections, the potential for galactic outflows to replenish gas reservoirs for future star formation, and the environments surrounding galaxies. The latter will influence our view of a galaxy’s environment and how it may interact with nearby neighbors such as our Milky Way galaxy.

Fluorescence anisotropy is a common measurement that helps provides important information on molecular mobility, solvent (environment) viscosity, or/and molecular size. Fluorescence anisotropy involves measurement of two orthogonally polarized light emission intensities. One of the common issues of fluorescence anisotropy measurements is that most optical detection systems respond differently to the parallel and perpendicular polarization of light. The challenging task is to estimate the calibration curve, often called as the instrumental G-factor (grating factor), a parameter indicates the contributions and/or distortion of the optical detection system to the parallel and/or perpendicular light polarization, so that one can correct their polarized emission intensity and obtain a proper fluorescence anisotropy result. Here we present novel techniques that we have been developed in our laboratory that help achieve the G-Factor curves for different instruments.